Generalized Löb's Theorem.Strong Reflection Principles and Large Cardinal Axioms. Consistency Results in Topology. Jaykov Foukzon jaykovfoukzon@list.ru Israel Institute of Technology,Haifa,Israel -------------------------------------- Abstract: In this article we proved so-called strong reflection principles corresponding to formal theories Th which has omega-models. A posible generalization of Lob's theorem is considered.Main results are: (i) ConZFC  MstZFC, (ii) ConZFC2, (iii) let k be an inaccessible cardinal then ConZFC  . Keywords: Gödel encoding, Completion of ZFC, Russell's paradox, -model, Henkin semantics, full second-order semantic,strongly inaccessible cardinal Part I.There is no standard model of ZFC 1.Introduction. 1.1.Main results. 2.Derivation of the inconsistent definable set in set theory ZFC2 Hs, ZFCst and ZFCNst. 2.1.Derivation of the inconsistent definable set in set theory ZFC2 Hs. 2.2.Derivation of the inconsistent definable set in set theory ZFCst. 2.3.Derivation of the inconsistent definable set in set theoryZFCNst. 3.Derivation of the inconsistent provably definable set in set theory ZFC2 Hs, ZFCst and ZFCNst. 3.1.Derivation of the inconsistent provably definable set in ZFC2 Hs. 3.2.Derivation of the inconsistent provably definable set in ZFCst. 3.3.Derivation of the inconsistent provably definable set in ZFCNst. 3.4.Generalized Tarski's undefinability lemma. 3.5.Generalized Tarski's undefinability theorem. 3.6 Avoiding the contradictions from set theory ZFC2 Hs, ZFCst and set theory ZFCNst using Quinean approach. 3.6.1.Quinean set theory NF. 3.6.2.Set theory ZFC2 Hs, ZFCstand set theory ZFCNst with stratified axiom schema of replacement. Part II.Generalized Löbs Theorem. 1.Introduction. 2.1.Generalized Löbs Theorem. 2.2.Derivation inconsistent countable set in set theory ZFC2Hs  MZFC2 Hs and in set theory ZFC  MstZFC using Generalized Tarski's undefinability theorem. 2.3.Proof of the inconsistensy of the set theory ZFC2Hs  MZFC2 Hs using Generalized Tarski's undefinability theorem. I.Introduction. 1.1.Main results. Let us remind that accordingly to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox. In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and Zermelo set theory, the first constructed axiomatic set theory. Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo–Fraenkel set theory ZFC. "But how do we know that ZFC is a consistent theory, free of contradictions? The short answer is that we don't; it is a matter of faith (or of skepticism)"- E.Nelson wrote in his paper [1]. However, it is deemed unlikely that even ZFC2 which is significantly stronger than ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC and ZFC2 were inconsistent, that fact would have been uncovered by now. This much is certain -ZFC and ZFC2 is immune to the classic paradoxes of naive set theory: Russell's paradox, the Burali-Forti paradox, and Cantor's paradox. Remark 1.1.1.Note that in this paper we view (i) the first order set theory ZFC under the canonical first order semantics (ii) the second order set theory ZFC2 under the Henkin semantics [2],[3] and (iii) the second order set theory ZFC2under the full second-order semantics [4],[5],[6]. Remark 1.1.2.Second-order logic essantially differs from the usual first-order predicate calculus in that it has variables and quantifiers not only for individuals but also for subsets of the universe and variables for n-ary relations as well [7],[8].The deductive calculus DED2 of second order logic is based on rules and axioms which guarantee that the quantifiers range at least over definable subsets [7]. As to the semantics, there are two tipes of models: (i) Suppose U is an ordinary first-order structure and S is a set of subsets of the domain A of U. The main idea is that the set-variables range over S, i.e. U, S  XX  SS  SU, S  S. We call U, S a Henkin model, if U, S satisfies the axioms of DED2 and truth in U, S is preserved by the rules of DED2. We call this semantics of second-order logic the Henkin semantics and second-order logic with the Henkin semantics the Henkin second-order logic. There is a special class of Henkin models, namely those U, S where S is the set of all subsets of A. We call these full models. We call this semantics of second-order logic the full semantics and second-order logic with the full semantics the full second-order logic. Remark 1.1.3.We emphasize that the following facts are the main features of second-order logic: 1.The Completeness Theorem: A sentence is provable in DED2 if and only if it holds in all Henkin models [7]. 2.The Löwenheim-Skolem Theorem: A sentence with an infinite Henkin model has a countable Henkin model. 3.The Compactness Theorem: A set of sentences, every finite subset of which has a Henkin model, has itself a Henkin model. 4.The Incompleteness Theorem: Neither DED2 nor any other effectively given deductive calculus is complete for full models, that is, there are always sentences which are true in all full models but which are unprovable. 5.Failure of the Compactness Theorem for full models. 6.Failure of the Löwenheim-Skolem Theorem for full models. 7.There is a finite second-order axiom system 2 such that the semiring  of natural numbers is the only full model (up to isomorphism) of 2. 8. There is a finite second-order axiom system RCF2 such that the field  of real numbers is the only (up to isomorphism) full model of RCF2. Remark 1.1.4.For let second-order ZFC be, as usual, the theory that results obtained from ZFC when the axiom schema of replacement is replaced by its second-order universal closure,i.e. XFuncX  urr    ss  u  s, r  X, 1. 1. 1 where X is a second-order variable, and where FuncX abbreviates " X is a functional relation",see [7]. Thus we interpret the wff's of ZFC2 language with the full second-order semantics as required in [4],[5],[6],[7]. Designation 1.1.1. We will denote (i) by ZFC2Hs set theory ZFC2 with the Henkin semantics, (ii) by ZFC2 fss set theory ZFC2 with the full second-order semantics,(iii) by ZFC2 Hs set theory ZFC2Hs  Mst ZFC2 Hs and (iv) by ZFCst set theory ZFC  MstZFC, where MstTh is a standard model of the theory Th. Remark 1.1.3.There is no completeness theorem for second-order logic with the full second-order semantics. Nor do the axioms of ZFC2 fss imply a reflection principle which ensures that if a sentence Z of second-order set theory is true, then it is true in some model MZFC2 fss of ZFC2 fss [5]. Let Z be the conjunction of all the axioms of ZFC2 fss. We assume now that: Z is true,i.e. Con ZFC2 fss . It is known that the existence of a model for Z requires the existence of strongly inaccessible cardinals, i.e. under ZFC it can be shown that κ is a strongly inaccessible if and only if Hκ, is a model of ZFC2 fss. Thus ConZFC2 fss  ConZFC  . In this paper we prove that: (i) ZFCst  ZFC  MstZFC (ii) ZFC2 Hs  ZFC2 Hs  Mst ZFC2 Hs and (iii) ZFC2 fss is inconsistent, where MstTh is a standard model of the theory Th. Axiom MZFC. [8]. There is a set MZFC and a binary relation   MZFC  MZFC which makes MZFC a model for ZFC. Remark 1.1.3.(i) We emphasize that it is well known that axiom MZFC a single statement in ZFC see [7],Ch.II,section 7.We denote this statement throught all this paper by symbol ConZFC; MZFC.The completness theorem says that MZFC  ConZFC. (ii) Obviously there exists a single statement in ZFC2Hs such that MZFC2 Hs  ConZFC2Hs. We denote this statement throught all this paper by symbol Con ZFC2Hs; MZFC2 Hs and there exists a single statement MZ2 Hs in Z2Hs. We denote this statement throught all this paper by symbol Con Z2Hs; MZ2 Hs . Axiom MstZFC. [8].There is a set MstZFC such that if R is x, y|x  y  x  MstZFC  y  MstZFC

then MstZFC is a model for ZFC under the relation R. Definition 1.1.1.[8].The model MstZFC is called a standard model since the relation  used is merely the standard relation. Remark 1.1.4.[8].Note that axiom MZFC doesn't imply axiom MstZFC. Remark 1.1.5.We remind that in Henkin semantics, each sort of second-order variable has a particular domain of its own to range over, which may be a proper subset of all sets or functions of that sort. Leon Henkin (1950) defined these semantics and proved that Gödel's completeness theorem and compactness theorem, which hold for first-order logic, carry over to second-order logic with Henkin semantics. This is because Henkin semantics are almost identical to many-sorted first-order semantics, where additional sorts of variables are added to simulate the new variables of second-order logic. Second-order logic with Henkin semantics is not more expressive than first-order logic. Henkin semantics are commonly used in the study of second-order arithmetic.Väänänen [6] argued that the choice between Henkin models and full models for second-order logic is analogous to the choice between ZFC and V (V is von Neumann universe), as a basis for set theory: "As with second-order logic, we cannot really choose whether we axiomatize mathematics using V or ZFC. The result is the same in both cases, as ZFC is the best attempt so far to use V as an axiomatization of mathematics." Remark 1.1.6.Note that in order to deduce: (i) ~ConZFC2Hs from ConZFC2Hs, (ii) ~ConZFC from ConZFC,by using Gödel encoding, one needs something more than the consistency of ZFC2Hs, e.g., that ZFC2Hs has an omega-model M ZFC2 Hs or an standard model Mst ZFC2 Hs i.e., a model in which the integers are the standard integers.To put it another way, why should we believe a statement just because there's a ZFC2Hs-proof of it? It's clear that if ZFC2Hs is inconsistent, then we won't believe ZFC2Hs-proofs. What's slightly more subtle is that the mere consistency of ZFC2 isn't quite enough to get us to believe arithmetical theorems of ZFC2Hs; we must also believe that these arithmetical theorems are asserting something about the standard naturals. It is "conceivable" that ZFC2 Hs might be consistent but that the only nonstandard models MNst ZFC2 Hs it has are those in which the integers are nonstandard, in which case we might not "believe" an arithmetical statement such as "ZFC2Hs is inconsistent" even if there is a ZFC2Hs-proof of it. Remark 1.1.7. However assumption Mst ZFC2 Hs is not necessary. Note that in any nonstandard model MNst Z2 Hs of the second-order arithmetic Z2Hs the terms 0, S0  1, SS0  2, comprise the initial segment isomorphic to Mst Z2 Hs  MNst Z2 Hs . This initial segment is called the standard cut of the MNst Z2 Hs . The order type of any nonstandard model of MNst Z2 Hs is equal to   A   for some linear order A [9]. Thus one can to choose Gödel encoding inside Mst Z2 Hs . Remark 1.1.8. However there is no any problem as mentioned above in second order set theory ZFC2 with the full second-order semantics becouse corresponding second order arithmetic Z2 fss is categorical. Remark 1.1.9. Note if we view second-order arithmetic Z2 as a theory in first-order predicate calculus. Thus a model MZ2 of the language of second-order arithmetic Z2 consists of a set M (which forms the range of individual variables) together with a constant 0 (an element of M), a function S from M to M, two binary operations  and  on M, a binary relation  on M, and a collection D of subsets of M, which is the range of the set variables. When D is the full powerset of M, the model MZ2 is called a full model. The use of full second-order semantics is equivalent to limiting the models of second-order arithmetic to the full models. In fact, the axioms of second-order arithmetic have only one full model. This follows from the fact that the axioms of Peano arithmetic with the second-order induction axiom have only one model under second-order semantics, i.e. Z2, with the full semantics, is categorical by Dedekind's argument, so has only one model up to isomorphism. When M is the usual set of natural numbers with its usual operations, MZ2 is called an ω-model. In this case we may identify the model with D, its collection of sets of naturals, because this set is enough to completely determine an ω-model. The unique full omega-model M Z2 fss , which is the usual set of natural numbers with its usual structure and all its subsets, is called the intended or standard model of second-order arithmetic. 2.Derivation of the inconsistent definable set in set theory ZFC2 Hs and in set theory ZFCst. 2.1.Derivation of the inconsistent definable set in set theory ZFC2 Hs. Designation 2.1.1.Let XHs be the collection of the all 1-place open wff of the set theory ZFC2 Hs. Definition 2.1.1.Let 1X,2X be 1-place open wff's of the set theory ZFC2 Hs. (i) We define now the equivalence relation  X   XHs  XHs by 1X X 2X  X1X  2X 2. 1. 1 (ii) A subset XHs of XHs such that 1X X 2X holds for all 1X and 2X in XHs, and never for 1X in XHs and 2X outside XHs, is called an equivalence class of XHs by X . (iii)The collection of all possible equivalence classes of XHs by ~X, denoted XHs/ X XHs/ X  XHs|X  X Hs , 2. 1. 2 is the quotient set of XHs by X . (iv) For any X  XHs let XHs  X  X Hs|X X denote the equivalence class to which X belongs. All elements of XHs equivalent to each other are also elements of the same equivalence class. Definition 2.1.2.[9].Let Th be any theory in the recursive language Th PA,where PA is a language of Peano arithmetic.We say that a number-theoretic relation Rx1, . . . , xn of n arguments is expressible in Th if and only if there is a wff Rx1, . . . , xn of Th with the free variables x1, . . . , xn such that,for any natural numbers k1, . . . , kn, the following hold: (i) If Rk1, . . . , kn is true, then Th Rk1, . . . , kn. (ii) If Rk1, . . . , kn is false, then Th Rk1, . . . , kn. Designation 2.1.2.(i) Let gZFC2Hsu be a Gödel number of given an expression u of the set theory ZFC2 Hs  ZFC2 Hs  Mst ZFC2 Hs . (ii) Let Fr2Hsy, v be the relation : y is the Gödel number of a wff of the set theoryZFC2 Hs that contains free occurrences of the variable X with Gödel number v [8]-[9]. (iii) Note that the relation Fr2Hsy, v is expressible in ZFC2 Hs by a wff Fr2Hsy, v (iv) Note that for any y, v   by definition of the relation Fr2Hsy, v follows that Fr2Hsy, v  !X gZFC2HsX  y  gZFC2HsX   , 2. 1. 3 where X is a unique wff of ZFC2 Hs which contains free occurrences of the variable X with Gödel number v.We denote a unique wff X defined by using equivalence (1.2.3) by symbol y,X, i.e. Fr2Hsy, v  !y,X gZFC2Hsy,X  y  gZFC2HsX   , 2. 1. 4 (v) Let 2Hsy, v,1 be a Gödel number of the following wff: !XX  Y  X,where gZFC2HsX  y, gZFC2HsX  , gZFC2HsY  1. (vi) Let PrZFC2Hsz be a predicate asserting provability in ZFC2 Hs, which defined by formula (2.6) in section 2, see Remark 2.2 and Designation 2.3,(see also [9]-[10]). Remark 2.1.0.Note that this function gZFC2Hsy,X  y is expressible in set theory ZFC2 Hs by a wff of the set theory ZFC2 Hs that contains free occurrences of the variable y  . Note that formula y,X is given by an expression u0u1. . uj. . . ur, i.e. y,X : u0u1. . uj. . . ur,where each uj is a symbol of ZFC2 Hs. We introduce now a functions y,X; j : y,X uj, j  0, 1, . . . , i.e. y,X; j : uj and revrite expression u0u1. . uj. . . ur in the following equivalent form y,X; 0y,X; 1. . . y,X; j. . . y,X; r. By definitions we obtain that gZFCsty,X  y  y  2gy,X;03gy,X;1. . . pj gy,X;j. . . pr gy,X;r. Let us denote by y j the exponent gy,X; j in this factorization y  2gy,X;03gy,X;1. . . pj gy,X;j. . . pr gy,X;r. If y  1, y j  1 for all j. If x  0, we arbitrarily let y j  0 for all j. Then the functions y j, j  0, 1, . . . is primitive recursive, since y j  zypjz|y  pjz1|y, [8]. Thus the function y j is expressible in set theory ZFC2 Hs by formula denoted below by jy, gy,X; j. For y 0, let lhy be the number of non-zero exponents in the factorization of y into powers of primes, or, equivalently, the number of distinct primes that divide y.Let lh0  0, then lhy is primitive recursive.Thus function gZFC2Hsy,X  y is expressible in set theory ZFC2 Hs by formula y,X, y y,X, y  jlhy jy, gy,X; j. Definition 2.1.3. Let XHs be the countable collection of the all 1-place open wff's of the set theoryZFC2 Hs that contains free occurrences of the variable X. Definition 2.1.4. Let gZFC2HsX  .Let  Hs be a set of the all Gödel numbers of the 1-place open wff's of the set theoryZFC2 Hs that contains free occurrences of the variable X with Gödel number v, i.e. Hs  y  |y,  Fr2Hsy, v , 2. 1. 5 or in the following equivalent form: yy   y  Hs  y    Fr2Hsy, v . 2. 1. 6 Remark 2.1.1.Note that from the axiom of separation it follows directly that Hs is a set in the sense of the set theory ZFC2 Hs. Definition 2.1.5.(i)We define now the equivalence relation     Hs  Hs 2. 1. 7 in the sense of the set theory ZFC2 Hs by y1  y2  Xy1,X  y2,X 2. 1. 8 Note that from the axiom of separation it follows directly that the equivalence relation    is a relation in the sense of the set theory ZFC2 Hs. (ii) A subset Hs of Hs such that y1  y2 holds for all y1 and y1 in Hs,and never for y1 in Hs and y2 outside Hs, is an equivalence class of Hs. (iii) For any y  Hs let yHs  z   Hs|y  z denote the equivalence class to which y belongs. All elements of Hs equivalent to each other are also elements of the same equivalence class. (iv)The collection of all possible equivalence classes of Hs by ~, denoted Hs/  Hs/   yHs|y   Hs . 2. 1. 9 Remark 2.1.2. Note that from the axiom of separation it follows directly that Hs/  is a set in the sense of the set theory ZFC2 Hs. Definition 2.1.6.Let 2Hs be the countable collection of the all sets definable by 1-place open wff of the set theory ZFC2 Hs, i.e. Y	Y  2Hs  XXHs  X Hs/ X   !XX  Y  X . 2. 1. 10 Definition 2.1.7.We rewrite now (2.1.10) in the following equivalent form Y	Y  2Hs  XXHs  X

Hs/ X   Y  X , 2. 1. 11 where the countable collection X Hs/ X is defined by X	X  X Hs/ X  X  XHs/ X   !XX 2. 1. 12 Definition 2.1.8. Let 2Hs be the countable collection of the all sets such that XX  2HsX  2Hs  X  X. 2. 1. 13 Remark 2.1.3. Note that 2Hs  2Hs since 2Hs is a collection definable by 1-place open wff Z, 2Hs  XX  2HsX  Z  X  X. From (2.1.13) one obtains 2Hs  2Hs  2Hs  2Hs. 2. 1. 14 But (2.1.14) gives a contradiction 2Hs  2Hs  2Hs  2Hs. 2. 1. 15 However contradiction (2.1.15) it is not a contradiction inside ZFC2 Hs for the reason that the countable collection 2Hs is not a set in the sense of the set theory ZFC2 Hs. In order to obtain a contradiction inside ZFC2 Hs we introduce the following definitions. Definition 2.1.9.We define now the countable set  Hs/  by y yHs  

Hs/   yHs   Hs/    Fr2Hsy, v  !Xy,X . 2. 1. 16 Remark 2.1.4. Note that from the axiom of separation it follows directly that  / is a set in the sense of the set theory ZFC2 Hs. Definition 2.1.10.We define now the countable set 2 Hs by formula Y Y  2 Hs  y y   Hs/    gZFC2HsX    Y  X . 2. 1. 17 Note that from the axiom schema of replacement (1.1.1) it follows directly that 2 Hs is a set in the sense of the set theory ZFC2 Hs. Definition 2.1.12.We define now the countable set 2 Hs by formula XX  2 HsX  2 Hs  X  X. 2. 1. 18 Note that from the axiom schema of separation it follows directly that 2 Hs is a set in the sense of the set theory ZFC2 Hs. Remark 2.1.5.Note that 2 Hs  2 Hs since 2 Hs is a definable by the following formula  Z  XX  2 HsX  Z  X  X. 2. 1. 19 Theorem 2.1.1.Set theory ZFC2 Hs is inconsistent. Proof. From (2.1.18) and Remark 2.1.5 we obtain 2 Hs  2 Hs  2 Hs  2 Hs from which immediately one obtains a contradiction 2 Hs  2 Hs  2 Hs  2 Hs. 2.2.Derivation of the inconsistent definable set in set theory ZFCst. Designation 2.2.1.(i) Let gZFCstu be a Gödel number of given an expression u of the set theory ZFCst  ZFC  MstZFC. (ii) Let Frsty, v be the relation : y is the Gödel number of a wff of the set theoryZFCst that contains free occurrences of the variable X with Gödel number v [9]. (iii) Note that the relation Frsty, v is expressible in ZFCst by a wff Frsty, v (iv) Note that for any y, v   by definition of the relation Frsty, v follows that Frsty, v  !XgZFCstX  y  gZFCstX  , 2. 2. 1 where X is a unique wff of ZFCst which contains free occurrences of the variable X with Gödel number v.We denote a unique wff X defined by using equivalence (2.2.1) by symbol y,X, i.e. Frsty, v  !y,XgZFCsty,X  y  gZFCstX  , 2. 2. 2 (v) Let sty, v,1 be a Gödel number of the following wff: !XX  Y  X,where gZFCstX  y, gZFCstX  , gZFCstY  1. (vi) Let PrZFCstz be a predicate asserting provability in ZFCst, which defined by formula (2.6) in section 2, see Remark 2.2 and Designation 2.3,(see also [8]-[9]). Remark 2.2.0.Note that this function gZFCsty,X  y is expressible in set theory ZFCst by a wff of the set theory ZFCst that contains free occurrences of the variable y  . Note that formula y,X is given by an expression u0u1. . uj. . . ur, i.e. y,X : u0u1. . uj. . . ur,where each uj is a symbol of ZFCst. We introduce now a functions y,X; j : y,X uj, j  0, 1, . . . , i.e. y,X; j : uj and revrite expression u0u1. . uj. . . ur in the following equivalent form y,X; 0y,X; 1. . . y,X; j. . . y,X; r. By definitions we obtain that gZFCsty,X  y  y  2gy,X;03gy,X;1. . . pj gy,X;j. . . pr gy,X;r. Let us denote by y j the exponent gy,X; j in this factorization y  2gy,X;03gy,X;1. . . pj gy,X;j. . . pr gy,X;r. If y  1, y j  1 for all j. If x  0, we arbitrarily let y j  0 for all j. Then the functions y j, j  0, 1, . . . is primitive recursive, since y j  zypjz|y  pjz1|y, [8]. Thus the function y j is expressible in set theory ZFCst by formula denoted below by jy, gy,X; j. For y 0, let lhy be the number of non-zero exponents in the factorization of y into powers of primes, or, equivalently, the number of distinct primes that divide y.Let lh0  0, then lhy is primitive recursive. Thus function gZFCsty,X  y is expressible in set theory ZFCst by formula y,X, y y,X, y  jlhy jy, gy,X; j. Definition 2.2.1. Let Xst be the countable collection of the all 1-place open wff's of the set theory ZFCst that contains free occurrences of the variable X. Definition 2.2.2. Let gZFCstX  .Let st be a set of the all Gödel numbers of the 1-place open wff's of the set theory ZFCst that contains free occurrences of the variable X with Gödel number v, i.e. st  y  |y,  Frsty, v , 2. 2. 3 or in the following equivalent form: yy   y  st  y    Frsty, v . Remark 2.2.1.Note that from the axiom of separation it follows directly that st is a set in the sense of the set theory ZFCst. Definition 2.2.3.(i)We define now the equivalence relation  X   Xst  Xst by 1X X 2X  X1X  2X 2. 2. 4 (ii) A subcollection Xst of Xst such that 1X X 2X holds for all 1X and 2X in Xst, and never for 1X in Xst and 2X outside Xst, is an equivalence class of Xst. (iii) For any X  Xst let Xst  X  X st|X X X denote the equivalence class to which X belongs. All elements of Xst equivalent to each other are also elements of the same equivalence class. (iv) The collection of all possible equivalence classes of Xst by ~X, denoted Xst/ X Xst/ X  Xst|X  X st . 2. 2. 5 Definition 2.2.4.(i)We define now the equivalence relation     st  st in the sense of the set theory ZFCst by y1  y2  Xy1,X  y2,X 2. 2. 6 Note that from the axiom of separation it follows directly that the equivalence relation    is a relation in the sense of the set theory ZFCst. (ii) A subset st of st such that y1  y2 holds for all y1 and y1 in st,and never for y1 in st and y2 outside st, is an equivalence class of st. (iii) For any y  st let yst  z   st|y  z denote the equivalence class to which y belongs. All elements of st equivalent to each other are also elements of the same equivalence class. (iv)The collection of all possible equivalence classes of st by ~, denoted st/  st/   yst|y   st . 2. 2. 7 Remark 2.2.2. Note that from the axiom of separation it follows directly that st/  is a set in the sense of the set theory ZFCst. Definition 2.2.5.Let st be the countable collection of the all sets definable by 1-place open wff of the set theory ZFCst, i.e. Y	Y  st  XXst  X st/ X   !XX  Y  X . 2. 2. 8 Definition 2.2.6.We rewrite now (2.2.8) in the following equivalent form Y	Y  st  XXst  X

st/ X   Y  X , 2. 2. 9 where the countable collection X st/ X is defined by X	Xst  X

st/ X  Xst  X st / X   !XX 2. 2. 10 Definition 2.2.7. Let st be the countable collection of the all sets such that XX  stX  st  X  X. 2. 2. 11 Remark 2.2.3. Note that st  st since st is a collection definable by 1-place open wff Z, st  XX  stX  Z  X  X. From (2.2.11) and Remark 2.2.3 one obtains directly st  st  st  st. 2. 2. 12 But (2.2.12) immediately gives a contradiction st  st  st  st. 2. 2. 13 However contradiction (2.2.13) it is not a true contradiction inside ZFCst for the reason that the countable collection st is not a set in the sense of the set theory ZFCst. In order to obtain a true contradiction inside ZFCst we introduce the following definitions. Definition 2.2.8.We define now the countable set  st/  by formula y yst  

st/   yst   st/    Frsty, v  !Xy,X . 2. 2. 14 Remark 2.2.4. Note that from the axiom of separation it follows directly that  st/  is a set in the sense of the set theory ZFCst. Definition 2.2.9.We define now the countable set st by formula Y	Y  st  yyst  

st/    gZFCstX    Y  X . 2. 2. 15 Note that from the axiom schema of replacement it follows directly that st is a set in the sense of the set theory ZFCst. Definition 2.2.10.We define now the countable set st by formula XX  st X  st  X  X. 2. 2. 16 Note that from the axiom schema of separation it follows directly that st is a set in the sense of the set theory ZFCst. Remark 2.2.5.Note that st  st since st is a definable by the following formula  Z  XX  st X  Z  X  X. 2. 2. 17 Theorem 2.2.1.Set theory ZFCst is inconsistent. Proof. From (2.2.17) and Remark 2.2.5 we obtain st  st  st  st from which immediately one obtains a contradiction st  st   st  st . 2.3.Derivation of the inconsistent definable set in ZFCNst. Definition 2.3.1.Let PA be a first order theory which contain usual postulates of Peano arithmetic [9] and recursive defining equations for every primitive recursive function as desired.So for any (n  1)-place function f defined by primitive recursion over any n-place base function g and (n  2)-place iteration function h there would be the defining equations: (i) f0, y1, . . . , yn  gy1, . . . , yn, (ii) fx  1, y1, . . . , yn  hx, fx, y1, . . . , yn, y1, . . . , yn. Designation 2.3.1.(i) Let MNstZFC be a nonstandard model of ZFC and let MstPA be a standard model of PA.We assume now that MstPA  MNstZFC and denote such nonstandard model of the set theory ZFC by MNstZFCPA. (ii) Let ZFCNst be the theory ZFCNst  ZFC  MNstZFCPA. Designation 2.3.2.(i) Let gZFCNstu be a Gödel number of given an expression u of the set theory ZFCNst  ZFC  MNstZFCPA. (ii) Let FrNsty, v be the relation : y is the Gödel number of a wff of the set theory ZFCNst that contains free occurrences of the variable X with Gödel number v [9]. (iii) Note that the relation FrNsty, v is expressible in ZFCNst by a wff FrNsty, v (iv) Note that for any y, v   by definition of the relation FrNsty, v follows that FrNsty, v  !XgZFCNstX  y  gZFCNstX  , 2. 3. 1 where X is a unique wff of ZFCst which contains free occurrences of the variable X with Gödel number v.We denote a unique wff X defined by using equivalence (2.3.1) by symbol y,X, i.e. FrNsty, v  !y,XgZFCNsty,X  y  gZFCNstX  , 2. 3. 2 (v) Let Nsty, v,1 be a Gödel number of the following wff: !XX  Y  X,where gZFCNstX  y, gZFCNstX  , gZFCNstY  1. (vi) Let PrZFCNstz be a predicate asserting provability in ZFCNst, which defined by formula (2.6) in section 2, see Remark 2.2 and Designation 2.3,(see also [9]-[10]). Definition 2.3.2. Let XNst be the countable collection of the all 1-place open wff's of the set theory ZFCNst that contains free occurrences of the variable X. Definition 2.3.3. Let gZFCNstX  .Let Nst be a set of the all Gödel numbers of the 1-place open wff's of the set theory ZFCNst that contains free occurrences of the variable X with Gödel number v, i.e. Nst  y  |y,  Fr Nsty, v , 2. 3. 3 or in the following equivalent form: yy   y  Nst  y    Fr Nsty, v . Remark 2.3.1.Note that from the axiom of separation it follows directly that st is a set in the sense of the set theory ZFCNst. Definition 2.3.3.(i)We define now the equivalence relation  X   XNst  XNst by 1X X 2X  X1X  2X 2. 3. 4 (ii) A subcollection Xst of Xst such that 1X X 2X holds for all 1X and 2X in Xst, and never for 1X in XNst and 2X outside XNst, is an equivalence class of XNst. (iii) For any X  XNst let XNst  X  X Nst|X X X denote the equivalence class to which X belongs. All elements of Xst equivalent to each other are also elements of the same equivalence class. (iv) The collection of all possible equivalence classes of XNst by ~X, denoted XNst/ X XNst/ X  XNst|X  X Nst . 2. 3. 5 Definition 2.3.4.(i)We define now the equivalence relation     Nst  Nst in the sense of the set theory ZFCNst by y1  y2  Xy1,X  y2,X 2. 3. 6 Note that from the axiom of separation it follows directly that the equivalence relation    is a relation in the sense of the set theory ZFCNst. (ii) A subset Nst of Nst such that y1  y2 holds for all y1 and y1 in Nst,and never for y1 in Nst and y2 outside Nst, is an equivalence class of Nst. (iii) For any y  Nst let yNst  z   Nst|y  z denote the equivalence class to which y belongs. All elements of Nst equivalent to each other are also elements of the same equivalence class. (iv)The collection of all possible equivalence classes of Nst by ~, denoted Nst/  Nst/   yNst|y   Nst . 2. 3. 7 Remark 2.3.2. Note that from the axiom of separation it follows directly that Nst/  is a set in the sense of the set theory ZFCNst. Definition 2.3.5.Let Nst be the countable collection of the all sets definable by 1-place open wff of the set theory ZFCNst, i.e. Y	Y  Nst  XXNst  X Nst/ X   !XX  Y  X . 2. 3. 8 Definition 2.3.6.We rewrite now (2.3.8) in the following equivalent form Y	Y  Nst  XXNst  X

Nst/ X   Y  X , 2. 3. 9 where the countable collection X Nst/ X is defined by X	XNst  X

Nst/ X  XNst  X Nst/ X   !XX 2. 3. 10 Definition 2.3.7. Let Nst be the countable collection of the all sets such that XX  NstX  Nst  X  X. 2. 3. 11 Remark 2.3.3.Note that Nst  Nst since Nst is a collection definable by 1-place open wff Z, Nst  XX  NstX  Z  X  X. From (2.3.11) one obtains Nst  Nst  Nst  Nst. 2. 3. 12 But (2.3.12) gives a contradiction Nst  Nst  Nst  Nst. 2. 3. 13 However a contradiction (2.3.13) it is not a true contradiction inside ZFCNst for the reason that the countable collection Nst is not a set in the sense of the set theory ZFCNst. In order to obtain a true contradiction inside ZFCNst we introduce the following definitions. Definition 2.3.8.We define now the countable set  Nst/  by formula y yNst  

Nst/   yNst   Nst/    FrNsty, v  !Xy,X . 2. 3. 14 Remark 2.3.4. Note that from the axiom of separation it follows directly that  Nst/  is a set in the sense of the set theory ZFCst. Definition 2.3.9.We define now the countable set Nst by formula Y	Y  Nst  yyNst  

Nst/    gZFCNstX    Y  X . 2. 3. 15 Note that from the axiom schema of replacement it follows directly that st is a set in the sense of the set theory ZFCNst. Definition 2.3.10.We define now the countable set Nst by formula XX  Nst X  Nst  X  X. 2. 3. 16 Note that from the axiom schema of separation it follows directly that Nst is a set in the sense of the set theory ZFCNst. Remark 2.3.5.Note that Nst  Nst since Nst is a definable by the following formula  Z  XX  Nst X  Z  X  X. 2. 3. 17 Theorem 2.3.1.Set theory ZFCNst is inconsistent. Proof. From (2.3.16) and Remark 2.3.5 we obtain Nst  Nst  Nst  Nst from which one obtains a contradiction Nst  Nst   Nst  Nst . 3.Derivation of the inconsistent provably definable set in set theory ZFC2 Hs, ZFCst and ZFCNst. 3.1.Derivation of the inconsistent provably definable set in set theory ZFC2 Hs. Definition 3.1.1. Let 2 Hs be the countable collection of all provable definable sets X such that ZFC2 Hs !XX,where X is a 1-place open wff i.e., Y Y  2 Hs  ZFC2 Hs XX  XHs/ X   !XX  Y  X . 3. 1. 1 Let X  ZFC2 Hs Y be a predicate such that X  ZFC2 Hs Y  ZFC2 Hs X  Y.Let 2 Hs be the countable collection of all sets such that X X  2 Hs X  2 Hs  X  ZFC2 Hs X . 3. 1. 2 From (3.1.2) one obtains 2 Hs  2 Hs  2 Hs  ZFC2 Hs 2 Hs . 3. 1. 3 But obviously this is a contradiction. However contradiction (3.1.3) it is not a contradiction inside ZFC2 Hs for the reason that predicate X  ZFC2 Hs Y is not a predicate of ZFC2 Hs and therefore countable collections 2 Hs and 2 Hs are not a sets of ZFC2 Hs. Nevertheless by using Gödel encoding the above stated contradiction can be shipped in special consistent extensions of ZFC2 Hs. Remark 3.1.1.More formally I can to explain the gist of the contradictions deriveded in this paper (see Proposition 2.5.(i)-(ii)) as follows. Let M be Henkin model of ZFC2Hs. Let 2 Hs be the set of the all sets of M provably definable in ZFC2 Hs, and let 2 Hs  x  2 Hs : x  x where A means 'sentence A derivable in ZFC2 Hs', or some appropriate modification thereof. We replace now formula (3.1.1) by the following formula Y Y  2 Hs  XX  XHs/ X   !XX  Y  X . 3. 1. 4 and we replace formula (3.1.2) by the following formula X X  2 Hs X  2 Hs  X  X . 3. 1. 5 Definition 3.1.2.We rewrite now (3.1.4) in the following equivalent form Y Y  2 Hs  XXHs  X Hs/ X   Y  X , 3. 1. 6 where the countable collection XHs/ X is defined by the following formula X	X  XHs/ X  XHs  X Hs/ X   !XX 3. 1. 7 Definition 3.1.3.Let 2 Hs be the countable collection of the all sets such that X X  2 Hs X  2 Hs  X  X . 3. 1. 8 Remark 3.1.2.Note that 2 Hs  2 Hs since 2 Hs is a collection definable by 1-place open wff  Z,2 Hs  X X  2 Hs X  Z  X  X. From (3.1.8) one obtains 2 Hs  2 Hs  2 Hs  2 Hs . 3. 1. 9 But (3.1.9) immediately gives a contradiction ZFC2 Hs 2 Hs  2 Hs  2 Hs  2 Hs . 3. 1. 10 However contradiction (3.1.10) it is not a true contradiction inside ZFC2 Hs for the reason that the countable collection 2 Hs is not a set in the sense of the set theory ZFC2 Hs. In order to obtain a true contradiction inside ZFC2 Hs we introduce the following definitions. Definition 3.1.4.We define now the countable set Hs/  by y yHs   Hs/   yHs   Hs/    Fr2 Hs y, v   !Xy,X . 3. 1. 11 Definition 3.1.5.We choose now A in the following form

A  BewZFC2Hs#A  BewZFC2Hs#A  A . 3. 1. 12 Here BewZFC2Hs#A is a canonycal Gödel formula which says to us that there exists proof in ZFC2 Hs of the formula A with Gödel number #A. Remark 3.1.3. Notice that the Definition 3.1.5 holds as definition of predicate really asserting provability in ZFC2 Hs. Definition 3.1.7.Using Definition 3.1.5, we replace now formula (3.1.7) by the following formula X	X  XHs/ X  XX  XHs/ X    BewZFC2Hs#!XX  Y  X   BewZFC2Hs#!XX  Y  X  !XX  Y  X . 3. 1. 13 Definition 3.1.8.Using Definition 3.1.5, we replace now formula (3.1.8) by the following formula X X  2 Hs X  2 Hs  BewZFC2Hs#X  X   BewZFC2Hs#X  X  X  X . 3. 1. 14 Definition 3.1.9.Using Definition1.3.5,we replace now formula (3.1.11) by the following formula y	yHs   Hs/   yHs   Hs/    Fr2Hsy, v  BewZFC2Hs#!Xy,X  Y  X   BewZFC2Hs#!Xy,X  Y  X  !Xy,X  Y  X . 3. 1. 15 Definition 3.1.10.Using Definitions 3.1.4-3.1.7, we define now the countable set 2 Hs by formula Y Y  2 Hs  y y  Hs/    gZFC2HsX   . 3. 1. 16 Remark 3.1.4.Note that from the axiom schema of replacement (1.1.1) it follows directly that 2 Hs is a set in the sense of the set theory ZFC2 Hs. Definition 3.1.11.Using Definition 3.1.8 we replace now formula (3.1.14) by the following formula X X  2 Hs X  2 Hs  BewZFC2Hs#X  X  BewZFC2Hs#X  X  X  X . 3. 1. 17 Remark 3.1.5. Notice that the expression (3.1.18) BewZFC2Hs#X  X  BewZFC2Hs#X  X  X  X 3. 1. 18 obviously is a well formed formula of ZFC2 Hs and therefore collection 2 Hs is a set in the sense of ZFC2 Hs. Remark 3.1.6.Note that 2 Hs  2 Hs since 2 Hs is a collection definable by 1-place open wff  Z,2 Hs  X X  2 Hs X  Z  BewZFC2Hs#X  X  BewZFC2Hs#X  X  X  X . 3. 1. 19 Theorem 3.1.1.Set theory ZFC2 Hs  ZFC2 Hs  Mst ZFC2 Hs is inconsistent. Proof. From (3.1.17) we obtain 2 Hs  2 Hs  BewZFC2Hs # 2 Hs  2 Hs   BewZFC2Hs # 2 Hs  2 Hs  2 Hs  2 Hs . 3. 1. 20 (a) Assume now that: 2 Hs  2 Hs . 3. 1. 21 Then from (3.1.20) we obtain ZFC2Hs BewZFC2Hs # 2 Hs  2 Hs and ZFC2Hs BewZFC2Hs # 2 Hs  2 Hs  2 Hs  2 Hs , therefore ZFC2Hs 2 Hs  2 Hs and so ZFC2Hs 2 Hs  2 Hs  2 Hs  2 Hs . 3. 1. 22 From (3.1.21)-(3.1.22) we obtain 2 Hs  2 Hs ,2 Hs  2 Hs  2 Hs  2 Hs 2 Hs  2 Hs and thus ZFC2Hs 2 Hs  2 Hs  2 Hs  2 Hs . (b) Assume now that BewZFC2Hs # 2 Hs  2 Hs   BewZFC2Hs # 2 Hs  2 Hs  2 Hs  2 Hs . 3. 1. 23 Then from (3.1.23) we obtain 2 Hs  2 Hs .From (3.1.23) and (3.1.20) we obtain ZFC2Hs 2 Hs  2 Hs ,so ZFC2Hs 2 Hs  2 Hs ,2 Hs  2 Hs which immediately gives us a contradiction ZFC2Hs 2 Hs  2 Hs  2 Hs  2 Hs . Definition 3.1.12.We choose now A in the following form

A  BewZFC2Hs#A, 3. 1. 24 or in the following equivalent form

A  BewZFC2Hs#A  BewZFC2Hs#A  A similar to (3.1.5).Here BewZFC2Hs#A is a Gödel formula (see Chapt. II section 2, Definition) which really asserts provability in ZFC2 Hs of the formula A with Gödel number #A. Remark 3.1.7. Notice that the Definition 3.1.12 with formula (3.1.24) holds as definition of predicate really asserting provability in ZFC2 Hs. Definition 3.1.13.Using Definition 3.1.12 with formula (3.1.24), we replace now formula (3.1.7) by the following formula X X  X Hs/ X  XX  XHs/ X    BewZFC2Hs#!XX  Y  X . 3. 1. 25 Definition 3.1.14.Using Definition 3.1.12 with formula (3.1.24), we replace now formula (3.1.8) by the following formula X X  2 Hs X  2 Hs  BewZFC2Hs#X  X 3. 1. 26 Definition 3.1.15.Using Definition 3.1.12 with formula (3.1.24),we replace now formula (3.1.11) by the following formula y	yHs   Hs/   yHs   Hs/    Fr2 Hs y, v  BewZFC2Hs#!Xy,X  Y  X . 3. 1. 27 Definition 3.1.16.Using Definitions 3.1.13-3.1.17, we define now the countable set

Hs by formula Y Y  2 Hs  y y  Hs/    gZFC2HsX   . 3. 1. 28 Remark 3.1.8.Note that from the axiom schema of replacement (1.1.1) it follows directly that 2 Hs is a set in the sense of the set theory ZFC2 Hs. Definition 3.1.17.Using Definition 3.1.16 we replace now formula (3.1.26) by the following formula X X  2 Hs X  2 Hs  BewZFC2Hs#X  X . 3. 1. 29 Remark 3.1.9. Notice that the expressions (3.1.30) BewZFC2Hs#X  X and BewZFC2Hs#X  X  BewZFC2Hs#X  X  X  X 3. 1. 30 obviously is a well formed formula of ZFC2 Hs and therefore collection 2 Hs is a set in the sense of ZFC2 Hs. Remark 3.1.10.Note that 2 Hs  2 Hs since 2 Hs is a collection definable by 1-place open wff  Z,2 Hs  X X  2 Hs X  Z  BewZFC2Hs#X  X . 3. 1. 31 Theorem 3.1.2.Set theory ZFC2 Hs  ZFC2 Hs  Mst ZFC2 Hs is inconsistent. Proof. From (3.1.29) we obtain 2 Hs  2 Hs  BewZFC2Hs # 2 Hs  2 Hs . 3. 1. 32 (a) Assume now that: 2 Hs  2 Hs . 3. 1. 33 Then from (3.1.32) we obtain ZFC2Hs BewZFC2Hs # 2 Hs  2 Hs and therefore ZFC2Hs 2 Hs  2 Hs thus we obtain ZFC2Hs 2 Hs  2 Hs  2 Hs  2 Hs . 3. 1. 34 From (3.1.33)-(3.1.34) we obtain 2 Hs  2 Hs and 2 Hs  2 Hs  2 Hs  2 Hs thus ZFC2Hs 2 Hs  2 Hs and finally we obtain ZFC2Hs 2 Hs  2 Hs  2 Hs  2 Hs . (b) Assume now that BewZFC2Hs # 2 Hs  2 Hs . 3. 1. 23 Then from (3.1.35) we obtain ZFC2Hs 2 Hs  2 Hs .From (3.1.35) and (3.1.32) we obtain ZFC2Hs 2 Hs  2 Hs , thus ZFC2Hs 2 Hs  2 Hs and ZFC2Hs 2 Hs  2 Hs which immediately gives us a contradiction ZFC2Hs 2 Hs  2 Hs  2 Hs  2 Hs . 3.2.Derivation of the inconsistent provably definable set in ZFCst. Let st be the countable collection of all sets X such that ZFCst !XX,where X is a 1-place open wff i.e., Y	Y  st  ZFCst XX  Xst/ X   !XX  Y  X . 3. 2. 1 Let X ZFCst Y be a predicate such that X ZFCst Y  ZFCst X  Y.Let  be the countable collection of all sets such that X X  st  X  st  X ZFCst X . 3. 2. 2 From (3.2.1) one obtains st  st  st ZFCst st. 3. 2. 3 But (3.2.3) gives a contradiction st  st  st  st. 3. 2. 4 However contradiction (3.2.4) it is not a contradiction inside ZFCst for the reason that predicate X ZFCst Y is not a predicate of ZFCst and therefore countable collections st and st are not a sets of ZFCst. Nevertheless by using Gödel encoding the above stated contradiction can be shipped in special consistent extensions of ZFCst. Designation 3.2.1 (i) Let MstZFC be a standard model of ZFC and (ii) let ZFCst be the theory ZFCst  ZFC  MstZFC, (iii) let st be the set of the all sets of MstZFC provably definable in ZFCst,and let st  X  st : stX  X where stA means: 'sentence A derivable in ZFCst', or some appropriate modification thereof. We replace now (3.2.1) by formula Y	Y  st  st!XX  Y  X , 3. 2. 5 and we replace (3.2.2) by formula X X  st  X  st  st X  X . 3. 2. 6 Assume that ZFCst st  st. Then, we have that: st  st if and only if

stst  st, which immediately gives us st  st if and only if st  st.But this is a contradiction, i.e., ZFCst st  st  st  st.We choose now stA in the following form

stA  BewZFCst#A  BewZFCst#A  A. 3. 2. 7 Here BewZFCst#A is a canonycal Gödel formula which says to us that there exists proof in ZFCst of the formula A with Gödel number #A  MstPA. Remark 3.2.1. Notice that definition (3.2.7) holds as definition of predicate really asserting provability in ZFCst. Definition 3.2.2.We rewrite now (3.2.5) in the following equivalent form Y Y  st  XXst  X st/ X   Y  X , 3. 2. 8 where the countable collection XHs/ X is defined by the following formula X	Xst  X st/ X  Xst  X st/ X   st!XX 3. 2. 9 Definition 3.2.3.Let st be the countable collection of the all sets such that X X  st X  st  stX  X . 3. 2. 10 Remark 3.2.2.Note that 2 Hs  2 Hs since 2 Hs is a collection definable by 1-place open wff  Z,st  X X  st X  Z  stX  X. 3. 2. 11 Definition 3.2.4.By using formula (3.2.7) we rewrite now (3.2.8) in the following equivalent form Y Y  st  XXst  X st/ X   Y  X , 3. 2. 12 where the countable collection XHs/ X is defined by the following formula X	Xst  X st/ X  Xst  X st/ X   BewZFCst#!XX  BewZFCst#!XX  !XX

3. 2. 13 Definition 3.2.5.Using formula (3.2.7), we replace now formula (3.2.10) by the following formula X X  st X  st  BewZFCst#X  X  BewZFCst#X  X. 3. 2. 14 Definition 3.2.6.Using Definition1.3.5,we replace now formula (3.2.11) by the following formula y	yst   st/   yst   st/    Frsty, v  BewZFCst#!Xy,X  Y  X  BewZFCst#!Xy,X  Y  X  !Xy,X  Y  X . 3. 2. 15 Definition 3.2.7.Using Definitions 3.2.4-3.2.6, we define now the countable set st  by formula Y Y  st   yyst   st/    gZFCstX   . 3. 2. 16 Remark 3.2.3.Note that from the axiom schema of replacement it follows directly that

st  is a set in the sense of the set theory ZFCst. Definition 3.2.8.Using Definition 3.2.7 we replace now formula (3.2.14) by the following formula X X  st  X  st   BewZFCst#X  X  BewZFCst#X  X  X  X . 3. 2. 17 Remark 3.2.4. Notice that the expression (3.2.18) BewZFCst#X  X  BewZFCst#X  X  X  X 3. 2. 18 obviously is a well formed formula of ZFCst and therefore collection st  is a set in the sense of ZFC2 Hs. Remark 3.2.5.Note that st   st  since st  is a collection definable by 1-place open wff  Z,st   X X  st  X  Z  BewZFCst#X  X  BewZFCst#X  X  X  X. 3. 2. 19 Theorem 3.2.1.Set theory ZFCst  ZFC  MstZFC is inconsistent. Proof. From (3.2.17) we obtain st   st   BewZFCst # st   st    BewZFCst # st   st   st   st  . 3. 2. 20 (a) Assume now that: st   st  . 3. 2. 21 Then from (3.2.20) we obtain BewZFCst # st   st  and BewZFCst # st   st   st   st  , therefore st   st  and so ZFCst st   st   st   st  . 3. 2. 22 From (3.2.21)-(3.2.22) we obtain st   st  ,st   st   st   st  st   st  and therefore ZFCst st   st   st   st  . (b) Assume now that BewZFCst # st   st    BewZFCst # st   st   st   st  . 3. 2. 23 Then from (3.2.23) we obtain 2 Hs  2 Hs .From (3.2.23) and (3.2.20) we obtain ZFC2Hs 2 Hs  2 Hs ,so ZFC2Hs 2 Hs  2 Hs ,2 Hs  2 Hs which immediately gives us a contradiction ZFC2Hs 2 Hs  2 Hs  2 Hs  2 Hs . 3.3.Derivation of the inconsistent provably definable set in ZFCNst. Designation 3.3.1.(i) Let PA be a first order theory which contain usual postulates of Peano arithmetic [8] and recursive defining equations for every primitive recursive function as desired. (ii) Let MNstZFC be a nonstandard model of ZFC and let MstPA be a standard model of PA.We assume now that MstPA  MNstZFC and denote such nonstandard model of ZFC by MNst ZFCPA. (iii) Let ZFCNst be the theory ZFCNst  ZFC  MNstZFCPA. (iv) Let Nst be the set of the all sets of MstZFCPA provably definable in ZFCNst,and let Nst  X  Nst : NstX  X where NstA means 'sentence A derivable in ZFCNst', or some appropriate modification thereof. We replace now (3.1.4) by formula Y	Y  Nst  Nst!XX  Y  X , 3. 3. 1 and we replace (3.1.5) by formula X X  Nst  X  Nst  Nst X  X . 3. 3. 2 Assume that ZFCNst Nst  Nst. Then, we have that: Nst  Nst if and only if

NstNst  Nst, which immediately gives us Nst  Nst if and only if Nst  Nst.But this is a contradiction, i.e., ZFCNst Nst  Nst  Nst  Nst.We choose now NstA in the following form

NstA  BewZFCNst#A  BewZFCNst#A  A. 3. 3. 3 Here BewZFCNst#A is a canonycal Gödel formula which says to us that there exists proof in ZFCNst of the formula A with Gödel number #A  MstPA. Remark 3.3.1. Notice that definition (3.3.3) holds as definition of predicate really asserting provability in ZFCNst. Designation 3.3.2.(i) Let gZFCNstu be a Gödel number of given an expression u of ZFCNst. (ii) Let FrNsty, v be the relation : y is the Gödel number of a wff of ZFCNst that contains free occurrences of the variable with Gödel number v [10]. (iii) Let Nsty, v,1 be a Gödel number of the following wff: !XX  Y  X,where gZFCNstX  y, gZFCNstX  , gZFCNstY  1. (iv) Let PrZFCNstz be a predicate asserting provability in ZFCNst, which defined by formula (2.6), see Chapt. II, section 2, Remark 2.2 and Designation 2.3,(see also [10]-[11]). Remark 3.3.2.Let Nst be the countable collection of all sets X such that ZFCNst !XX,where X is a 1-place open wff i.e., Y	Y  Nst  ZFCNst X!XX  Y  X . 3. 3. 4 We rewrite now (3.3.4) in the following form Y	Y  Nst  gZFCNstY  1  yFrNsty, v  gZFCNstX    PrZFCNst	Nsty, v,1  PrZFCNst	Nsty, v,1  !XX  Y  X

3. 3. 5 Designation 3.3.3.Let Nstz be a Gödel number of the following wff: Z  Z, where gZFCNstZ  z. Remark 3.3.3.Let Nst above by formula (3.3.2), i.e., Z Z  Nst  Z  Nst  Nst Z  Z . 3. 3. 6 We rewrite now (3.3.6) in the following form . ZZ  Nst  Z  Nst   gZFCNstZ  z  PrZFCNst	Nstz   PrZFCNst	Nstz  Z  Z . 3. 3. 7 Theorem 3.3.1.ZFCNst Nst  Nst  Nst  Nst . 3.4.Generalized Tarski's undefinability lemma. Remark 3.4.1.Remind that: (i) if Th is a theory, let TTh be the set of Godel numbers of theorems of Th,[10],(ii) the property x  TTh is said to be is expressible in Th by wff Truex1 if the following properties are satisfies [10]: (a) if n  TTh then Th Truen, (b) if n  TTh then Th Truen. Remark 3.4.2.Notice it follows from (a)(b) that Th  Truen  Th  Truen. Theorem 3.4.1. (Tarski's undefinability Lemma) [10].Let Th be a consistent theory with equality in the language  in which the diagonal function D is representable and let gThu be a Gödel number of given an expression u of Th.Then the property x  TTh is not expressible in Th. Proof.By the diagonalization lemma applied to Truex1 there is a sentence  such that: (c)Th   Trueq,where q is the Godel number of , i.e. gTh  q. Case 1.Suppose that Th , then q  TTh. By (a), Th Trueq. But, from Th  and (c), by biconditional elimination, one obtains Th Trueq.Hence Th is inconsistent, contradicting our hypothesis. Case 2. Suppose that Th  . Then q  TTh. By (b), Th Trueq. Hence, by (c) and biconditional elimination, Th .Thus, in either case a contradiction is reached. Definition 3.4.1.If Th is a theory, let TTh be the set of Godel numbers of theorems of Th and let gThu be a Gödel number of given an expression u of Th.The property x  TTh is said to be is a strongly expressible in Th by wff True x1 if the following properties are satisfies: (a) if n  TTh then Th True n  True n  gTh 1 n, (b) if n  TTh then Th True n. Theorem3.4.2.(Generalized Tarski's undefinability Lemma).Let Th be a consistent theory with equality in the language  in which the diagonal function D is representable and let gThu be a Gödel number of given an expression u of Th.Then the property x  TTh is not strongly expressible in Th. Proof.By the diagonalization lemma applied to True x1 there is a sentence  such that: (c)Th   True q,where q is the Godel number of  , i.e. gTh   q. Case 1.Suppose that Th  , then q  TTh. By (a), Th True q. But, from Th  and (c), by biconditional elimination, one obtains Th True q.Hence Th is inconsistent, contradicting our hypothesis. Case 2. Suppose that Th   . Then q  TTh. By (b), Th True q. Hence, by (c) and biconditional elimination, Th  .Thus, in either case a contradiction is reached. Remark 3.4.3.Notice that it is widely believed on ubnormal part of the mathematical comunity that Tarski's undefinability theorems 3.4.1-3.4.2 blocking any possible definitions of the sets , st, Nst,mentioned in subsection 1.2 and therefore these theorems blocking definitions of the sets ,st,Nst, and correspondingly Tarski's undefinability theorem blocking the biconditionals        ,st  st  st  st , Nst  Nst  Nst  Nst. 3. 4. 1 3.5.Generalized Tarski's undefinability theorem. Remark 3.5.1.(I) Let Th1# be the theory Th1#  ZFC2 Hs. In addition under assumption ConTh1 #, we establish a countable sequence of the consistent extensions of the theory Th1 # such that: (i)Th1 # . . . Thi #  Thi1 # . . . Th # , where (ii) Thi1 # is a finite consistent extension of Thi #, (iii) Th #  i Thi #, (iv) Th # proves the all sentences of Th1 #, which valid in M, i.e.,M  A  Th # A, see Part II, section 2,Proposition 2.1.(i). (II) Let Th1,st# be Th1,st#  ZFCst. In addition under assumption ConTh1,st # , we establish a countable sequence of the consistent extensions of the theory Th1 # such that: (i) Th1,st # . . . Thi,st #  Thi1,st # . . . Th,st # , where (ii) Thi1,st # is a finite consistent extension of Thi,st # , (iii) Th,st #  i Thi,st # , (iv) Th,st # proves the all sentences of Th1,st # , which valid in MstZFC, i.e., MstZFC  A  Th,st # A, see Part II, section 2, Proposition 2.1.(ii). (III) Let Th1,Nst# be Th1,Nst#  ZFCNst. In addition under assumption ConTh1,Nst # , we establish a countable sequence of the consistent extensions of the theory Th1 # such that: (i)Th1,Nst # . . . Thi,Nst #  Thi1,st # . . . Th,Nst # , where (ii) Thi1,Nst # is a finite consistent extension of Thi,Nst # , (iii) Th,st #  i Thi,st # (iv) Th,st # proves the all sentences of Th1,st # , which valid in MNstZFCPA, i.e., MNst ZFCPA  A  Th,Nst # A, see Part II, section 2, Proposition 2.1.(iii). Remark 3.5.2.(I)Let i, i  1, 2, . . . be the set of the all sets of M provably definable in Thi #, Y	Y  i  i!XX  Y  X . 3. 5. 1 and let i  x  i : ix  x where iA means sentence A derivable in Thi #.Then we have that i  i if and only if ii  i, which immediately gives us i  i if and only if i  i.We choose now iA, i  1, 2, . . . in the following form

iA  Bewi#A  Bewi#A  A. 3. 5. 2 Here Bewi#A, i  1, 2, . . . is a canonycal Gödel formulae which says to us that there exist proof in Thi #, i  1, 2, . . .of the formula A with Gödel number #A. (II) Let i,st, i  1, 2, . . . be the set of the all sets of MstZFC provably definable in Thi,st# , Y	Y  i,st  i,st!XX  Y  X . 3. 5. 3 and let i,st  x  i,st : i,stx  x where i,stA means sentence A derivable in Thi,st # . Then we have that i,st  i,st if and only if i,sti,st  i,st, which immediately gives us i,st  i,st if and only if i,st  i,st.We choose now i,stA, i  1, 2, . . . in the following form

i,stA  Bewi,st#A  Bewi,st#A  A. 3. 5. 4 Here Bewi,st#A, i  1, 2, . . . is a canonycal Gödel formulae which says to us that there exist proof in Thi,st # , i  1, 2, . . .of the formula A with Gödel number #A. (III) Let i,Nst, i  1, 2, . . . be the set of the all sets of MNstZFCPA provably definable in Thi,Nst # , Y	Y  i,Nst  i,Nst!XX  Y  X . 3. 5. 5 and let i,Nst  x  i,Nst : i,Nstx  x where i,NstA means sentence A derivable in Thi,Nst # .Then we have that i,Nst  i,Nst if and only if i,Nsti,Nst  i,Nst, which immediately gives us i,Nst  i,Nst if and only if i,Nst  i,Nst. We choose now i,NstA, i  1, 2, . . . in the following form

i,NstA  Bewi,Nst#A  Bewi,Nst#A  A. 3. 5. 6 Here Bewi,Nst#A, i  1, 2, . . . is a canonycal Gödel formulae which says to us that there exist proof in Thi,Nst # , i  1, 2, . . .of the formula A with Gödel number #A. Remark 3.5.3 Notice that definitions (3.5.2),(3.5.4) and (3.5.6) hold as definitions of predicates really asserting provability in Thi #, Thi,st # and Thi,Nst # , i  1, 2, . . . correspondingly. Remark 3.5.4.Of course the all theories Thi#, Thi,st# , Thi,Nst# , i  1, 2, . . . are inconsistent,see Part II,Proposition 2.10.(i)-(iii). Remark 3.5.5.(I)Let  be the set of the all sets of M provably definable in Th# , Y	Y    !XX  Y  X . 3. 5. 7 and let   x   : x  x where A means 'sentence A derivable in Th # .Then, we have that    if and only if   , which immediately gives us    if and only if   .We choose now A, i  1, 2, . . . in the following form

A  iBewi#A  Bewi#A  A. 3. 5. 8 (II) Let ,st be the set of the all sets of MstZFC provably definable in Th,st# , Y	Y  ,st  ,st!XX  Y  X . 3. 5. 9 and let ,st be the set ,st  x  ,st : ,stx  x , where ,stA means 'sentence A derivable in Th,st # .Then, we have that ,st  ,st if and only if ,st,st  ,st, which immediately gives us ,st  ,st if and only if ,st  ,st.We choose now

,stA, i  1, 2, . . . in the following form

,stA  iBewi,st#A  Bewi,st#A  A. 3. 5. 10 (III) Let ,Nst be the set of the all sets of MNstZFCPA provably definable in Th,Nst # , Y	Y  ,Nst  ,Nst!XX  Y  X . 3. 5. 11 and let ,Nst be the set ,Nst  x  ,Nst : ,Nstx  x where ,NstA means 'sentence A derivable in Th,Nst # .Then, we have that ,Nst  ,Nst if and only if

,Nst,Nst  ,Nst, which immediately gives us ,Nst  ,Nst if and only if ,Nst  ,Nst.We choose now ,NstA, i  1, 2, . . . in the following form

,NstA  iBewi,Nst#A  Bewi,Nst#A  A. 3. 5. 12 Remark 3.5.6.Notice that definitions (3.5.8),(3.5.10) and (3.5.12) holds as definitions of a predicate really asserting provability in Th # , Th,st # and Th,Nst # correspondingly. Remark 3.5.7.Of course all the theories Th# , Th,st# and Th,Nst# are inconsistent,see Part II,Proposition 2.14.(i)-(iii). Remark 3.5.8.Notice that under naive consideration the set  and  can be defined directly using a truth predicate,which of couse is not available in the language of ZFC2Hs (but iff ZFC2Hs is consistent) by well-known Tarski's undefinability theorem [10]. Theorem 3.5.1. Tarski's undefinability theorem: (I) Let Th be first order theory with formal language ,which includes negation and has a Gödel numbering g such that for every -formula Ax there is a formula B such that B  AgB holds. Assume that Th has a standard model Mst Th and ConTh,st where Th,st  Th  Mst Th . 3. 5. 13 Let T be the set of Gödel numbers of -sentences true in Mst Th . Then there is no -formula Truen (truth predicate) which defines T .That is, there is no -formula Truen such that for every -formula A, TruegA  A 3. 5. 14 holds. (II) Let ThHs be second order theory with Henkin semantics and formal language , which includes negation and has a Gödel numbering g such that for every -formula Ax there is a formula B such that B  AgB holds. Assume that Th Hs has a standard model Mst Th Hs and ConTh,st Hs ,where Th,st Hs  Th Hs  Mst Th Hs 3. 5. 15 Let T be the set of Gödel numbers of the all -sentences true in M. Then there is no -formula Truen (truth predicate) which defines T .That is, there is no -formula Truen such that for every -formula A, TruegA  A 3. 5. 16 holds. Remark 3.5.9.Notice that the proof of Tarski's undefinability theorem in this form is again by simple reductio ad absurdum. Suppose that an formula True(n) defines T . In particular, if A is a sentence of Th then TruegA holds in  if and only if A is true in Mst Th . Hence for all A, the Tarski T-sentence TruegA  A is true in Mst Th . But the diagonal lemma yields a counterexample to this equivalence, by giving a "Liar" sentence S such that S  TruegS holds in Mst Th . Thus no -formula Truen can define T . Remark 3.5.10.Notice that the formal machinery of this proof is wholly elementary except for the diagonalization that the diagonal lemma requires. The proof of the diagonal lemma is likewise surprisingly simple; for example, it does not invoke recursive functions in any way. The proof does assume that every -formula has a Gödel number, but the specifics of a coding method are not required. Remark 3.5.11.The undefinability theorem does not prevent truth in one consistent theory from being defined in a stronger theory. For example, the set of (codes for) formulas of first-order Peano arithmetic that are true in  is definable by a formula in second order arithmetic. Similarly, the set of true formulas of the standard model of second order arithmetic (or n-th order arithmetic for any n) can be defined by a formula in first-order ZFC. Remark1. 3. 5. 12.Notice that it is widely believed on ubnormal part of mathematical comunity that Tarski's undefinability theorem blocking any possible definition of the sets

i, , i,st, i,st, ,st, ,Nst, and the sets ,st. Correspondingly Tarski's undefinability theorem blocking the biconditionals i  i  i  i , i  ,       ,etc. 3. 5. 17 Thus in contrast with naive definition of the sets  and  there is no any problem which arises from Tarski's undefinability theorem. Remark 3.5.13.(I) We define again the set  but now by using generalized truth predicate True# gA, A such that TruegA, A  iBewi#A  Bewi#A  A  TruegA  TruegA  A  A, TruegA  iBewi#A. 3. 5. 18 holds. (II) We define the set ,st using generalized truth predicate True,st# gA, A such that True,stgA, A  iBewi,st#A  Bewi,st#A  A  True,stgA  True,stgA  A  A, True,stgA  iBewi,st#A 3. 5. 19 holds.Thus in contrast with naive definition of the sets  and  there is no any problem which arises from Tarski's undefinability theorem. (III) We define the set ,Nst using generalized truth predicate True,Nst# gA, A such that True,NstgA, A  iBewi,Nst#A  Bewi,Nst#A  A  True,NstgA  True,NstgA  A  A, True,NstgA  iBewi,Nst#A 3. 5. 20 holds.Thus in contrast with naive definition of the sets ,Nst and ,Nst there is no any problem which arises from Tarski's undefinability theorem. Remark 3.5.14.In order to prove that set theory ZFC2Hs  MZFC2 Hs is inconsistent without any refference to the set ,notice that by the properties of the extension Th # follows that definition given by formula (1.5.18) is correct, i.e.,for every ZFC2Hs-formula  such that MZFC2 Hs   the following equivalence   Trueg, holds. Theorem 3.5.2.(Generalized Tarski's undefinability theorem) (see Part II, section 2, Proposition 2.30).Let Th be a first order theory or the second order theory with Henkin semantics and with formal language ,which includes negation and has a Gödel encoding g such that for every -formula Ax there is a formula B such that the equivalence B  AgB  AgB  Bholds. Assume that Th has an standard Model MstTh. Then there is no -formula Truen, n  , such that for every -formula A such that M  A, the following equivalence A  TruegA  TruegA  A 3. 5. 21 holds. Theorem 3.5.3. (i) Set theory Th1# ZFC2Hs  MZFC2 Hs is inconsistent; (ii) Set theory Th1,st #  ZFC  MstZFC is inconsistent;(iii) Set theory Th1,Nst #  ZFC  MNstZFC is inconsistent; (see Part.II, section 2, Proposition 2.31.(i)-(iii)). Proof.(i) Notice that by the properties of the extension Th# of the theory ZFC2 Hs  MZFC2 Hs  Th1# follows that MZFC2 Hs    Th # . 3. 5. 22 Therefore formula (3.5.18) gives generalized "truth predicate" for the set theory Th1#.By Theorem 3.5.2 one obtains a contradiction. (ii) Notice that by the properties of the extension Th,Nst # of the theoryZFC  MstZFC  Th1,st # follows that MstZFC    Th,st # . 3. 5. 23 Therefore formula (3.5.19) gives generalized "truth predicate" for the set theory Th1,st # .By Theorem 3.5.2 one obtains a contradiction. (iii) Notice that by the properties of the extension Th,Nst # of the theory ZFC  MNstZFC  Th1,st # follows that MNst ZFC    Th,Nst # . 3. 5. 24 Therefore (3.5.20) gives generalized "truth predicate" for the set theory Th1,Nst # .By Theorem 3.5.2 one obtains a contradiction. 3.6. Avoiding the contradictions from set theory ZFC2 Hs, ZFCst and set theory ZFCNst using Quinean approach. In order to avoid difficultnes mentioned above we use well known Quinean approach. 3.6.1.Quinean set theory NF. Remind that the primitive predicates of Russellian unramified typed set theory (TST), a streamlined version of the theory of types, are equality  and membership . TST has a linear hierarchy of types: type 0 consists of individuals otherwise undescribed. For each (meta-) natural number n, type n  1 objects are sets of type n objects; sets of type n have members of type n  1. Objects connected by identity must have the same type. The following two atomic formulas succinctly describe the typing rules: xn  yn and xn  yn1. The axioms of TST are: Extensionality: sets of the same (positive) type with the same members are equal; Axiom schema of comprehension: If xn is a formula, then the set xn  xn n1 exists i.e., given any formula xn, the formula An1xnxn  An1  xn 3. 6. 1 is an axiom where An1 represents the set xn  xn n1 and is not free in xn. Quinean set theory.(New Foundations) seeks to eliminate the need for such superscripts. New Foundations has a universal set, so it is a non-well founded set theory.That is to say, it is a logical theory that allows infinite descending chains of membership such as  xn  xn1 x3  x2  x1. It avoids Russell's paradox by only allowing stratifiable formulae in the axiom of comprehension. For instance x  y is a stratifiable formula, but x  x is not (for details of how this works see below). Definition 3.6.1.In New Foundations (NF) and related set theories, a formula  in the language of first-order logic with equality and membership is said to be stratified if and only if there is a function σ which sends each variable appearing in  [considered as an item of syntax] to a natural number (this works equally well if all integers are used) in such a way that any atomic formula x  y appearing in  satisfies σx  1  σy and any atomic formula x  y appearing in  satisfies σx  σy. Quinean set theory NF. Axioms and stratification are: The well-formed formulas of New Foundations (NF) are the same as the well-formed formulas of TST, but with the type annotations erased. The axioms of NF are: Extensionality: Two objects with the same elements are the same object; A comprehension schema: All instances of TST Comprehension but with type indices dropped (and without introducing new identifications between variables). By convention, NF's Comprehension schema is stated using the concept of stratified formula and making no direct reference to types.Comprehension then becomes. Stratified Axiom schema of comprehension: x  s exists for each stratified formula s. Even the indirect reference to types implicit in the notion of stratification can be eliminated. Theodore Hailperin showed in 1944 that Comprehension is equivalent to a finite conjunction of its instances,so that NF can be finitely axiomatized without any reference to the notion of type.Comprehension may seem to run afoul of problems similar to those in naive set theory, but this is not the case. For example, the existence of the impossible Russell class x  x  x is not an axiom of NF, because x  x cannot be stratified. 3.6.2.Set theory ZFC2 Hs, ZFCst and set theory ZFCNst with stratified axiom schema of replacement. The stratified axiom schema of replacement asserts that the image of a set under any function definable by stratified formula of the theory ZFCst will also fall inside a set. Stratified Axiom schema of replacement: Let sx, y, w1, w2, , wn be any stratified formula in the language of ZFCst whose free variables are among x, y, A, w1, w2, , wn, so that in particular B is not free in s. Then Aw1w2. . .wnxx  A  !ysx, y, w1, w2, , wn   Bxx  A  yy  B  sx, y, w1, w2, , wn, 3. 6. 2 i.e.,if the relation sx, y, . . .  represents a definable function f, A represents its domain, and fx is a set for every x  A, then the range of f is a subset of some set B. Stratified Axiom schema of separation: Let sx, w1, w2, , wn be any stratified formula in the language of ZFCst whose free variables are among x, A, w1, w2, , wn, so that in particular B is not free in s. Then w1w2. . .wnABxx  B  x  A  sx, w1, w2, , wn, 3. 6. 3 Remark 3.6.1. Notice that the stratified axiom schema of separation follows from the stratified axiom schema of replacement together with the axiom of empty set. Remark 3.6.2. Notice that the stratified axiom schema of replacement (separation) obviously violeted any contradictions (2.1.20),(2.2.18) and (2.3.18) mentioned above. The existence of the countable Russell sets 2 Hs,st and Nst impossible,because x  x cannot be stratified. Designation 3.6.1. Part II.Generalized Löbs Theorem. 1. 2.Generalized Löbs Theorem Remark 2.1.In this section we use second-order arithmetic Z2Hs with Henkin semantics. Notice that any standard model Mst Z2 Hs of second-order arithmetic Z2Hs consists of a set  of usual natural numbers (which forms the range of individual variables) together with a constant 0 (an element of ), a function S from  to , two binary operations  and * on , a binary relation  on , and a collection D  2 of subsets of , which is the range of the set variables. Omitting D produces a model of the first order Peano arithmetic. When D  2 is the full powerset of , the model Mst Z2 is called a full model. The use of full second-order semantics is equivalent to limiting the models of second-order arithmetic to the full models. In fact, the axioms of second-order arithmetic Z2 fss have only one full model. This follows from the fact that the axioms of Peano arithmetic with the second-order induction axiom have only one model under second-order semantics, see section 3. Let Th be some fixed, but unspecified, consistent formal theory. For later convenience, we assume that the encoding is done in some fixed formal second order theory S and that Th contains S.We assume throughout this paper that formal second order theory S has an -model MS .The sense in which S is contained in Th is better exemplified than explained: if S is a formal system of a second order arithmetic Z2Hs and Th is, say, ZFC2Hs, then Th contains S in the sense that there is a well-known embedding, or interpretation, of S in Th. Since encoding is to take place in MS , it will have to have a large supply of constants and closed terms to be used as codes. (e.g. in formal arithmetic, one has 0, 1, . . . .) S will also have certain function symbols to be described shortly.To each formula, , of the language of Th is assigned a closed term, c, called the code of . We note that if x is a formula with free variable x, then xc is a closed term encoding the formula x with x viewed as a syntactic object and not as a parameter. Corresponding to the logical connectives and quantifiers are the function symbols, neg, imp, etc., such that for all formulae , : S negc  c, S impc, c   c etc. Of particular importance is the substitution operator, represented by the function symbol sub, . For formulae x, terms t with codes tc : S subxc, tc  tc. 2. 1 It is well known [8] that one can also encode derivations and have a binary relation ProvThx, y (read "x proves y " or "x is a proof of y") such that for closed t1, t2 : S ProvTht1, t2 iff t1 is the code of a derivation in Th of the formula with code t2 . It follows that Th  iff S ProvTht, c 2. 2 for some closed term t.Thus one can define PrThy  xProvThx, y, 2. 3 and therefore one obtain a predicate asserting provability. Remark 2.2. (I)We note that it is not always the case that [8]: Th  iff S PrThc, 2. 4 unless S is fairly sound,e.g. this is a case when S and Th replaced by S  S  MTh and Th  Th  MTh correspondingly (see Designation 2.1 below). (II)Notice that it is always the case that: Th  iff S PrTh  c, 2. 5 i.e. that is the case when predicate PrThy, y  MTh : PrThy  xx  MThProvThx, y 2. 6 really asserts provability. It is well known [8] that the above encoding can be carried out in such a way that the following important conditions D1, D2 and D3 are meet for all sentences [8]: D1. Th  implies S PrThc, D2. S PrThc PrThPrThcc, D3. S PrThc  PrTh c PrThc. 2. 7 Conditions D1, D2 and D3 are called the Derivability Conditions. Remark 2.3.From (2.5)-(2.6) follows that D4. Th  iff S PrTh  c, D5. S PrTh  c  PrThPrTh  cc, D6. S PrTh  c  PrTh   c PrTh  c. 2. 8 Conditions D4, D5 and D6 are called the Strong Derivability Conditions. Definition 2.1. Let  be well formed formula (wff) of Th. Then wff  is called Th-sentence iff it has no free variables. Designation 2.1.(i) Assume that a theory Th has an -model MTh and  is a Th-sentence, then: MTh    M Th (we will write  instead MTh) is a Th-sentence  with all quantifiers relativized to -model MTh [11] and Th  Th MTh is a theory Th relativized to model MTh, i.e., any Th-sentence has the form  for some Th-sentence . (ii) Assume that a theory Th has a standard model MstTh and  is a Th-sentence, then: (iii) Assume that a theory Th has a non-standard model MNstTh and  is a Th-sentence, then: MNstTh    MNst Th (we will write Nst instead MNstTh ) is a Th-sentence with all quantifiers relativized to non-standard model MNstTh ,and ThNst  Th MNstTh is a theory Th relativized to model MNstTh , i.e., any ThNst-sentence has a form Nst for some Th-sentence . (iv) Assume that a theory Th has a model M  MTh and  is a Th-sentence, then: MTh is a Th-sentence with all quantifiers relativized to model MTh,and ThM is a theory Th relativized to model MTh, i.e. any ThM-sentence has a form M for some Th-sentence . Designation 2.2. (i) Assume that a theory Th with a lenguage  has an -model MTh and there exists Th-sentence S such that: (a) S expressible by lenguage  and (b) Sasserts that Th has a model MTh;we denote such Th-sentence S by ConTh; MTh. (ii) Assume that a theory Th with a lenguage  has a non-standard model MNstTh and there exists Th-sentence S such that: (a) S expressible by lenguage  and (b) S asserts that Th has a non-standard model MNstTh ;we denote such Th-sentence S by ConTh; MNstTh . (iii) Assume that a theory Th with a lenguage  has an model MTh and there exists Th-sentence S such that: (a) S expressible by lenguage  and (b) S asserts that Th has a model MTh;we denote such Th-sentence S by ConTh; MTh Remark 2. 4. We emphasize that: (i) it is well known that there exist a ZFC-sentence ConZFC; MZFC [10],[11],(ii) obviously there exists a ZFC2Hs-sentence Con ZFC2 Hs; MZFC2 Hs and there exists a Z2Hs-sentence Con Z2Hs; MZ2 Hs . Designation 2.3. Let ConTh be the formula: ConTh  t1t1  MTht1 t1  MTht2t2  MTht2 t2  MTh ProvTht1, c  ProvTht2, negc, t1   c, t2  neg c or ConTh  t1t1  MTht2t2  MThProvTht1,  c  ProvTht2, negc 2. 9 and where t1, t1 , t2, t2 is a closed term. Lemma 2.1. (I) Assume that: (i) ConTh; MTh, (ii) MTh  ConTh and (iii) Th PrThc,where  is a closed formula.Then Th  PrThc, (II) Assume that: (i) ConTh; MTh (ii) MTh  ConTh and (iii) Th PrTh  c, where  is a closed formula.Then Th  PrTh  c. Proof. (I) Let ConTh be the formula : ConTh  t1t1  MTht2t2  MThProvTht1,  c  ProvTht2, negc, t1t1  MTht2t2  MThProvTht1,  c  ProvTht2, negc 	t1t1  MTht2t2  MThProvTht1,  c  ProvTht2, negc . 2. 10 where t1, t2 is a closed term. From (i)-(ii) follows that theory Th ConTh is consistent. We note that Th ConTh ConTh for any closed . Suppose that Th PrThc, then (iii) gives Th PrThc  PrThc. 2. 11 From (2.3) and (2.11) we obtain t1t2ProvTht1, c  ProvTht2, negc. 2. 12 But the formula (2.10) contradicts the formula (2.12). Therefore Th  PrThc. (II) This case is trivial becourse formula PrTh  c by the Strong Derivability Condition D4,see formulae (2.8), really asserts provability of the Th-sentence .But this is a contradiction. Lemma 2.2. (I) Assume that: (i) ConTh; MTh, (ii) MTh  ConTh and (iii) Th PrThc,where  is a closed formula.Then Th  PrThc, (II) Assume that: (i) ConTh; MTh (ii) MTh  ConTh and (iii) Th PrTh  c, where  is a closed formula.Then Th  PrTh  c. Proof. Similarly as Lemma 2.1 above. Example 2.1. (i) Let Th  PA be Peano arithmetic and   0  1. Then obviously by Löbs theorem PA PrPA0  1, and therefore by Lemma 2.1 PA  PrPA0  1. (ii) Let PA PA  ConPA and   0  1. Then obviously by Löbs theorem PA PrPA0  1, and therefore PA  PrPA0  1. However obviously PA PrPA0  1  PrPA0  1. Remark 2.5.Notice that there is no standard model of PA. Assumption 2.1. Let Th be a first order a second order theory with the Henkin semantics. We assume now that: (i) the language of Th consists of: numerals 0,1,... countable set of the numerical variables: v0, v1, . . .

countable set  of the set variables:   x, y, z, X, Y, Z, ,, . . .

countable set of the n-ary function symbols: f0n, f1n, . . . countable set of the n-ary relation symbols: R0n, R1n, . . . connectives: ,

quantifier:. (ii) Th contains ZFC2Hs or ZFC (iii) Th has an -model MTh or (iv) Th has a nonstandard model MNstTh PA. Definition 2.1. A Th-wff  (well-formed formula ) is closed i.e.  is a sentence if it has no free variables; a wff is open if it has free variables.We'll use the slang 'k-place open wff' to mean a wff with k distinct free variables. Definition 2.2.We will say that,Th# is a nice theory or a nice extension of the Th iff the following (i) Th # contains Th; (ii) Let  be any closed formula of Th, then Th PrThc implies Th # ; (iii) Let  be any closed formula of Th # , then MTh   implies Th # , i.e. ConTh  ; MTh implies Th # . Remark 2.6.Notice that formulae ConTh  ; MTh and ConTh#  ; MTh are expressible in Th # . Definition 2.3.Let L be a classical propositional logic L. Recall that a set Δ of L-wff's is said to be L-consistent, or consistent for short, if    and there are other equivalent formulations of consistency:(1) Δ is consistent, (2) DedΔ : A  Δ A is not the set of all wff's,(3) there is a formula such that Δ  A. (4) there are no formula A such that Δ A and Δ A. We will say that,Th # is a maximally nice theory or a maximally nice extension of the Th iff Th # is consistent and for any consistent nice extension Th # of the Th : DedTh #  DedTh # implies DedTh #   DedTh #. Remark 2.7. We note that a theory Th# depend on model MTh or MNstTh , i.e. Th #  Th # MTh  or Th #  Th # MNstTh  correspondingly. We will consider now the case Th #  Th # MTh  without loss of generality. Remark 2.8.a. Notice that in order to prove the statements: (i) ConZFC2Hs; MTh, (ii) ConZFC; MTh the following Proposition 2.1 is not necessary, see Proposition 2.18. Proposition 2.1.(Generalized Löbs Theorem). (I) Assume that: (i) ConTh,where predicate ConTh defined by formula 2.9 (ii) Th has an -model MTh, and (iii) the statement MTh is expressible by lenguage of Th as a single sentence of Th. Then theory Th can be extended to a maximally consistent nice theory Th,st #  Th,st # MTh .Below we write for short Th,st #  Th #  Th # MTh . Remark 2.8.b. We emphasize that (iii) valid for ZFC despite the fact that the axioms of ZFC are infinite, see [10] Chapter II,section 7,p.78. (II) Assume that: (i) ConTh ,where predicate ConTh defined by formula 2.9, (ii ) Th has an -model MTh and (iii) the statement MTh is expressible by lenguage of Th as a single sentence of Th. Then theory Th  Th MTh can be extended to a maximally consistent nice theory Th # . (III) Assume that: (i) ConTh ,where predicate ConTh defined by formula 2.9, (ii) Th has a nonstandard model MNstTh PA and (iii) the statement MNstTh PA is expressible by lenguage of Th as a single sentence of Th. Then theory Th can be extended to a maximally consistent nice theory Th,Nst #  Th,Nst # MNstTh . Remark 2.8.c. We emphasize that (iii) valid for ZFC despite the fact that the axioms of ZFC are infinite, see [10] Ch.II,section 7,p.78. Proof.(I) Let 1. . . i. . . be an enumeration of all closed wff's of the theory Th (this can be achieved if the set of propositional variables can be enumerated). Define a chain  Thi,st # |i   , Th1,st #  Th of consistent theories inductively as follows: assume that theory Thi,st # is defined. Notice that below we write for short Thi,st #  Thi #. (i) Suppose that the statement (2.13) is satisfied Thi #  PrTh i#i  c  Thi #  i  and MTh  i. 2. 13 Then we define a theory Thi1 # as follows Thi1 #  Thi #  i .We will rewrite the condition (2.13) using predicate PrTh i1# #  symbolically as follows: Thi1 # PrTh i1# # i c, PrTh i1# # i c  PrTh i#i  c  MTh  i , MTh  i  ConThi# i; MTh, i.e. PrTh i1# # i c  PrTh i#i  c  ConThii; MTh, PrTh i1# # i c  PrTh i1# i  c, PrTh i1# i  c  i, PrTh i1# # i c  i. 2. 14 (ii) Suppose that the statement (2.15) is satisfied Thi #  PrTh i#i  c  Thi #  i  and MTh  i. 2. 15 Then we define a theory Thi1 # as follows Thi1 #  Thi #  i .We will rewrite the condition (2.15) using predicate PrTh i1# # , symbolically as follows: Thi1 # PrTh i1# # i c, PrTh i1# # i c  PrTh i#i  c  MTh  i , MTh  i  ConThi #i; MTh, i.e. PrTh i1# # i c  PrTh i#i  c  ConThii; MTh, PrTh i1# # i c  PrTh i1# i  c, PrTh i1# i  c  i, PrTh i1# # i c  i. 2. 16 (iii) Suppose that the statement (2.17) is satisfied Thi # PrTh i#i  c and Thi #  i   MTh  i . 2. 17 Then we define a theory Thi1 # as follows Thi1 #  Thi #  i .Using Lemma 2.1and predicate PrTh i1# # ,we will rewrite the condition (2.17) symbolically as follows: Thi1 # PrTh i1# # i c, PrTh i1# # i c  PrTh i#i  c  MTh  i , MTh  i  ConThi #i; MTh, i.e. PrTh i1# # i c  PrTh i#i  c  ConThii; MTh, PrTh i1# # i c  PrTh i1# i  c, PrTh i1# i  c  i, PrTh i1# # i c  i. 2. 18 Remark 2.9.Notice that predicate PrTh i1# # i c is expressible in Thi # because Thi # is a finite extension of the recursive theory Th and ConThi #i; MTh  Thi #. (iv) Suppose that a statement (2.19) is satisfied Thi # PrTh i#i  c and Thi #  i   MTh  i . 2. 19 Then we define theory Thi1 # as follows: Thi1 #  Thi #  i . Using Lemma 2.2 and predicate PrTh i1# # ,we will rewrite the condition (2.15) symbolically as follows Thi # PrTh i# # i c , PrTh i# # i c  PrTh i#i  c  MTh  i , MTh  i  ConThi #i; MTh, i.e. PrTh i# # i c  PrTh i#i  c  ConThi #i; MTh, PrTh i1# # i c  PrTh i1# i  c, PrTh i1# i  c  i, PrTh i1# # i c  i. 2. 20 Remark 2.10. Notice that predicate PrTh i# # i c is expressible in Thi # because Thi # is a finite extension of the recursive theory Th and ConThi# i; MTh  Thi #. (v) Suppose that the statement (2.21) is satisfied Thi # PrTh i#i  c and Thi # PrTh i#i  c  i. 2. 21 We will rewrite now the conditions (2.21) symbolically as follows Thi # PrTh i#

i c PrTh i#

i c  PrTh i#i  c  PrTh i#i  c  i 2. 22 Then we define a theory Thi1 # as follows: Thi1 #  Thi #. (iv) Suppose that the statement (2.23) is satisfied Thi # PrTh i#i  c and Thi # PrTh i#i  c  i. 2. 23 We will rewrite now the condition (2.23) symbolically as follows Thi # PrTh i#

i c  PrTh i#i  c  PrTh i#i  c  i 2. 24 Then we define a theory Thi1 # as follows: Thi1 #  Thi #.We define now a theory Th # as follows: Th #   i Thi #. 2. 25 First, notice that each Thi # is consistent. This is done by induction on i and by Lemmas 2.1-2.2. By assumption, the case is true when i  1.Now, suppose Thi # is consistent. Then its deductive closure DedThi # is also consistent. If the statement (2.14) is satisfied,i.e. Thi1 # PrTh i1# # i c and Thi1 # i, then clearly Thi1 #  Thi #  i is consistent since it is a subset of closure DedThi1 # . If a statement (2.16) is satisfied,i.e. Thi1 # PrTh i1# # i c and Thi1 # i, then clearly Thi1 #  Thi #  i is consistent since it is a subset of closure DedThi1 # . If the statement (2.18) is satisfied,i.e. Thi # PrTh i#i  c and Thi #  i   MTh  i  then clearly Thi1 #  Thi #  i is consistent by Lemma 2.1 and by one of the standard properties of consistency:   A

is consistent iff   A. If the statement (2.20) is satisfied,i.e. Thi # PrTh i#i  c and Thi #  i   MTh  i  then clearly Thi1 #  Thi #  i is consistent by Lemma 2.2 and by one of the standard properties of consistency:   A is consistent iff   A.Next, notice DedTh #  is maximally consistent nice extension of the DedTh. DedTh #  is consistent because, by the standard Lemma 2.3 below, it is the union of a chain of consistent sets. To see that DedTh #  is maximal, pick any wff . Then  is some i in the enumerated list of all wff's. Therefore for any  such that Thi PrTh i c or Thi # PrTh i# c, either   Th # or   Th # .Since DedThi1 #  DedTh # , we have   DedTh #  or   DedTh # ,which implies that DedTh #  is maximally consistent nice extension of the DedTh. Proof.(II) Let ,1. . . ,i. . . be an enumeration of all closed wff's of the theory Th (this can be achieved if the set of propositional variables can be enumerated). Define a chain  Th,i # |i   , Th,1 #  Th of consistent theories inductively as follows: assume that theory Th,i # is defined. (i) Suppose that a statement (2.26) is satisfied Th,i #  PrTh,i# ,i  c and MTh  i. 2. 26 Then we define a theory Th,i1 # as follows Th,i1 #  Th,i #  ,i . 2. 27 We will rewrite now the conditions (2.26) and (2.27) symbolically as follows Th,i1 # PrTh,i1# ,i  c  Th,i1 # ,i, PrTh,i1# # i c  PrTh,i1# i  c  ,i. 2. 28 (ii) Suppose that a statement (2.29) is satisfied Th,i #  PrTh,i# ,i  c and MTh  i. 2. 29 Then we define theory Th,i1 # as follows: Th,i1 #  Th,i #  ,i . 2. 30 We will rewrite the conditions (2.25) and (2.26) symbolically as follows Th,i1 PrTh,i1,i  c  Th,i1 ,i, PrTh,i1 # i c  PrTh,i1i  c. 2. 31 (iii) Suppose that the following statement (2.32) is satisfied Th,i PrTh,i,i  c, 2. 32 and therefore by Derivability Conditions (2.8) Th,i ,i. 2. 33 We will rewrite now the conditions (2.28) and (2.29) symbolically as follows PrTh,i

,i c  Th,i PrTh,i,i  c 2. 34 Then we define a theory Th,i1 as follows: Th,i1  Th,i. (iv) Suppose that the following statement (2.35) is satisfied Th,i PrTh,i,i  c, 2. 35 and therefore by Derivability Conditions (2.8) Th,i ,i. 2. 36 We will rewrite now the conditions (2.35) and (2.36) symbolically as follows PrTh,i

,i c  Th,i PrTh,i,i  c 2. 37 Then we define a theory Th,i1 as follows: Th,i1  Th,i.We define now a theory Th; # as follows: Th; #   i Th,i. 2. 38 First, notice that each Th,i is consistent. This is done by induction on i.Now, suppose Th,i is consistent. Then its deductive closure DedTh,i is also consistent. If statement (2.22) is satisfied,i.e. Th,i  PrTh,i,i  c and MTh  i then clearly Th,i1  Th,i  ,i is consistent.If statement (2.25) is satisfied,i.e. Th,i PrTh,i,i  c and MTh  i, then clearly Th,i1  Th,i  ,i is consistent. If the statement (2.28) is satisfied,i.e. Th,i PrTh,i,i  c, then clearly Th,i1  Th,i is also consistent. If the statement (2.35) is satisfied,i.e. Th,i PrTh,i,i  c, then clearly Th,i1  Th,i is also consistent.Next, notice DedTh; #  is a maximally consistent nice extension of the DedTh;.The set DedTh; #  is consistent because, by the standard Lemma 2.3 belov, it is the union of a chain of consistent sets. Lemma 2.3. The union of a chain  i|i   of consistent sets i, ordered by , is consistent. Definition 2.4. (I) We define now predicate PrTh#  c and predicate PrTh#  c asserting provability in Th # by the following formulae PrTh#  c  i  Thi # PrTh i# # c  PrTh i#

c    Th #   ConTh# ; MTh, ConTh# ; MTh  2. 39 (II) We define now predicate PrTh;#   c and predicate PrTh;#   c asserting provability in Th; # by the following formulae PrTh;#   c  i  Th,i #  PrTh,i# #  c  PrTh,i#

 c    Th; #   ConTh;#  ; MTh, ConTh;#  ; MTh  2. 40 Remark 2.11.(I) Notice that both predicate PrTh#  c and predicate PrTh#  c are expressible in Th # because for any i  , Thi # is an finite extension of the recursive theory Th and ConThi #; MTh  Thi, ConThi #; MTh  Thi. (II) Notice that both predicate PrTh;#   c and predicate PrTh;#   c are expressible in Th; # because for any i  , Th,i # is an finite extension of the recursive theory Th and ConTh,i # ; MTh  Th,i # , ConTh,i # ; MTh  Th,i # . Definition 2.5.Let   x be one-place open Th-wff such that the following condition: Th  Th1 # !xx 2. 41 is satisfied. Remark 2.12.We rewrite now the condition (2.41) using only the language of the theory Th1 # : Th1 # !xx  PrTh1#!xx c   PrTh1#!xx c  !xx . 2. 42 Definition 2.6. We will say that, a set y is a Th1#-set if there exist one-place open wff x such that y  x. We write yTh1 #  iff y is a Th1 #-set. Remark 2.13. Note that yTh1 #    y  x  PrTh1#!xx c PrTh1#!xx c  !xx . 2. 43 Definition 2.7.Let 1 be a collection such that : x x  1  x is a Th1 #-set . 2. 44 Proposition 2.2. Collection 1 is a Th1#-set. Proof. Let us consider an one-place open wff x such that conditions (2.41) are satisfied, i.e. Th1 # !xx.We note that there exists countable collection  of the one-place open wff's   nx n such that: (i) x   and (ii) Th  Th1 # !xx  nn  x  nx  or in the equivalent form Th  Th1 # PrTh1#!xx c  PrTh1#!xx c  !xx  PrTh1#nn  x  nx c  PrTh1#nn  x  nx c  nn  x  nx 2. 45 or in the following equivalent form Th1 # !x11x1  nn  1x1  n,1x1  or Th1 # PrTh1#!x1x1 c  PrTh1#!x1x1 c  !x1x1  PrTh1#nn  x1  nx1 c  PrTh1#nn  x1  nx1 c  nn  x1  nx1, 2. 46 where we have set x  1x1,nx1  n,1x1 and x  x1. We note that any collection k  n,kx n, k  1, 2, . . . such as mentioned above, defines an unique set xk , i.e. k1 k2  iff xk1  xk2 .We note that collections k , k  1, 2, . . are not a part of the ZFC2Hs or ZFC,i.e. collection k is not a set in sense of ZFC2Hs or ZFC. However this is no problem, because by using Gödel numbering one can to replace any collection k , k  1, 2, . . by collection k  gk  of the corresponding Gödel numbers such that k  gk   gn,kxk n, k  1, 2, . . . . 2. 47 It is easy to prove that any collection k  gk , k  1, 2, . . is a Th1 #-set.This is done by Gödel encoding [7],[10] (2.47), by the statament (2.45) and by axiom schemata of separation [10]. Let gn,k  gn,kxk, k  1, 2, . . be a Gödel number of the wff n,kxk. Therefore gk  gn,k n, where we have set k  k , k  1, 2, . . and k1k2	gn,k1 n  gn,k2 n   xk1  xk2 . 2. 48 Let gn,k n k be a family of the sets gn,k n, k  1, 2, . . . .By the axiom of choice [10] one obtains unique set 1  gk k such that kgk  gn,k n .Finally one obtains a set 1 from the set 1 by the axiom schema of replacement [10]. Proposition 2.3. Any collection k  gk , k  1, 2, . . is a Th1 #-set. Proof. We define gn,k  gn,kxk  n,kxkc, vk  xk c. Therefore gn,k  gn,kxk  Frgn,k, vk (see [7]). Let us define now predicate gn,k, vk gn,k, vk  PrTh1#!xk1,kx1 c  !xkvk  xk c nn   PrTh1#1,kxk c  PrTh1#Frgn,k, vk . 2. 49 We define now a set k such that k  k   gk , nn  gn,k  k   gn,k, vk 2. 50 Obviously definitions (2.45) and (2.50) are equivalent. Definition 2.7.We define now the following Th1#-set 1  1 : x x  1  x  1  PrTh1#x  x c  PrTh1#x  x c  x  x . 2. 51 Proposition 2.4. (i) Th1# 1, (ii) 1 is a countable Th1#-set. Proof.(i) Statement Th1# 1 follows immediately from the statement  1 and the axiom schema of separation [4], (ii) follows immediately from countability of a set

.Notice that 1 is nonempty countable set such that   1, because for any n   : Th1 # n  n. Proposition 2.5. A set 1 is inconsistent. Proof.From formula (2.51) we obtain Th1 # 1  1  PrTh1#1  1  c  PrTh1#1  1  c  1  1 . 2. 52 From (2.52) we obtain Th1 # 1  1  1  1 2. 53 and therefore Th1 # 1  1  1  1. 2. 54 But this is a contradiction. Definition 2.8. Let   x be one-place open Th-wff such that the following condition: Thi # !xx 2. 55 is satisfied. Remark 2.14.We rewrite now the condition (2.55) using only the lenguage of the theory Thi # : Thi # !xx  PrTh i#!xx c   PrTh i#!xx c  !xx . 2. 56 Definition 2.9. We will say that, a set y is a Thi#-set if there exist one-place open wff x such that y  x. We write yThi #  iff y is a Thi #-set. Remark 2.15. Note that yThi #    y  x  PrTh i#!xx c PrTh i#!xx c  !xx . 2. 57 Definition 2.10.Let i be a collection such that : x x  i  x is a Thi #-set . 2. 58 Proposition 2.6. Collection i is a Thi#-set. Proof. Let us consider an one-place open wff x such that conditions (2.51) are satisfied, i.e. Thi # !xx.We note that there exists countable collection  of the one-place open wff's   nx n such that: (i) x   and (ii) Thi # !xx  nn  x  nx  or in the equivalent form Thi # PrTh i#!xx c  PrTh i#!xx c  !xx  PrTh i#nn  x  nx c  PrTh i#nn  x  nx c  nn  x  nx 2. 59 or in the following equivalent form Thi # !x11x1  nn  1x1  n,1x1  or Thi # PrTh i#!x1x1 c  PrTh i#!x1x1 c  !x1x1  PrTh i#nn  x1  nx1 c  PrTh i#nn  x1  nx1 c  nn  x1  nx1. 2. 60 where we have set x  1x1,nx1  n,1x1 and x  x1. We note that any collection k  n,kx n, k  1, 2, . . . such as mentioned above, defines an unique set xk , i.e. k1 k2  iff xk1  xk2 .We note that collections k , k  1, 2, . . are not a part of the ZFC2Hs, i.e. collection k there is no set in the sense of ZFC2Hs. However that is no problem, because by using Gödel numbering one can to replace any collection k , k  1, 2, . . by collection k  gk  of the corresponding Gödel numbers such that k  gk   gn,kxk n, k  1, 2, . . . . 2. 61 It is easy to prove that any collection k  gk , k  1, 2, . . is a Thi #-set.This is done by Gödel encoding [7],[10] (2.61), by the statament (2.55) and by the axiom schema of separation [10]. Let gn,k  gn,kxk, k  1, 2, . . be a Gödel number of the wff n,kxk. Therefore gk  gn,k n, where we have set k  k , k  1, 2, . . and k1k2	gn,k1 n  gn,k2 n   xk1  xk2 . 2. 62 Let gn,k n k be a family of the all sets gn,k n. By axiom of choice [10] one obtains a unique set i  gk k such that kgk  gn,k n .Finally for any i   one obtains a set i from the set i by the axiom schema of replacement [10]. Proposition 2.8. Any collection k  gk , k  1, 2, . . is a Thi #-set. Proof. We define gn,k  gn,kxk  n,kxkc, vk  xk c. Therefore gn,k  gn,kxk  Frgn,k, vk (see [7]). Let us define now predicate ign,k, vk ign,k, vk  PrTh i#!xk1,kx1 c  !xkvk  xk c nn   PrTh i#1,kxk c  PrTh i#Frgn,k, vk . 2. 63 We define now a set k such that k  k   gk , nn  gn,k  k   ign,k, vk. 2. 64 Obviously definitions (2.59) and (2.64) are equivalent. Definition 2.11.We define now the following Thi#-set i  i : x x  i  x  i  PrTh i#x  x c  PrTh i#x  x c  x  x . 2. 65 Proposition 2.9. (i) Thi# i, (ii) i is a countable Thi#-set,i  . Proof.(i) Statement Thi# i follows immediately by using statement  i and axiom schema of separation [4]. (ii) follows immediately from countability of a set i. Proposition 2.10. Any set i, i   is inconsistent. Proof.From the formula (2.65) we obtain Thi # i  i  PrTh i#i  i  c  PrTh i#i  i  c  i  i . 2. 66 From the formla (2.66) we obtain Thi # i  i  i  i 2. 67 and therefore Thi # i  i  i  i. 2. 68 But this is a contradiction. Definition 2.12. A Th# -wff  that is: (i) Th-wff  or (ii) well-formed formula  which contains predicate PrTh#  c given by formula (2.39).An Th # -wff  (well-formed formula ) is closed i.e.  is a sentence if it has no free variables; a wff is open if it has free variables. Definition 2.13.Let   x be one-place open Th# -wff such that the following condition: Th # !xx 2. 69 is satisfied. Remark 2.16.We rewrite now the condition (2.69) using only the lenguage of the theory Th # : Th # !xx  PrTh# !xx c  	PrTh# !xx c  !xx . 2. 70 Definition 2.14.We will say that, a set y is a Th# -set if there exists one-place open wff x such that y  x. We write yTh #  iff y is a Th # -set. Definition 2.15. Let  be a collection such that : x x    x is a Th# -set . Proposition 2.11. Collection  is a Th# -set. Proof. Let us consider an one-place open wff x such that condition (2.69) is satisfied, i.e. Th # !xx.We note that there exists countable collection  of the one-place open wff's   nx n such that: (i) x   and (ii) Th # !xx  nn  x  nx  or in the equivalent form Th # PrTh# !xx c  PrTh# !xx c  !xx  PrTh# nn  x  nx c  PrTh# nn  x  nx c  nn  x  nx 2. 71 or in the following equivalent form Th # !x11x1  nn  1x1  n,1x1  or Th # PrTh i#!x1x1 c  PrTh# !x1x1 c  !x1x1  PrTh i#nn  x1  nx1 c  PrTh i#nn  x1  nx1 c  nn  x1  nx1. 2. 72 where we set x  1x1,nx1  n,1x1 and x  x1. We note that any collection k  n,kx n, k  1, 2, . . . such as mentioned above defines a unique set xk , i.e. k1 k2  iff xk1  xk2 .We note that collections k , k  1, 2, . . are not a part of the ZFC2Hs, i.e. collection k there is no set in sense of ZFC2Hs. However that is not a problem, because by using Gödel numbering one can to replace any collection k , k  1, 2, . . by collection k  gk  of the corresponding Gödel numbers such that k  gk   gn,kxk n, k  1, 2, . . . . 2. 73 It is easy to prove that any collection k  gk , k  1, 2, . . is a Th #-set.This is done by Gödel encoding [8],[10] by the statament (2.66) and by axiom schema of separation [9]. Let gn,k  gn,kxk, k  1, 2, . . be a Gödel number of the wff n,kxk. Therefore gk  gn,k n, where we have set k  k , k  1, 2, . . and k1k2	gn,k1 n	gn,k2 n   xk1  xk2 . 2. 74 Let gn,k n k be a family of the sets gn,k n, k  1, 2, . . . . By axiom of choice [9] one obtains an unique set   gk k such that kgk  gn,k n .Finally one obtains a set

 from the set  by axiom schema of replacement [9].Thus one can define Th # -set    : xx    x    PrTh# x  x c  PrTh# x  x c  x  x . 2. 75 Proposition 2.12. Any collection k  gk , k  1, 2, . . is a Th # -set. Proof. We define gn,k  gn,kxk  n,kxkc, vk  xk c. Therefore gn,k  gn,kxk  Frgn,k, vk (see [10]). Let us define now predicate gn,k, vk gn,k, vk  PrTh# !xk1,kx1 c  PrTh# !xk1,kx1 c  !x1x1 !xkvk  xk cnn  PrTh# 1,kxk c  PrTh# Frgn,k, vk. 2. 76 We define now a set k such that k  k   gk , nn  gn,k  k   gn,k, vk 2. 77 Obviously definitions (2.70) and (2.77) are equivalent by Proposition 2.1. Proposition 2.13. (i) Th# , (ii)  is a countable Th# -set. Proof.(i) Statement Th#  follows immediately from the statement   and axiom schema of separation [9], (ii) follows immediately from countability of the set . Proposition 2.14. Set  is inconsistent. Proof.From the formula (2.75) we obtain Th #     PrTh#     c  PrTh#     c     . 2. 78 From (2.74) one obtains Th #        2. 79 and therefore Th #       . 2. 80 But this is a contradiction. Definition 2.16.An Th;# -wff ; that is: (i) Th-wff  or (ii) well-formed formula ; which contains predicate PrTh;#  c given by formula (2.36).An Th; # -wff ; (well-formed formula ;) is closed i.e. ; is a sentence if it has no free variables; a wff is open if it has free variables. Definition 2.17.Let   x be one-place open Th-wff such that the following condition: Th  Th,1 # !xx 2. 81 is satisfied. Remark 2.17.We rewrite now the condition (2.81) using only the lenguage of the theory Th,1 # : Th,1 # !xx  PrTh,1# !xx c. 2. 82 Definition 2.18. We will say that, a set y is a Th,1# -set if there exist one-place open wff x such that y  x. We write yTh,1 #  iff y is a Th,1 # -set. Remark 2.18. Note that yTh,1 #    y  x  PrTh,1# !xx c PrTh,1# !xx c  !xx . 2. 83 Definition 2.19.Let ,1 be a collection such that : x x  ,1  x is a Th,1 # -set . 2. 84 Proposition 2.15. Collection ,1 is a Th,1# -set. Proof. Let us consider an one-place open wff x such that conditions (2.37) are satisfied, i.e. Th,1 # !xx.We note that there exists countable collection  of the one-place open wff's   nx n such that: (i) x   and (ii) Th Th,1 # !xx  nn  x  nx  or in the equivalent form Th Th,1 # PrTh,1# !xx c  PrTh,1# nn  x  nx c , 2. 85 or in the following equivalent form Th,1 # !x11x1  nn  1x1  n,1x1  or Th,1 # PrTh,1# !x1x1 c  PrTh,1# nn  x1  nx1 c , 2. 86 where we have set x  1x1,nx1  n,1x1 and x  x1. We note that any collection k  n,kx n, k  1, 2, . . . such as mentioned above, defines an unique set xk , i.e. k1 k2  iff xk1  xk2 .We note that collections k , k  1, 2, . . are not a part of the ZFC2Hs, i.e. collection k is not a set in the sense of ZFC2. However that is not a problem, because by using Gödel numbering one can to replace any collection k , k  1, 2, . . by collection k  gk  of the corresponding Gödel numbers such that k  gk   gn,kxk n, k  1, 2, . . . . 2. 87 It is easy to prove that any collection k  gk , k  1, 2, . . is a Th,1 # -set.This is done by Gödel encoding [7],[10] (2.87), by the statament (2.85) and by the axiom schema of separation [7]. Let gn,k  gn,kxk, k  1, 2, . . be a Gödel number of the wff n,kxk. Therefore gk  gn,k n, where we have set k  k , k  1, 2, . . and k1k2	gn,k1 n  gn,k2 n   xk1  xk2 . 2. 88 Let gn,k n k be a family of the sets gn,k n, k  1, 2, . . . . By the axiom of choice [7] one obtains an unique set 1  gk k such that kgk  gn,k n .Finally one obtains a set ,1 from the set ,1 by the axiom schema of replacement [7]. Proposition 2.16. Any collection k  gk , k  1, 2, . . is a Th,1 # -set. Proof. We define gn,k  gn,kxk  n,kxkc, vk  xk c. Therefore gn,k  gn,kxk  Frgn,k, vk (see [10]). Let us define now predicate gn,k, vk gn,k, vk  PrTh,1# !xk1,kx1 c  !xkvk  xk c nn   PrTh,1# 1,kxk c  PrTh,1# Frgn,k, vk . 2. 89 We define now a set k such that k  k   gk , nn  gn,k  k   gn,k, vk 2. 90 Obviously definitions (2.85) and (2.90) are equivalent. Definition 2.20.We define now the following Th,1# -set ,1  ,1 : x x  ,1  x  ,1  PrTh,1# x  x c . 2. 91 Proposition 2.17. (i) Th,1# ,1, (ii) ,1 is a countable Th,1# -set. Proof.(i) Statement Th,1# ,1 follows immediately from the statement  ,1 and axiom schema of separation [7], (ii) follows immediately from countability of the set

,1. Proposition 2.18. A set ,1 is inconsistent. Proof.From formla (2.87) we obtain Th,1 # ,1  ,1  PrTh,1# ,1  ,1  c. 2. 92 From (2.92) we obtain Th,1 # ,1  ,1  ,1  ,1 2. 93 and therefore Th,1 # ,1  ,1  ,1  ,1. 2. 94 But this is a contradiction. Definition 2.21. Let   x be one-place open Th-wff such that the following condition: Th,i # !xx 2. 95 is satisfied. Remark 2.19.We rewrite now the condition (2.95) using only the lenguage of the theory Th,i # : Th,i # !xx  PrTh,i# !xx c. 2. 96 Definition 2.22. We will say that, a set y is a Th,i# -set if there exist one-place open wff x such that y  x. We write yTh,i #  iff y is a Th,i # -set. Remark 2.20. Note that yTh,i #    y  x  PrTh,i# !xx c . 2. 97 Definition 2.23.Let ,i be a collection such that : x x  ,i  x is a Th,i # -set . 2. 98 Proposition 2.19. Collection ,i is a Th,i# -set. Proof. Let us consider an one-place open wff x such that conditions (2.95) is satisfied, i.e. Th,i # !xx.We note that there exists countable collection  of the one-place open wff's   nx n such that: (i) x   and (ii) Th,i # !xx  nn  x  nx  or in the equivalent form Th,i # PrTh,i# !xx c  PrTh,i# nn  x  nx c , 2. 99 or in the following equivalent form Th,i # !x11x1  nn  1x1  n,1x1  or Th,i # PrTh,i# !x1x1 c  PrTh,i# nn  x1  nx1 c . 2. 100 where we have set x  1x1,nx1  n,1x1 and x  x1. We note that any collection k  n,kx n, k  1, 2, . . . such as mentioned above, defines an unique set xk , i.e. k1 k2  iff xk1  xk2 .We note that collections k , k  1, 2, . . is not a part of the ZFCst, i.e. collection k is not a set in the sense of ZFCst. However that is not a problem, because by using Gödel numbering one can to replace any collection k , k  1, 2, . . by collection k  gk  of the corresponding Gödel numbers such that k  gk   gn,kxk n, k  1, 2, . . . . 2. 101 It is easy to prove that any collection k  gk , k  1, 2, . . is a Th,i # -set.This is done by Gödel encoding [8],[10] (2.101), by the statament (2.95) and by axiom schema of separation [9]. Let gn,k  gn,kxk, k  1, 2, . . be a Gödel number of the wff n,kxk. Therefore gk  gn,k n, where we have set k  k , k  1, 2, . . and k1k2	gn,k1 n  gn,k2 n   xk1  xk2 . 2. 102 Let gn,k n k be the family of the sets gn,k n. By axiom of choice [9] one obtains an unique set i  gk k such that kgk  gn,k n .Finally one obtains a set ,i from the set i by axiom schema of replacement [9]. Proposition 2.20. Any collection k  gk , k  1, 2, . . is a Th,i # -set. Proof. We define gn,k  gn,kxk  n,kxkc, vk  xk c. Therefore gn,k  gn,kxk  Frgn,k, vk (see [10]). Let us define now predicate ,ign,k, vk ,ign,k, vk  PrTh,i# !xk1,kx1 c  !xkvk  xk c nn   PrTh,i# 1,kxk c  PrTh,i# Frgn,k, vk . 2. 103 We define now a set k such that k  k   gk , nn  gn,k  k   ,ign,k, vk. 2. 104 Obviously definitions (2.95) and (2.104) are equivalent. Definition 2.24.We define now the following Th,i# -set ,i  ,i : x x  ,i  x  ,i  PrTh,i# x  x c . 2. 105 Proposition 2.21. (i) Th,i# ,i, (ii) ,i is a countable Th,i# -set,i  . Proof.(i) Statement Th,i# ,i follows immediately by using statement  ,i and axiom schema of separation [9]. (ii) follows immediately from countability of a set ,i. Proposition 2.22. Any set ,i, i   is inconsistent. Proof.From formla (2.105) we obtain Th,i # ,i  ,i  PrTh,i# ,i  ,i  c. 2. 106 From (2.106) we obtain Th,i # ,i  ,i  ,i  ,i 2. 107 and therefore Th,i # ,i  ,  ,i  ,i. 2. 108 But this is a contradiction. Definition 2.25.Let   x be one-place open Th;# -wff such that the following condition: Th; # !xx 2. 109 is satisfied. Remark 2.20.We rewrite now the condition (2.109) using only the lenguage of the theory Th # in the following equivalent form 1. Th; # !xx  Th; # PrTh;# !xx c or 2. Th; # !xx  Th; # PrTh;# !xx c   PrTh;# !xx c  !xx 2. 110 Definition 2.26.We will say that: (i) a set y is a Th;# -set if there exist one-place open wff x such that y  x, i.e. Th; # PrTh;# !xx c  y  x; (ii) a set y is a Th; # -set if there exist one-place open wff We write yTh; #  iff y is a Th; # -set. Definition 2.27. Let ; be a collection such that : x x  ;  x Th;#  . Proposition 2.23. Collection ; is a Th;# -set. Proof. Let us consider an one-place open wff x such that condition (2.109) is satisfied, i.e. Th; # !xx.We note that there exists countable collection  of the one-place open wff's   nx n such that: (i) x   and (ii) Th; # !xx  nn  x  nx  or in the equivalent form Th; # PrTh;# !xx c  PrTh;# nn  x  nx c , 2. 111 or in the following equivalent form Th; # !x11x1  nn  1x1  n,1x1  or Th; # PrTh;# !x1x1 c  PrTh;# nn  x1  nx1 c , 2. 112 where we set x  1x1,nx1  n,1x1 and x  x1. We note that any collection k  n,kx n, k  1, 2, . . . such as mentioned above defines unique set xk , i.e. k1 k2  iff xk1  xk2 .We note that the collections k , k  1, 2, . . is not a part of the ZFC, i.e. collection k is not a set in the sense of ZFC. However that is not a problem, because by using Gödel numbering one can to replace any collection k , k  1, 2, . . by collection k  gk  of the corresponding Gödel numbers such that k  gk   gn,kxk n, k  1, 2, . . . . 2. 113 It is easy to prove that any collection k  gk , k  1, 2, . . is a Th; # -set.This is done by Gödel encoding [8],[10] by the statament (2.109) and by axiom schema of separation [9]. Let gn,k  gn,kxk, k  1, 2, . . be a Gödel number of the wff n,kxk. Therefore gk  gn,k n, where we have set k  k , k  1, 2, . . and k1k2	gn,k1 n	gn,k2 n   xk1  xk2 . 2. 114 Let gn,k n k be the family of thesets gn,k n. By axiom of choice [9] one obtains unique set   gk k such that kgk  gn,k n .Finally one obtains a set ; from the set ; by axiom schema of replacement [9].Thus one can define Th; # -set ;  ; : x x  ;  x  ;  PrTh;# x  x c . 2. 115 Proposition 2.24. Any collection k  gk , k  1, 2, . . is a Th; # -set. Proof. We define gn,k  gn,kxk  n,kxkc, vk  xk c. Therefore gn,k  gn,kxk  Frgn,k, vk (see [7]). Let us define now predicate ;gn,k, vk ;gn,k, vk  PrTh;# !xk1,kx1 c  !xkvk  xk c nn   PrTh;# 1,kxk c  PrTh;# Frgn,k, vk . 2. 116 We define now a set k such that k  k   gk , nn  gn,k  k   ;gn,k, vk 2. 117 Obviously definitions (2.114) and (2.117) is equivalent by Proposition 2.1. Proposition 2.25. (i) Th;# ;, (ii) ; is a countable Th;# -set. Proof.(i) Statement Th;# ; follows immediately from the statement  and axiom schema of separation [9], (ii) follows immediately from countability of the set . Proposition 2.26. Set ; is inconsistent. Proof.From the formula (2.119) we obtain Th; # ;  ;  PrTh;# ;  ;  c. 2. 118 From the formula (2.118) and Proposition 2.1 we obtain Th; # ;  ;  ;  ; 2. 115 and therefore Th; # ;  ;  ;  ;. 2. 116 But this is a contradiction. Proposition 2.26.Assume that (i) ConTh and (ii) Th has an nonstandard model MNstTh and MZ2  MNstTh .Then theory Th can be extended to a maximally consistent nice theory Th #  Th # MNstTh . Proof. Let 1. . . i. . . be an enumeration of all wff's of the theory Th (this can be achieved if the set of propositional variables can be enumerated). Define a chain  ThNst,i # |i   , ThNst,1 #  Th of consistent theories inductively as follows: assume that theory Thi is defined. (i) Suppose that a statement (2.117) is satisfied ThNst,i # PrThNst,i# i  c and ThNst,i #  i   MNstTh  i . 2. 117 Then we define a theory ThNst,i1 as follows ThNst,i1  ThNst,i  i .Using Lemma 2.1 we will rewrite the condition (2.117) symbolically as follows ThNst,i # PrThNst,i# # i c, PrTh i # i c  PrThNst,i# i  c  MNstTh  i . 2. 118 (ii) Suppose that the statement (2.119) is satisfied ThNst,i # PrThNst,i# i  c and ThNst,i #  i   MNstTh  i . 2. 119 Then we define theory Thi1 as follows: Thi1  Thi  i . Using Lemma 2.2 we will rewrite the condition (2.119) symbolically as follows ThNst,i # PrThNst,i# # i c , PrThNst,i# # i c  PrThNst,i# i  c  MTh  i . 2. 120 (iii) Suppose that a statement (2.121) is satisfied ThNst,i # PrThNst,i# i  c and ThNst,i # PrThNst,i# i  c  i. 2. 121 We will rewrite the condition (2.121) symbolically as follows ThNst,i # PrThNst,i#

i c, PrThNst,i#

i c  PrTh ii  c  PrTh ii  c  i  2. 122 Then we define a theory ThNst,i1 # as follows: ThNst,i1 #  ThNst,i # . (iv) Suppose that the statement (2.123) is satisfied ThNst,i1 # PrThNst,i# i  c and ThNst,i # PrThNst,i# i  c  i. 2. 123 We will rewrite the condition (2.123) symbolically as follows ThNst,i # PrThNst,i#

i c, PrThNst,i#

i c  PrThNst,i# i  c  PrThNst,i# i  c  i 2. 124 Then we define a theory ThNst,i1 # as follows: ThNst,i1 #  ThNst,i # .We define now a theory Th;Nst # as follows: Th;Nst #   i ThNst,i # . 2. 125 First, notice that each ThNst,i # is consistent. This is done by induction on i and by Lemmas 2.1-2.2. By assumption, the case is true when i  1.Now, suppose ThNst,i # is consistent. Then its deductive closure DedThNst,i #   A|ThNst,i # A is also consistent. If a statement (2.121) is satisfied,i.e. ThNst,i # PrThNst,i# i  c and ThNst,i # i, then clearly ThNst,i1 #  ThNst,i #  i is consistent since it is a subset of closure DedThNst,i # . If a statement (2.123) is satisfied,i.e. ThNst,i # PrThNst,i# i  c and ThNst,i # i, then clearly ThNst,i1 #  ThNst,i #  i is consistent since it is a subset of closure DedThNst,i # . If a statement (2.117) is satisfied,i.e. ThNst,i # PrThNst,i# i  c and ThNst,i #  i   MNstTh  i  then clearly ThNst,i1 #  ThNst,i #  i is consistent by Lemma 2.1 and by one of the standard properties of consistency:   A is consistent iff   A. If a statement (2.119) is satisfied,i.e. ThNst,i # PrThNst,i# i  c and ThNst,i #  i   MNstTh  i  then clearly ThNst,i1 #  ThNst,i #  i is consistent by Lemma 2.2 and by one of the standard properties of consistency:   A is consistent iff   A.Next, notice DedTh;Nst #  is maximally consistent nice extension of the DedTh. DedTh;Nst #  is consistent because, by the standard Lemma 2.3 above, it is the union of a chain of consistent sets. To see that DedTh;Nst #  is maximal, pick any wff . Then  is some i in the enumerated list of all wff's. Therefore for any  such that ThNst,i # PrThNst,i#  c or ThNst,i # PrThNst,i#  c, either   Th;Nst # or   Th;Nst # .Since DedThNst,i1 #  DedTh;Nst # , we have   DedTh;Nst #  or   DedTh;Nst # ,which implies that DedTh;Nst #  is maximally consistent nice extension of the DedTh. Definition 2.28. We define now predicate PrTh#i  c asserting provability in Th;Nst # : PrTh;Nst# i  c  PrTh;Nst# # i c  PrTh;Nst#

i c . 2. 126 Definition 2.29. Let   x be one-place open wff such that the conditions:   Th;Nst # !xx or   Th;Nst # PrTh;Nst# !xx c and MNstTh  !xx is satisfied. Then we said that, a set y is a Th#-set iff there is exist one-place open wff x such that y  x. We write yTh;Nst #  iff y is a Th;Nst # -set. Remark 2.21. Note that       Th;Nst# !xx. Remark 2.22. Note that yTh;Nst#    y  x  PrTh;Nst# !xx c Definition 2.30.Let ;Nst# be a collection such that : x x  ;Nst#  x is a Th #-set . Proposition 2.27.Collection ;Nst# is a Th;Nst # -set. Proof. Let us consider an one-place open wff x such that conditions ( ) or ( ) is satisfied, i.e. Th# !xx.We note that there exists countable collection  of the one-place open wff's   nx n such that: (i) x   and (ii) Th;Nst # !x x  n n  M Z2 Hs x  nx or Th;Nst # !x PrTh;Nst# x c  n n  M Z2 Hs PrTh;Nst# x  nx c and MNst Th  !x x  n n  M Z2 Hs x  nx 2. 127 or of the equivalent form Th;Nst # !x1 1x1  n n  M Z2 Hs 1x1  n,1x1 or Th;Nst # !x PrTh;Nst# x1 c  n n  M Z2 Hs PrTh;Nst# x1  nx1 c and MNst Th  !x x1  n n  M Z2 Hs x1  nx1 2. 128 where we set x  1x1,nx1  n,1x1 and x  x1. We note that any collection k  n,kx n, k  1, 2, . . . such above defines an unique set xk , i.e. k1 k2  iff xk1  xk2 .We note that collections k , k  1, 2, . . is no part of the ZFC2 Hs, i.e. collection k there is no set in sense of ZFC2Hs. However that is no problem, because by using Gödel numbering one can to replace any collection k , k  1, 2, . . by collection k  gk  of the corresponding Gödel numbers such that k  gk   gn,kxk n, k  1, 2, . . . . 2. 129 It is easy to prove that any collection k  gk , k  1, 2, . . is a Th;Nst # -set.This is done by Gödel encoding [8],[10] (2.129) and by axiom schema of separation [9]. Let gn,k  gn,kxk, k  1, 2, . . be a Gödel number of the wff n,kxk.Therefore gk  gn,k n, where we set k  k , k  1, 2, . . and k1k2	gn,k1 n	gn,k2 n   xk1  xk2 . 2. 130 Let gn,k n k be a family of the all sets gn,k n. By axiom of choice [9] one obtain unique set ;Nst#  gk k such that kgk  gn,k n .Finally one obtain a set ;Nst # from a set ;Nst# by axiom schema of replacement [9].Thus we can define a Th;Nst # -set ;Nst#  ;Nst# : x x  ;Nst#  x  ;Nst#   PrTh;Nst# x  x c  PrTh;Nst# x  x c  x  x . 2. 131 Proposition 2.28. Any collection k  gk , k  1, 2, . . is a Th;Nst # -set. Proof. We define gn,k  gn,kxk  n,kxkc, vk  xk c. Therefore gn,k  gn,kxk  Frgn,k, vk (see [10]). Let us define now predicate gn,k, vk gn,k, vk  PrTh;Nst# !xk1,kx1 c  !xkvk  xk c n n  Mst Z2 Hs PrTh;Nst# 1,kxk c  PrTh;Nst# Frgn,k, vk . 2. 132 We define now a set k such that k  k   gk , nn  gn,k  k   gn,k, vk 2. 133 But obviously definitions (2.29) and (2.133) is equivalent by Proposition 2.26. Proposition 2.28. (i) Th;Nst# ;Nst# , (ii) ;Nst# is a countable Th;Nst # -set. Proof.(i) Statement Th# c follows immediately from the statement  ;Nst# and axiom schema of separation [9]. (ii) follows immediately from countability of the set ;Nst# . Proposition 2.29. A set ;Nst# is inconsistent. Proof.From formla (2.131) we obtain Th;Nst # ;Nst#  ;Nst#  ;Nst#  ;Nst#c . 2. 134 From formula (2.41) and Proposition 2.6 one obtains Th;Nst # ;Nst#  ;Nst#  ;Nst#  ;Nst# 2. 135 and therefore Th;Nst # ;Nst#  ;Nst#   ;Nst#  ;Nst# . 2. 136 But this is a contradiction. 2.3.Proof of the inconsistensy of the set theory ZFC2 Hs  MZFC2 Hs using Generalized Tarski's undefinability theorem. In this section we will prove that a set theory ZFC2Hs  MZFC2 Hs is inconsistent, without any refference to the set  and inconsistent set . Proposition 2.30.(Generalized Tarski's undefinability theorem).Let ThHs be second order theory with Henkin semantics and with formal language , which includes negation and has a Gödel encoding g such that for every -formula Ax there is a formula B such that B  AgB  AgB  B holds. Assume that Th Hs has an standard Model M. Then there is no -formula Truen such that for every -formula A such that M  A, the following equivalence A  TruegA  TruegA  A 2. 137 holds. Proof.The diagonal lemma yields a counterexample to this equivalence, by giving a "Liar" sentence S such that S  TruegS holds. Remark 2.23. Above we defined the set  (see Definition 2.10) in fact using generalized "truth predicate" True#  c, such that True#  c,  PrTh#  c  PrTh#  c   . 2. 138 In order to prove that set theory ZFC2Hs  MZFC2 Hs is inconsistent without any refference to the set ,notice that by the properties of the nice extension Th # follows that definition given by (2.138) is correct, i.e.,for every ZFC2Hs-formula  such that MZFC2 Hs   the following equivalence   PrTh#  c  PrTh#  c   . 2. 139 holds. Proposition 2.31.Set theory Th1#  ZFC2Hs  MZFC2 Hs is inconsistent. Proof.Notice that by the properties of the nice extension Th# of theTh1# follows that MZFC2 Hs    Th # . 2. 140 Therefore (2.138) gives generalized "truth predicate" for set theory Th1#.By Proposition 2.30 one obtains a contradiction. Remark 2.24.A cardinal  is inaccessible if and only if  has the following reflection property: for all subsets U  Vκ, there exists α  κ such that Vα,, U  Vα is an elementary substructure of Vκ,, U. (In fact, the set of such α is closed unbounded in κ.) Equivalently, κ is Πn0 -indescribable for all n  0. Remark 2.25.Under ZFC it can be shown that κ is inaccessible if and only if Vκ, is a model of second order ZFC, [5]. Remark 2.26. By the reflection property, there exists α  κ such that Vα, is a standard model of (first order) ZFC. Hence, the existence of an inaccessible cardinal is a stronger hypothesis than the existence of the standard model of ZFC2Hs. 3.Derivation inconsistent countable set in set theory ZFC2 with the full semantics. Let Th  Thfss be an second order theory with the full second order semantics.We assume now that Th contains ZFC2 fss.We will write for short Th, instead Thfss. Remark 3.1.Notice that M is a model of ZFC2 fss if and only if it is isomorphic to a model of the form Vκ, Vκ  Vκ, for κ a strongly inaccessible ordinal. Remark 3.2.Notice that a standard model for the language of first-order set theory is an ordered pair D, I .Its domain, D, is a nonempty set and its interpretation function, I, assigns a set of ordered pairs to the two-place predicate "" .A sentence is true in D, I

just in case it is satisfied by all assignments of first-order variables to members of D and second-order variables to subsets of D; a sentence is satisfiable just in case it is true in some standard model; finally, a sentence is valid just in case it is true in all standard models. Remark 3.3.Notice that: (I)The assumption that D and I be sets is not without consequence. An immediate effect of this stipulation is that no standard model provides the language of set theory with its intended interpretation. In other words, there is no standard model D, I in which D consists of all sets and I assigns the standard element-set relation to "" . For it is a theorem of ZFC that there is no set of all sets and that there is no set of ordered-pairs x, y for x an element of y. (II)Thus, on the standard definition of model: (1) it is not at all obvious that the validity of a sentence is a guarantee of its truth; (2) similarly, it is far from evident that the truth of a sentence is a guarantee of its satisfiability in some standard model. (3)If there is a connection between satisfiability, truth, and validity, it is not one that can be "read off" standard model theory. (III) Nevertheless this is not a problem in the first-order case since set theory provides us with two reassuring results for the language of first-order set theory. One result is the first order completeness theorem according to which first-order sentences are provable, if true in all models. Granted the truth of the axioms of the first-order predicate calculus and the truth preserving character of its rules of inference, we know that a sentence of the first-order language of set theory is true, if it is provable. Thus, since valid sentences are provable and provable sentences are true, we know that valid sentences are true. The connection between truth and satisfiability immediately follows: if φ is unsatisfiable, then φ, its negation, is true in all models and hence valid. Therefore, φ is true and φ is false. Definition 3.1. The language of second order arithmetic Z2 is a two-sorted language: there are two kinds of terms, numeric terms and set terms. 0 is a numeric term, 1.There are in nitely many numeric variables, x0, x1, . . . , xn, . . . each of which is a numeric term; 2.If s is a numeric term then Ss is a numeric term; 3.If s, t are numeric terms then st and st are numeric terms (abbreviated s  t and s t); 3.There are infinitely many set variables, X0, X1, . . . , Xn. . . each of which is a set term; 4.If t is a numeric term and S then  tS is an atomic formula (abbreviated t  S); 5.If s and t are numeric terms then  st and  st are atomic formulas (abbreviated s  t and s  t correspondingly). The formulas are built from the atomic formulas in the usual way. As the examples in the definition suggest, we use upper case letters for set variables and lower case letters for numeric terms. (Note that the only set terms are the variables.) It will be more convenient to work with functions instead of sets, but within arithmetic, these are equivalent: one can use the pairing operation, and say that X represents a function if for each n there is exactly one m such that the pair n, m belongs to X. We have to consider what we intend the semantics of this language to be. One possibility is the semantics of full second order logic: a model consists of a set M, representing the numeric objects, and interpretations of the various functions and relations (probably with the requirement that equality be the genuine equality relation), and a statement XX is satisfied by the model if for every possible subset of M, the corresponding statement holds. Remark 3.1.Full second order logic has no corresponding proof system. An easy way to see this is to observe that it has no compactness theorem. For example, the only model (up to isomorphism) of Peano arithmetic together with the second order induction axiom: X0  X  xx  X  Sx  X  xx  X is the standard model . This is easily seen: any model of Peano arithmetic has an initial segment isomorphic to ; applying the induction axiom to this set, we see that it must be the whole of the model. Remark 3.2.There is no completeness theorem for second-order logic. Nor do the axioms of second-order ZFC imply a reflection principle which ensures that if a sentence of second-order set theory is true, then it is true in some standard model. Thus there may be sentences of the language of second-order set theory that are true but unsatisfiable, or sentences that are valid, but false. To make this possibility vivid, let Z be the conjunction of all the axioms of second-order ZFC. Z is surely true. But the existence of a model for Z requires the existence of strongly inaccessible cardinals. The axioms of second-order ZFC don't entail the existence of strongly inaccessible cardinals, and hence the satisfiability of Z is independent of second-order ZFC. Thus, Z is true but its unsatisfiability is consistent with second-order ZFC [5]. Thus with respect to ZFC2 fss, this is a semantically defined system and thus it is not standard to speak about it being contradictory if anything, one might attempt to prove that it has no models, which to be what is being done in section 2 for ZFC2Hs. Definition 3.2. Using formula (2.3) one can define predicate PrTh# y really asserting provability in Th  ZFC2 fss PrTh # y  PrThy  PrThy  , PrThy  x x  M Z2 fss ProvThx, y, y  c. 3. 1 Theorem 3.1.[12].(Löb's Theorem for ZFC2 fss) Let  be any closed formula with code y  c  MZ2 , then Th PrThc implies Th  (see [12] Theorem 5.1). Proof. Assume that (#) Th PrThc. Note that (1) Th  . Otherwise one obtains Th PrThc  PrThc, but this is a contradiction. (2) Assume now that (2.i) Th PrThc and (2.ii) Th  . From (1) and (2.ii) follows that (3) Th   and Th  . Let Th be a theory (4)Th  Th 	 .From (3) follows that (5) ConTh. From (4) and (5) follows that (6) Th PrTh c. From (4) and (#) follows that (7) Th PrTh c. From (6) and (7) follows that (8) Th PrTh c  PrTh c,but this is a contradiction. Definition 3.3. Let   x be one-place open wff such that: Th !xx 3. 2 Then we will says that, a set y is a Th-set iff there is exist one-place open wff x such that y  x. We write yTh iff y is a Th-set. Remark 3.2. Note that yTh  y  x  PrTh!xxc  PrTh!xxc  !xx 3. 3 Definition 3.4. Let be a collection such that : x x   x is a Th-set . Proposition 3.1. Collection is a Th-set. Definition 3.4. We define now a Th-set c  : xx  c  x    PrThx  xc  PrThx  xc  x  x. 3. 4 Proposition 3.2. (i) Th c, (ii) c is a countable Th-set. Proof.(i) Statement Th c follows immediately by using statement  and axiom schema of separation [4], (ii) follows immediately from countability of a set . Proposition 3.3. A set c is inconsistent. Proof.From formla (3.2) one obtains Th c  c  PrThc  c c  PrThc  c c  c  c . 3. 5 From formula (3.4) and definition 3.5 one obtains Th c  c  c  c 3. 6 and therefore Th c  c  c  c. 3. 7 But this is a contradiction. Thus finally we obtain: Theorem 3.2.[12].ConZFC2 fss. It well known that under ZFC it can be shown that κ is inaccessible if and only if Vκ, is a model of ZFC2 [5],[11].Thus finally we obtain. Theorem 3.3.[12].ConZFC  MstZFCMstZFC  Hk. 4.Consistency Results in Topology. Definition 4.1.[19].A Lindelöf space is indestructible if it remains Lindelöf after forcing with any countably closed partial order. Theorem 4.1.[20].If it is consistent with ZFC that there is an inaccessible cardinal, then it is consistent with ZFC that every Lindelöf T3 indestructible space of weight 1 has size 1. Corollary 4.1.[20] The existence of an inaccessible cardinal and the statement: T3, 1, 1   "every Lindelöf T3 indestructible space of weight 1 has size 1" are equiconsistent. Theorem 4.2.[12].ConZFC  T3, 1, 1 . Proof.Theorem 4.2 immediately follows from Theorem 3.3 and Corollary 4.1. Definition 4.2.The 1-Borel Conjecture is the statement: BC1   "a Lindelöf space is indestructible if and only if all of its continuous images in 0; 11 have cardinality 1". Theorem 4.3.[12]. If it is consistent with ZFC that there is an inaccessible cardinal, then it is consistent with ZFC that the 1-Borel Conjecture holds. Corollary 4.2.The 1-Borel Conjecture and the existence of an inaccessible cardinal are equiconsistent. Theorem 4.4.[12] ConZFC  BC1 . Proof.Theorem 4.4 immediately follows from Theorem 3.3 and Corollary 4.2. Theorem 4.5.[20]. If 2 is not weakly compact in L, then there is a Lindelöf T3 indestructible space of pseudocharacter 1 and size 2. Corollary 4.3.The existence of a weakly compact cardinal and the statement: T3, 1,2   "there is no Lindelöf T3 indestructible space of pseudocharacter 1 and size 2 are equiconsistent. Theorem 4.6.[12].There is a Lindelöf T3 indestructible space of pseudocharacter 1 and size 2 in L. Proof.Theorem 4.6 immediately follows from Theorem 3.3 and Theorem 4.5. Theorem 4.7.[12]. Con ZFC  T3, 1,2  . Proof.Theorem 3.7 immediately follows from Theorem 3.3 and Corollary 4.3. 5.Conclusion. In this paper we have proved that the second order ZFC with the full second-order semantic is inconsistent,i.e. ConZFC2 fss.Main result is: let k be an inaccessible cardinal and Hk is a set of all sets having hereditary size less then k, then ConZFC  V  Hk.This result also was obtained in [7],[12],[13] by using essentially another approach. For the first time this result has been declared to AMS in [14],[15]. An important applications in topology and homotopy theory are obtained in [16],[17],[18]. 5.Acknowledgments A reviewers provided important clarifications. References. [1] E. Nelson.Warning Signs of a Possible Collapse of Contemporary Mathematics. https://web.math.princeton.edu/~nelson/papers/warn.pdf In Infinity: New Research Frontiers, by Michael Heller (Editor), W. Hugh Woodin Pages 75–85, 2011. Published February 14th 2013 by Cambridge University Press Hardcover, 311 pages. ISBN: 1107003873 (ISBN13: 9781107003873) [2] L. Henkin, "Completeness in the theory of types". Journal of Symbolic Logic 15 (2): 81–91. doi:10.2307/2266967. JSTOR 2266967 [3] M. Rossberg, "First-Order Logic, Second-Order Logic, and Completeness". In V.Hendricks et al., eds. First-order logic revisited. Berlin: Logos-Verlag. [4] S. Shapiro, Foundations without Foundationalism: A Case for Second-order Logic. Oxford University Press. ISBN 0-19-825029-0 [5] A. Rayo and G. Uzquiano,Toward a Theory of Second-Order Consequence, Notre Dame Journal of Formal Logic Volume 40, Number 3, Summer 1999. [6] J. Vaananen, Second-Order Logic and Foundations of Mathematics, The Bulletin of Symbolic Logic, Vol.7, No. 4 (Dec., 2001), pp. 504-520. [7] L. Henkin. Completeness in the theory of types. J. Symbolic Logic, 15:81–91, 1950. [8] G. Uzquiano, Quantification without a domain. New Waves in Philosophy of Mathematics. Springer, 29 Sep 2009 Philosophy 327 pp. ISBN 0230245196, 9780230245198 [9] P. Cohen, Set Theory and the continuum hypothesis.Reprint of the W. A. Benjamin,Inc.,New York,1966 edition. ISBN-13: 978-0486469218 [10] E. Mendelson,Introduction to mathematical logic.June1,1997. ISBN-10: 0412808307. ISBN-13: 978-0412808302 [11] C. Smorynski, Handbook of mathematical logic, Edited by J. Barwise. North-Holland Publishing Company, 1977 [12] G. Takeuti,Proof Theory: Second Edition (Dover Books on Mathematics) 2013 ISBN-13: 978-0486490731; ISBN-10: 0486490734 [11] A. Bovykin, "On order-types of models of arithmetic". Ph.D. thesis pp.109, University of Birmingham 2000.On order-types of models of arithmetic. (with R.Kaye) (2001). Contemporary Mathematics Series of the AMS, 302, pp. 275-285. [12] H. Friedman, Countable models of set theories, Cambridge Summer School in Mathematical Logic, 1971, Lecture Notes in Mathematics, Springer, 337 (1973), 539–573 [13] M. Magidor, S. Shelah and J. Stavi, On the Standard Part of Nonstandard Models of Set Theory,The Journal of Symbolic Logic Vol. 48, No. 1 (Mar., 1983), pp. 33-38 [12] P. Lindstrom, "First Order Predicate Logic with Generalized Quantifiers," Theoria, Vol. 32, No. 3, 1966, pp. 186-195. [12] P. Cohen, Set Theory and the continuum hypothesis.Reprint of the W. A. Benjamin, Inc., New York, 1966 edition. ISBN-13: 978-0486469218 [13] G. Takeuti and W. M, Zaring.Introduction to Axiomatic Set Theory, Springer-Verlag, 1971. [7] J.Foukzon, Strong Reflection Principles and Large Cardinal Axioms, Pure and Applied Mathematics Journal, Vol.2, Issue Number 3. pp.119-127 http://www.sciencepublishinggroup.com/j/pam DOI: 10.11648/j.pamj.20130203.12 [10] P. Cohen, Set Theory and the continuum hypothesis.Reprint of the W. A. Benjamin, Inc., New York, 1966 edition. ISBN-13: 978-0486469218 [11] A proof that cannot be formalized in ZFC. [FOM] https://cs.nyu.edu/pipermail/fom/2007-October/012013.html [12] J. Foukzon,Generalized Lob's Theorem.Strong Reflection Principles and Large Cardinal Axioms. Consistency Results in Topology. http://arxiv.org/abs/1301.5340v10 [13] J.Foukzon,E. R. Men'kova,Generalized Löb's Theorem. Strong Reflection Principles and Large Cardinal Axioms, Advances in Pure Mathematics, Vol.3 No.3, 2013. http://dx.doi.org/10.4236/apm.2013.33053 [14] J. Foukzon, "An Possible Generalization of the Lob's Theorem," AMS Sectional Meeting AMS Special Session. Spring Western Sectional Meeting University of Colorado Boulder, Boulder, CO 13-14 April 2013. Meeting # 1089. http://www.ams.org/amsmtgs/2210_abstracts/1089-03-60.pdf [15] J. Foukzon, Strong Reflection Principles and Large Cardinal Axioms. Fall Southeastern Sectional Meeting University of Louisville, Louisville, KY October 5-6, 2013 (Saturday -Sunday) Meeting #1092 http://www.ams.org/amsmtgs/2208_abstracts/1092-03-13.pdf [16] J. Foukzon, Consistency Results in Topology and Homotopy Theory, Pure and Applied Mathematics Journal, 2015; 4(1-1): 1-5 Published online October 29, 2014 doi: 10.11648/j.pamj.s.2015040101.11 ISSN: 2326-9790 (Print); ISSN: 2326-9812 (Online) [17] J. Foukzon, Generalized Lob's Theorem. Strong Reflection Principles and Large Cardinal Axioms. Consistency Results in Topology,IX Iberoamerican Conference on Topology and its Applications 24-27 June, 2014 Almeria, Spain. Book of abstracts, p.66. [18] J. Foukzon, Generalized Lob's Theorem. Strong Reflection Principles and Large Cardinal Axioms. Consistency Results in Topology, International Conference on Topology and its Applications, July 3-7, 2014, Nafpaktos, Greece. Book of abstracts, p.81. [19] F.D. Tall, On the cardinality of Lindelöf spaces with points Gδ, Topology and its Applications 63 (1995),21–38. [20] R.R. Dias, F. D. Tall, Instituto de Matemática e Estatística Universidade de São Paulo,15-th Galway Topology Colloquium, Oxford, 2012. [21] F. Larusson, A proof that cannot be formalized in ZFC. [FOM] Fri Oct 5 23:35:03 EDT 2007. https://www.cs.nyu.edu/pipermail/fom/2007-October/012009.html [22] P. Cohen, Set Theory and the continuum hypothesis. ISBN-13: 978-0486469218 [23] P. Lindstrom, "First Order Predicate Logic with Generalized Quantifiers," Theoria, Vol. 32, No. 3, 1966, pp. 186-195. [24] G. Takeuti,Proof Theory: Second Edition (Dover Books on Mathematics) 2013 ISBN-13: 978-0486490731; ISBN-10: