There is No Standard Model of ZFC and ZFC2 Jaykov Foukzon1 and Elena Men'kova2 jaykovfoukzon@list.ru E_Menkova@mail.ru 1Israel Institute of Technology,Haifa,Israel 2 Lomonosov Moscow State University, Moscow, Russia -------------------------------------- Abstract: In this article we proved so-called strong reflection principles corresponding to formal theories Th which has omega-models or nonstandard model with standard part. An posible generalization of Lob's theorem is considered.Main results are: (i) ConZFC  MstZFC, (ii) ConZF  V  L, (iii) ConNF  MstNF, (iv) ConZFC2, (v) let k be inaccessible cardinal then ConZFC  . Keywords: Gödel encoding, Completion of ZFC, Russell's paradox, -model, Henkin semantics, full second-order semantic,strongly inaccessible cardinal 1.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 consistent, 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.The inconsistency of the second order set theory ZFC2 originally have been uncovered in [2] and officially announced in [3],see also ref.[4],[5],[6]. Remark 1.1.2.In order to derive a contradiction in second order set theory ZFC2 with the Henkin semantics [7],we remind the definition given in P.Cohen handbook [8], (see [8] Ch.III,sec.1,p.87). P.Cohen wroted: "A set which can be obtained as the result of a transfinite sequence of predicative definitions Godel called "constructible". His result then is that the con-structible sets are a model for ZF and that in this model GCH and AC hold.The notion of a predicative construction must be made more precise,of course, but there is essentially only one way to proceed. Another way to explain constructibility is to remark that the constructible sets are those sets which jnust occur in any model in which one admits all ordinals.The definition we now give is the one used in [9]. Definition 1.1.1. [8]. Let X be a set. The set X  is defined as the union of X and the set Y of all sets у for which there is a formula Az, t1, . . . , tk in ZF such that if AX denotes A with all bound variables restricted to X, then for some ti, i  1, . . . , k. in X, у  z  X | AXz, t1, . . . , tk . 1. 1. 1 Observe X   Px  X, X   X if X is infinite (and we assume AC). It should be clear to the reader that the definition of X , as we have given it, can be done entirely within ZF and that Y  X  is a single formula AX, Y in ZF. In general, one's intuition is that all normal definitions can be expressed in ZF, except possibly those which involve discussing the truth or falsity of an infinite sequence of statements. Since this is a very important point we shall give a rigorous proof in a later section that the construction of X  is expressible in ZF. " Remark 1.1.3.We will say that a set y is definable by the formula Az, t1, . . . , tk relative to a given set X. Remark 1.1.4.Note that a simple generalsation of the notion of of the definability which has been by Definition1.1.1 immediately gives Russell's paradox in second order set theory ZFC2 with the Henkin semantics [7]. Definition 1.1.2.[6].(i)We will say that a set y is definable relative to a given set X iff there is a formula Az, t1, . . . , tk in ZFC then for some ti  X, i  1, . . . , k. in X there exists a set z such that the condition Az, t1, . . . , tk is satisfied and y  z or symbolically zAz, t1, . . . , tk  y  z. 1. 1. 2 It should be clear to the reader that the definition of X , as we have given it, can be done entirely within second order set theory ZFC2 with the Henkin semantics [7] denoted by ZFC2Hs and that Y  X  is a single formula AX, Y in ZFC2Hs. (ii)We will denote the set Y of all sets у definable relative to a given set X by Y  2Hs. Definition 1.1.3. Let 2Hs be a set of the all sets definable relative to a given set X by the first order 1-place open wff's and such that xx  2Hsx  2Hs  x  x. 1. 1. 3 Remark 1.1.5.(a) Note that 2Hs  2Hs since 2Hs is a set definable by the first order 1-place open wff Z,2Hs : Z,2Hs  xx  2Hsx  Z  x  x, 1. 1. 4 Theorem 1.1.1.[6].Set theory ZFC2Hs is inconsistent. Proof. From (1.1.3) and Remark 1.1.2 one obtains 2Hs  2Hs  2Hs  2Hs . 1. 1. 5 From (1.1.5) one obtains a contradiction 2Hs  2Hs  2Hs  2Hs. 1. 1. 6 Remark 1.1.6.Note that in paper [6] we dealing by using following definability condition: a set у is definable if there is a formula Az in ZFC such that zAz  y  z. 1. 1. 7 Obviously in this case a set Y  2Hs is a countable set. Definition 1.1.4. Let 2Hs be the countable set of the all sets definable by the first order 1-place open wff's and such that xx  2Hsx  2Hs  x  x. 1. 1. 8 Remark 1.1.7.(a) Note that 2Hs  2Hs since 2Hs is a ZFC-set definable by the first order 1-place open wff Z,2Hs : Z,2Hs  xx  2Hsx  Z  x  x, 1. 1. 9 one obtains a contradiction 2Hs  2Hs  2Hs  2Hs. In this paper we dealing by using following definability condition. Definition 1.1.5.(i) Let Mst  MstZFC be a standard model of ZFC. We will say that a set y is definable relative to a given standard model Mst of ZFC if there is a formula Az, t1, . . . , tk in ZFC such that if AMst denotes A with all bound variables restricted to Mst, then for some ti  Mst, i  1, . . . , k. in Mst there exists a set z such that the condition AMstz, t1, . . . , tk is satisfied and y  z or symbolically zAMstz, t1, . . . , tk  y  z. 1. 1. 10 It should be clear to the reader that the definition of Mst , as we have given it, can be done entirely within second order set theory ZFC2 with the Henkin semantics. (ii) In this paper we assume for simplicity but without loss of generality that AMstz, t1, . . . , tk  AMstz. 1. 1. 11 Remark 1.1.8.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 [7] and (iii) the second order set theory ZFC2under the full second-order semantics [8],[9],[10],[11],[12] but also with a proof theory bused on formal Urlogic [13]. Remark 1.1.9.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]-[13].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 1.1.10.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]-[13]. 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 of 2 up to isomorphism. 8. There is a finite second-order axiom system RCF2 such that the field  of the real numbers is the only full model of RCF2 up to isomorphism. Remark 1.1.11.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. 12 where X is a second-order variable, and where FuncX abbreviates " X is a functional relation",see [12]. Thus we interpret the wff's of ZFC2 language with the full second-order semantics as required in [12],[13] but also wit a proof theory bused on formal urlogic [13]. 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 1.1.12.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 [11]. 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  . 1. 1. 13 In this paper we prove that: (i) ZFCst  ZFC  MstZFC (ii) ZFC2 Hs  ZFC2Hs  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 1.1.13.(i) We emphasize that it is well known that axiom MZFC a single statement in ZFC see [8],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 1.1.6.[8].The model MstZFC is called a standard model since the relation  used is merely the standard relation. Remark 1.1.14.Note that axiom MZFC doesn't imply axiom MstZFC,see ref. [8]. Remark 1.1.15.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 [13] 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 1.1.16.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 and the all wff of ZFC2Hs, ZFC,etc. represented by standard objects.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 ZFC2 Hs; we must also believe that these arithmetical theorems are asserting something about the standard naturals. It is "conceivable" that ZFC2Hs 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 1.1.17. Note that assumption Mst ZFC2 Hs is not necessary if nonstandard model MNst ZFC2 Hs is a transtive or has an standard part Mst Z2 Hs

MNst Z2 Hs ,see [14],[15]. Remark 1.1.18.Remind that if M is a transitive model, then ωM is the standard ω. This implies that the natural numbers, integers, and rational numbers of the model are also the same as their standard counterparts. Each real number in a transitive model is a standard real number, although not all standard reals need be included in a particular transitive model. 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  ,see ref. [16], for some linear order A Thus one can to choose Gödel encoding inside the standard model Mst Z2 Hs . Remark 1.1.19. However there is no any problem as mentioned above in second order set theory ZFC2 with the full second-order semantics because corresponding second order arithmetic Z2 fss is categorical. Remark 1.1.20. 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.Generalized Löb's theorem.Remarks on the Tarski's undefinability theorem. 2.1.Remarks on the Tarski's undefinability theorem. Remark 2.1.1.In paper [17] under the following assumption ConZFC  MstZFC 2. 1. 1 has been proved that there exists countable Russell's set  such that the followng statement is satisfied: ZFC  MstZFC   MstZFC  card  0  MstZFC xx    x  x . 2. 1. 2 From (2.1.2) immediately follows a contradiction MstZFC       . 2. 1. 3 From (2.1.3) and (2.1.1) by reductio ad absurdum it follows ConZFC  MstZFC 2. 1. 4 Theorem 2.1.1.(Tarski's undefinability theorem) 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 . 2. 1. 5 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 Mst Th , 2. 1. 6 where the abbraviation A Mst Thmeans that A holds in standard model Mst Th , i.e. A Mst Th  Mst Th A.Therefore ConTh,st implies that Truex TruegA  A Mst Th 2. 1. 7 Thus Tarski's undefinability theorem reads ConTh,st  Truex TruegA  AMst Th . 2. 1. 8 Remark 2.1.2.(i) By the other hand the Theorem 2.1.1 says that given some really consistent formal theory Th,st that contins formal arithmetic, the concept of truth in that formal theory Th,st is not definable using the expressive means that that arithmetic affords. This implies a major limitation on the scope of "self-representation." It is possible to define a formula Truen,but only by drawing on a metalanguage whose expressive power goes beyond that of .To define a truth predicate for the metalanguage would require a still higher metametalanguage, and so on. (ii) However if formal theory Th,st is inconsistent this is not surpriising if we define a formula Truen  Truen; Th,st by drawing only on a language . (iii) Note that if under assumption ConTh,st we define a formula Truen; Th,st by drawing only on a language  by reductio ad absurdum it follows ConTh,st. 2. 1. 9 Remark 2.1.3. (i) Let ZFCst be a theory ZFCst  ZFC  MstZFC. In this paper under assumption ConZFCst we define a formula Truen; ZFCst by drawing only on a language ZFCst by using Generalized Löb's theorem [4],[5]. Thus by reductio ad absurdum it follows ConZFC  MstZFC. 2. 1. 10 (ii) However note that in this case we obtain ConZFCst by using approach that completely different in comparison with approach based on derivation of the countable Russell's set  with conditions (2.1.2). 2.2.Generalized Löb's theorem. Definition 2.2.1. Let Th# be first order theory and ConTh# . A theory Th# is complete if, for every formula A in the theory's language , that formula A or its negation A is provable in Th # , i.e., for any wff A, always Th # A or Th # A. Definition 2.2.2.Let Th be first order theory and ConTh.We will say that a theory Th # is completion of the theory Th if (i) Th Th # , (ii) a theory Th # is complete. Theorem 2.2.1.[4],[5]. Assume that:ConZFCst,where ZFCst  ZFC  MstZFC.Then there exists completion ZFCst# of the theory ZFCst such that the following condtions holds: (i) For every formula A in the language of ZFC that formula AMstZFC or formula AMstZFC is provable in ZFCst# i.e., for any wff A, always ZFCst# AMstZFC or ZFCst # AMstZFC . (ii) ZFCst#  m Thm,where for any m a theory Thm1 is finite extension of the theory Thm. (iii) Let Prmsty, x be recursive relation such that: y is a Gödel number of a proof of the wff of the theory Thm and x is a Gödel number of this wff.Then the relation Prmsty, x is expressible in the theory Thm by canonical Gödel encoding and really asserts provability in Thm. (iv) Let Prst# y, x be relation such that: y is a Gödel number of a proof of the wff of the theory ZFCst# and x is a Gödel number of this wff.Then the relation Prst# y, x is expressible in the theory ZFCst# by the following formula Prst# y, x  mm  Prmsty, x 2. 2. 1 (v) The predicate Prst# y, x really asserts provability in the set theory ZFCst# . Remark 2.2.1.Note that the relation Prmsty, x is expressible in the theory Thm since a theory Thm is an finite extension of the recursively axiomatizable theory ZFC and therefore the predicate Prmsty, x exists since any theory Thm is recursively axiomatizable. Remark 2.2.2.Note that a theory ZFCst# obviously is not recursively axiomatizable nevertheless Gödel encoding holds by Remark 2.2.1. Theorem 2.2.2.Assume that:ConZFCst,where ZFCst  ZFC  MstZFC.Then truth predicate Truen is expressible by using only first order language by the following formula TruegA  yy  mm  Prmsty, gA. 2. 2. 2 Proof.Assume that: ZFCst# AMstZFC . 2. 2. 3 It follows from (2.2.3) there exists m  m gA such that Thm AMstZFC and therefore by (2.2.1) we obtain Prst# y, gA  Prm st y, gA. 2. 2. 4 From (2.2.4) immediately by definitions one obtains (2.2.2). Remark 2.2.3.Note that Theorem 2.1.1 in tis case reads ConZFCst  Truex TruegA  AMstZFC . 2. 2. 5 Theorem 2.2.3. ConZFCst. Proof.Assume that: ConZFCst.From (2.2.2) and (2.2.5) one obtains a condradiction ConZFCst  ConZFCst (see Remark 2.1.3) and therefore by reductio ad absurdum it follows ConZFCst. Theorem 2.2.4.[4],[5]. 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  MNstZFCPA.Let ZFCNst be the theory ZFCNst  ZFC  MNstZFCPA.Assume that:ConZFCNst,where ZFCst  ZFC  MNstZFC.Then there exists completion ZFCNst# of the theory ZFCNst such that the following condtions holds: (i) For every formula A in the language of ZFC that formula AMNstZFC or formula AMNstZFC is provable in ZFCNst# i.e., for any wff A, always ZFCNst# AMNstZFC or ZFCNst # AMNstZFC . (ii) ZFCNst#  m Thm,where for any m a theory Thm1 is finite extension of the theory Thm. (iii) Let PrmNsty, x be recursive relation such that: y is a Gödel number of a proof of the wff of the theory Thm and x is a Gödel number of this wff.Then the relation PrmNsty, x is expressible in the theory Thm by canonical Gödel encoding and really asserts provability in Thm. (iv) Let PrNst# y, x be relation such that: y is a Gödel number of a proof of the wff of the theory ZFCNst# and x is a Gödel number of this wff.Then the relation PrNst# y, x is expressible in the theory ZFCNst# by the following formula PrNst# y, x  mm  MstPAPrmNsty, x 2. 2. 6 (v) The predicate PrNst# y, x really asserts provability in the set theory ZFCNst# . Remark 2.2.4.Note that the relation PrmNsty, x is expressible in the theory Thm since a theory Thm is an finite extension of the recursively axiomatizable theory ZFC and therefore the predicate PrmNsty, x exists since any theory Thm is recursively axiomatizable. Remark 2.2.5.Note that a theory ZFCNst# obviously is not recursively axiomatizable nevertheless Gödel encoding holds by Remark 2.2.1. Theorem 2.2.5.Assume that:ConZFCNst,where ZFCNst  ZFC  MNstZFC, MstPA MNstZFC Then truth predicate Truen is expressible by using first order language by the following formula TruegA  yy  MstPAmm  MstPAPrmNsty, gA. 2. 2. 7 Proof.Assume that: ZFCNst # AMNstZFC . 2. 2. 8 It follows from (2.2.6) there exists m  m gA such that Thm AMNstZFC and therefore by (2.2.8) we obtain PrNst# y, gA  Prm Nsty, gA. 2. 2. 9 From (2.2.9) immediately by definitions one obtains (2.2.7). Remark 2.2.6.Note that Theorem 2.1.1 in tis case reads ConZFCNst  Truex TruegA  AMNstZFC . 2. 2. 10 Theorem 2.2.6. ConZFCNst. Proof.Assume that: ConZFCNst.From (2.2.15) and (2.2.10) one obtains a condradiction ConZFCNst  ConZFCNst and therefore by reductio ad absurdum it follows ConZFCNst. Theorem 2.2.7.Assume that:Con ZFC2 Hs ,where ZFC2 Hs  ZFC2Hs  Mst ZFC2 Hs .Then there exists completion ZFC2 Hs# of the theory ZFC2 Hs such that the following condtions holds: (i) For every first order wff formula A (wff1 A) in the language of ZFC2Hs that formula A Mst ZFC2 Hs or formula A Mst ZFC2 Hs is provable in ZFC2 Hs# i.e., for any wff1 A, always ZFC2 Hs# A Mst ZFC2 Hs or ZFC2 Hs# A Mst ZFC2 Hs . (ii) ZFC2 Hs#  m Thm,where for any m a theory Thm1 is finite extension of the theory Thm. (iii) Let Prmsty, x be recursive relation such that: y is a Gödel number of a proof of the wff1 of the theory Thm and x is a Gödel number of this wff1.Then the relation Prmsty, x is expressible in the theory Thm by canonical Gödel encoding and really asserts provability in Thm. (iv) Let Prst# y, x be relation such that: y is a Gödel number of a proof of the wff of the set theory ZFC2 Hs# and x is a Gödel number of this wff1.Then the relation Prst# y, x is expressible in the set theory ZFC2 Hs# by the following formula Prst# y, x  mm  Prmsty, x 2. 2. 11 (v) The predicate Prst# y, x really asserts provability in the set theory ZFC2 Hs#. Remark 2.2.7.Note that the relation Prmsty, x is expressible in the theory Thm since a theory Thm is an finite extension of the finite axiomatizable theory ZFC2Hs and therefore the predicate PrmNsty, x exists since any theory Thm is recursively axiomatizable. Remark 2.2.8.Note that a theory ZFCNst# obviously is not recursively axiomatizable nevertheless Gödel encoding holds by Remark 2.2.1. Theorem 2.2.8.Assume that:Con ZFC2 Hs ,where ZFC2 Hs  ZFC2Hs  Mst ZFC2 Hs .Then truth predicate Truen is expressible by using first order language by the following formula TruegA  yy  mm  Prmsty, gA, 2. 2. 12 where A is wff1. Proof.Assume that: ZFC2 Hs# A Mst ZFC2 Hs . 2. 2. 13 It follows from (2.2.11) there exists m  m gA such that Thm A Mst ZFC2 Hs and therefore by (2.2.13) we obtain Prst# y, gA  Prm st y, gA. 2. 2. 14 From (2.2.22) immediately by definitions one obtains (2.2.12). Remark 2.2.13.Note that in considerd case Tarski's undefinability theorem reads Con ZFC2 Hs#  Truex TruegA  A Mst ZFC2 Hs , 2. 2. 15 where A is wff1. Theorem 2.2.9. Con ZFC2 Hs# . Proof.Assume that: Con ZFC2 Hs# .From (2.2.12) and (2.2.15) one obtains a condradiction Con ZFC2 Hs#  Con ZFC2 Hs# and therefore by reductio ad absurdum it follows Con ZFC2 Hs# . 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 set of the all first order provable definable sets X, i.e. a sets such that ZFC2 Hs !XX,where X  MstX is a first order 1-place open wff that contains only first order variables (we will be denoted such wff for short by wff1), with all bound variables restricted to standard model Mst  Mst ZFC2 Hs , i.e. Y Y  2 Hs  ZFC2 Hs MstXMstX  X,Mst Hs / X   !XMstX  Y  X , 3. 1. 1 or in a short notatons Y Y  2 Hs  ZFC2 Hs XX  XHs/ X   !XX  Y  X . 3. 1. 1. a Notation 3.1.1. In this subsection we often write for short X,XHs,XHs instead MstX, X,Mst Hs ,X,Mst Hs but this should not lead to a confusion. Assumption 3.1.1.We assume now for simplicity but without loss of generality that X,Mst Hs  Mst 3. 1. 1. b and therefore by definition of model Mst  Mst ZFC2 Hs one obtains X,Mst Hs  Mst ZFC2 Hs . Let X  ZFC2 Hs Y be a predicate such that X  ZFC2 Hs Y ZFC2 Hs X  Y.Let 2 Hs be the countable set of the 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 (3.1.3) immediately gives a contradiction 2 Hs  2 Hs  2 Hs  ZFC2 Hs 2 Hs . 3. 1. 3 Remark 3.1.1.Note that a contradiction (3.1.3) in fact is a contradiction inside ZFC2 Hs for the reason that predicate X  ZFC2 Hs Y is expressible by first order lenguage as predicate of ZFC2 Hs(see subsection1.2,Theorem 1.2.8 (ii)-(iii) and therefore countable sets 2 Hs and 2 Hs are sets in the sense of the set theory ZFC2 Hs. Remark 3.1.2.Note that by using Gödel encoding the above stated contradiction can be shipped in special completion ZFC2 Hs# of ZFC2 Hs,see subsection1.2,Theorem 1.2.8. Remark 3.1.3.(i) Note that Tarski's undefinability theorem cannot bloked the equivalence (3.1.3) since this theorem is no longer holds by Proposition 2.2.1.(Generalized Löbs Theorem). (ii) In additional note that: since Tarski's undefinability theorem has been proved under the same assumption Mst ZFC2 Hs by reductio ad absurdum it follows again ConZFCNst, see Theorem 1.2.10. Remark 3.1.4.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 set 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 set of the all sets such that X X  2 Hs X  2 Hs  X  X . 3. 1. 8 Remark 3.1.5.Note that 2 Hs  2 Hs since 2 Hs is a set definable by the first order 1-place open wff1 :  Z,2 Hs  X X  2 Hs X  Z  X  X. 3. 1. 8 From (3.1.8) and Remark 3.1.4 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 Remark 3.1.6.Note that contradiction (3.1.10) is a contradiction inside ZFC2 Hs for the reason that the countable set 2 Hs is a set in the sense of the set theory ZFC2 Hs. In order to obtain a contradiction inside ZFC2 Hs without any refference to Assumption 3.1.1 we introduce the following definitions. Definition 3.1.4.We define now the countable set Hs/  by the following formula 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 3.1.7. Note that the Definition 3.1.5 holds as definition of predicate really asserting provability of the first order sentence A 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 3.1.8.Note that from the second order axiom schema of replacement (1.1.12) 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 3.1.9. 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 a set 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 set 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  ZFC2Hs  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 3.1.11. 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 2 Hs by formula Y Y  2 Hs  y y  Hs/    gZFC2HsX   . 3. 1. 28 Remark 3.1.12.Note that from the axiom schema of replacement (1.1.12) 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 3.1.13. 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 3.1.14.Note that 2 Hs  2 Hs since 2 Hs is a set definable by 1-place open wff1  Z,2 Hs  X X  2 Hs X  Z  BewZFC2Hs#X  X . 3. 1. 31 Theorem 3.1.2.Set theory ZFC2 Hs  ZFC2Hs  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. 35 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 set of all sets X such that ZFCst !XX,where X is a 1-place open wff of ZFC 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 set of the all sets such that X X  st  X  st  X ZFCst X . 3. 2. 2 From (3.2.2) one obtains st  st  st ZFCst st. 3. 2. 3 But (3.2.3) immediately gives a contradiction st  st  st  st. 3. 2. 4 Remark 3.2.1.Note that a contradiction (3.2.4) is a contradiction inside ZFCst for the reason that predicate X ZFCst Y is expressible by using first order leguage as predicate of ZFCst (see subsection1.2,Theorem 1.2.2(ii)-(iii)) and therefore countable sets st and st are sets in the sense of the set theory ZFCst. Remark 3.2.2.Note that by using Gödel encoding the above stated contradiction can be shipped in special completion ZFCst# of ZFCst,see subsection1.2,Theorem 1.2.2 (i). 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 3.2.2. 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. 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 Tarski's undefinability theorem cannot 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 Tarski's undefinability theorem cannot blocking the biconditionals i  i  i  i , i  ,       ,etc. 3. 5. 17 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 (3.5.18) 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 subsection 4.2, Proposition 4.2.1).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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 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 subsection 4.2, Proposition 4.2.2). 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. IV.Generalized Löbs Theorem. IV.1.Generalized Löbs Theorem. Second-Order theories with Henkin semantics. Remark 4.1.1.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. 4. 1. 1 It well known 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 4. 1. 2 for some closed term t.Thus we can define PrThy xProvThx, y, 4. 1. 3 and therefore we obtain a predicate really asserting provability. Remark 4.1.2. (I)We note that it is not always the case that: Th  iff S PrThc, 4. 1. 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, 4. 1. 5 i.e. that is the case when predicate PrThy, y  MTh : PrThy xx  MThProvThx, y 4. 1. 6 really asserts provability. It well known 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: D1. Th  implies S PrThc, D2. S PrThc  PrThPrThcc, D3. S PrThc  PrTh  c  PrThc. 4. 1. 7 Conditions D1, D2 and D3 are called the Derivability Conditions. Remark 4.1.3.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. 4. 1. 8 Conditions D4, D5 and D6 are called the Strong Derivability Conditions. Definition4.1.1. Let  be well formed formula (wff) of Th. Then wff  is called Th-sentence iff it has no free variables. Designation 4.1.1.(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 4.1.2. (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 4.1.4. We emphasize that: (i) it is well known that there exist a ZFC-sentence ConZFC; MZFC [8],(ii) obviously there exists a ZFC2Hs-sentence Con ZFC2Hs; MZFC2 Hs and there exists a Z2Hs-sentence Con Z2Hs; MZ2 Hs . Designation 4.1.3.Assume that ConTh; MTh.Let ConTh; MTh be the formula: ConTh; MTh  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; MTh  t1t1  MTht2t2  MThProvTht1,  c  ProvTht2, negc 4. 1. 9 and where t1, t1 , t2, t2 is a closed term. Lemma 4.1.1. (I) Assume that: (i) a theory Th is recursively axiomatizable. (ii) ConTh; MTh, (iii) MTh  ConTh; MTh and (iv) Th PrThc,where  is a closed formula. Then Th  PrThc, (II) Assume that: (i) a theory Th is recursively axiomatizable. (ii) ConTh; MTh (iii) MTh  ConTh; MTh and (iv) Th PrTh  c, where  is a closed formula. Then Th  PrTh  c. Proof. (I) Let ConTh; MTh be the formula : ConTh; MTh  t1t1  MTht2t2  MThProvTht1, c  ProvTht2, negc, i.e. t1t1  MTht2t2  MThProvTht1, c  ProvTht2, negc t1t1  MTht2t2  MThProvTht1, c  ProvTht2, negc . 4. 1. 10 where t1, t2 is a closed term. From (i)-(ii) follows that theory Th ConTh; MTh is consistent. We note that Th ConTh; MTh ConTh; MTh for any closed . Suppose that Th PrThc, then (iii) gives Th PrThc  PrThc. 4. 1. 11 From ( 4.1.3) and ( 4.1.11) we obtain t1t2ProvTht1, c  ProvTht2, negc. 4. 1. 12 But the formula (4.1.10) contradicts the formula (4.1.12). Therefore Th  PrThc. Remark 4.1.5.In additional note that under the following conditions: (i) a theory Th is recursively axiomatizable, (ii) ConTh; MstTh, and (iii) MstTh  ConTh; MstTh predicate PrTh c really asserts provability, one obtains Th   . 4. 1. 13 and therefore by reductio ad absurdum again one obtains Th  PrThc. (II) Let ConTh; MTh be the formula : ConTh; MTh  t1t1  MTht2t2  MThProvTht1,  c  ProvTht2, negc, i.e. 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 . 4. 1. 14 This case is trivial becourse formula PrTh  c by the Strong Derivability Condition D4,see formulae ( 4.1.8),really asserts provability of the Th-sentence .But this is a contradiction. Lemma 4.1.2. (I) Assume that: (i) a theory Th is recursively axiomatizable. (ii) ConTh; MTh, (iii) MTh  ConTh and (iv) Th PrThc,where  is a closed formula.Then Th  PrThc, (II) Assume that: (i) a theory Th is recursively axiomatizable. (ii) ConTh; MTh (iii) MTh  ConTh and (iv) Th PrTh  c, where  is a closed formula.Then Th  PrTh  c. Proof. Similarly as Lemma 4.1.1 above. Example 4.1.1. (i) Let Th  PA be Peano arithmetic and   0  1. Assume that: (i) ConPA; MPA (ii) MPA  ConPA; MPA where MPA is a model of PA. Then obviously PA PrPA0  1 since PA 0  1 and therefore by Lemma 4.1.1 PA  PrPA0  1. (ii) Let ConPA; MPA , MPA  ConPA; MPA and let PA be a theory PA PA  ConPA; MPA and   0  1. Then obviously PA PrPA0  1  PrPA0  1. 4. 1. 15 and therefore PA PrPA0  1, 4. 1. 16 and PA PrPA0  1. 4. 1. 17 However by Löbs theorem PA  0  1. 4. 1. 18 (iii) Let ConPA; MPA   , MPA   ConPA; MPA   and   0  1. Then obviously PA PrPA0  1 since PA 0  1 and therefore by Lemma 4.1.1 we obtain. PA  PrPA0  1. Remark 4.1.6.Notice that there is no standard model of PA. Assumption 4.1.1. Let Th be a second order theory with 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 1 of the first order variables, i.e. a set of variables: 1  x, y, z, X, Y, Z,,, . . .

countable set 2 of the first order variables, i.e. a set of variables: 2  f0n, R0n, f1n, R1n, . . .

countable set of the n-ary function symbols: f0n, f1n, . . . countable set of the n-ary relation symbols: R0n, R1n, . . . connectives: , quantifier:. (ii) A theory Th is recursively axiomatizable. (iii) Th contains ZFC2Hs or ZFC or NF and ConTh; MTh is expressible in Th by a single statement of Th; (iv) Th has an -model MTh and MTh  ConTh; MTh;or (v) Th has an nonstandard model MNstTh  MNstTh PA  MstPA and MNstTh  ConTh; MNstTh . Definition 4.1.1. A Th-wff  (well-formed formula ) is closed, i.e.  is a sentence, i.e. 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 4.1.2.We will say that Th# is a nice theory or a nice extension of the Th iff the following properties holds: (i) Th # contains Th; (ii) Let  be any first order closed formula of Th, then Th PrThc implies Th # ; (iii) Let  be any first order closed formula of Th # , then MTh   implies Th # , i.e. ConTh  ; MTh implies Th # . (iv) Let  be any first order closed formula of Th # , thenformulae ConTh  ; MTh and ConTh #  ; MTh are expressible in Th # . Definition 4.1.3.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. Definition 4.1.4.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 4.1.7. 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 4.1.8. Notice that in order to prove the statements: (i) ConNF2Hs; MTh, (ii) ConNF; MTh the following Proposition 4.1.1 is necessary. Proposition 4.1.1.(Generalized Löbs Theorem). (I) Assume that: (i) A theory Th is recursively axiomatizable. (ii) Th is a second order theory with Henkin semantics. (iii) Th contains ZFC2Hs. (iv) Th has an -model MTh,and (v) the statement MTh is expressible by lenguage of Th as a single sentence of Th. (vi) MTh  ConTh; MTh,where predicate ConTh; MTh is defined by formula 4.1.9. 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 4.1.9. We emphasize that (v) valid for ZFC despite the fact that the axioms of ZFC are infinite, see [8] Chapter II,section 7,p.78. (II) Assume that: (i) A theory Th is recursively axiomatizable. (ii) Th is a first order theory. (iii) Th contains ZFC. (iv) Th has an -model MTh and (v) the statement MTh is expressible by lenguage of Th as a single sentence of Th. (vi) MTh  ConTh; MTh ,where predicate ConTh; MTh defined by formula 4.1.9, Then theory Th  Th MTh can be extended to a maximally consistent nice theory Th # . (III) Assume that: (i) A theory Th is recursively axiomatizable. (ii) Th is a first order theory. (iii) Th contains ZFC. (iv) Th has a nonstandard model MNstTh  MNstTh PA and (v) the statement MNstTh PA is expressible by lenguage of Th as a single sentence of Th. (vi) MNstTh  ConTh; MNstTh  ,where predicate ConTh; MNstTh  defined by formula 4.1.10. Then theory Th can be extended to a maximally consistent nice theory Th,Nst #  Th,Nst # MNstTh . Remark 4.1.10. We emphasize that (v) valid for ZFC despite the fact that the axioms of ZFC are infinite, see [8] Ch.II,section 7,p.78. Proof.(I) Let 1. . . i. . . be an enumeration of the all first order closed wff's of the theory Th (this can be achieved if the set of propositional variables,etc. 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 following statement (4.1.19) is satisfied Thi #  PrTh i#i  c  Thi #  PrTh i#i  c  MTh  i. 4. 1. 19 Note that Thi #  PrTh i#i  c  Thi #  i, Thi #  PrTh i#i  c  Thi #  i, 4. 1. 19. a since predicate PrTh i#i  c really aserts provability in Thi #. Then we define a theory Thi1 # as follows Thi1 #  Thi #  i . 4. 1. 19. b Remark 4.1.11.Note that the predicate PrTh i1# i  c is expressible in Thi1 # since a theory Thi1 # is an finite extension of the recursively axiomatizable theory Th. We will rewrite the conditions (4.1.19)-(4.1.19.b) using predicate PrTh i1# #   symbolically as follows: Thi1 # PrTh i1# # i c, PrTh i1# # 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1# # i c  PrTh i#i  c  PrTh i#i  c  ConThii; MTh, PrTh i1# # i c  PrTh i#i  c  PrTh i#i  c PrTh i1# i  c  Thi1 # i, Thi1 # PrTh i1# # i c  i. 4. 1. 20 (ii) Suppose that the following statement (2.2.21) is satisfied Thi #  PrTh i#i  c  Thi #  PrTh i#i  c  MTh  i. 4. 1. 21 Note that Thi #  PrTh i#i  c  Thi #  i, Thi #  PrTh i#i  c  Thi #  i, 4. 1. 21. a since predicate PrTh i#i  c really aserts provability in Thi #. Then we define a theory Thi1 # as follows Thi1 #  Thi #  i . 4. 1. 21. b We will rewrite the conditions (4.1.21)-(4.1.21.b) 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, Thi1 # PrTh i1# # i c  i. 4. 1. 22 (iii) Suppose that the following statement (4.1.23) is satisfied Thi # PrTh i#i  c 4. 1. 23 and therefore Thi # i   MTh  i .Then we define a theory Thi1 # as follows Thi1 #  Thi #. 4. 1. 24 Remark 4.1.12.Note 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; MTh  Thi1 # . (iv) Suppose that the following statement (4.1.25) is satisfied Thi # PrTh i#i  c 4. 1. 25 and therefore Thi #  i   MTh  i . Then we define theory Thi1 # as follows: Thi1 #  Thi #. 4. 1. 26 We define now a theory Th # as follows: Th #   i Thi #. 4. 1. 27 (1) First, notice that each Thi# is consistent. This is done by induction on i and by Lemmas 4.1.1-4.1.2. By assumption, the case is true when i  1.Now, suppose Thi # is consistent. Then its deductive closure DedThi # is also consistent. (2) If a statements (4.1.19)-(4.1.19.b) is satisfied,i.e. Thi1# PrTh i1# # i c and Thi1 # i, then clearly a theory Thi1 #  Thi #  i is consistent since it is a subset of closure DedThi1 # . (3) If a statements (4.1.21)-(4.1.21.b) 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 # . (4) If the statement (4.1.23) is satisfied,i.e. Thi# PrTh i#i  c then clearly Thi1 #  Thi # is consistent (5) If the statement (4.1.25) is satisfied,i.e. Thi# PrTh i#i  c then clearly Thi1 #  Thi # is consistent. (6) Next, notice DedTh#  is maximally consistent nice extension of the DedTh. DedTh #  is consistent because, by the standard Lemma 4.1.3 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. Definition 4.1.5.We define now predicate PrTh#  c really asserting provability in Th # by the following formula PrTh#  c  i  Thi # PrTh i# # c . 4. 1. 28 Proof.(II) and (III) similarly to (I). Lemma 4.1.3. The union of a chain  i|i   of consistent sets i, ordered by  is consistent. Definition 4.1.6.Let   x be one-place open Th-wff such that the following condition: Th  Th1 # !xx 4. 1. 29 is satisfied. Remark 4.1.13.We rewrite now the condition (4.1.28) using only the language of the theory Th1 # : Th1 # !xx  PrTh1#!xx c   PrTh1#!xx c  !xx . 4. 1. 30 Definition 4.1.7. We will say that, a set y is a Th1#-set if there exist one-place open wff x such that y  x. We will be write yTh1 #  iff y is a Th1 #-set. Remark 4.1.14. Note that yTh1 #    y  x  PrTh1#!xx c PrTh1#!xx c  !xx . 4. 1. 31 Definition 4.1.7.Let 1 be a set such that : x x  1 x is a Th1 #-set . 4. 1. 32 Proposition 4.1.2. 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 4. 1. 33 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, 4. 1. 34 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, . . . . 4. 1. 35 It is easy to prove that any set k  gk , k  1, 2, . . is a Th1 #-set.This is done by Gödel encoding (4.1.35), by the statament (4.1.33) and by axiom schemata of separation. 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 . 4. 1. 36 Let gn,k n k be a family of the sets gn,k n, k  1, 2, . . . .By the axiom of choice 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. Proposition 4.1.3. Any set 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. 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 . 4. 1. 37 We define now a set k such that k  k   gk , nn  gn,k  k  gn,k, vk 4. 1. 38 Obviously definitions (4.1.37) and (4.1.38) are equivalent. Definition 4.1.8.We define now the following Th1#-set 1  1 : x x  1  x  1  PrTh1#x  x c  . 4. 1. 39 Proposition 4.1.4. (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, (ii) follows immediately from countability of a set 1.Notice that 1 is nonempty countable set such that  1, because for any n   : Th1 # n  n. Proposition 4.1.5. A set 1 is inconsistent. Proof.From formula (4.1.39) we obtain Th1 # 1  1  PrTh1#1  1  c. 4. 1. 40 From (4.1.40) we obtain Th1 # 1  1  1  1 4. 1. 41 and therefore Th1 # 1  1  1  1. 4. 1. 42 But this is a contradiction. Definition 4.1.9. Let   x be one-place open Th-wff such that the following condition is satisfied: Thi # !xx 4. 1. 43 Remark 4.1.15.We rewrite now the condition (4.1.43) in the following equivalent form using only the lenguage of the theory Thi # : Thi # !xx  PrTh i#!xx c 4. 1. 44 Definition 4.1.10. We will say that, a set y is a Thi#-set if there exist one-place open wff x such that y  x. We will be write for short yThi #  iff y is a Thi #-set. Remark 4.1.16. Note that yThi #    y  x  PrTh i#!xx c . 4. 1. 45 Definition 4.1.11.Let i be a set such that : x x  i x is a Thi #-set . 4. 1. 46 Proposition 4.1.6. i is a Thi#-set. Proof. Let us consider an one-place open wff x such that conditions (4.1.43) 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 4. 1. 47 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. 4. 1. 48 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, . . . . 4. 1. 49 It is easy to prove that any collection k  gk , k  1, 2, . . is a Thi #-set.This is done by Gödel encoding, by the statament (4.1.43) and by the axiom schema of separation .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 . 4. 1. 50 Let gn,k n k be a family of the all sets gn,k n. By axiom of choice 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. Proposition 4.1.8. 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. 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 . 4. 1. 51 We define now a set k such that k  k   gk , nn  gn,k  k  ign,k, vk. 4. 1. 52 Obviously definitions (4.1.51) and (4.1.52) are equivalent. Definition 4.1.12.We define now the following Thi#-set i  i : x x  i  x  i  PrTh i#x  x c . 4. 1. 53 Proposition 4.1.9. (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. (ii) follows immediately from countability of a set i. Proposition 4.1.10. Any set i, i   is inconsistent. Proof.From the formula (4.1.53) we obtain Thi # i  i  PrTh i#i  i  c. 4. 1. 54 From the formla (2.66) we obtain Thi # i  i  i  i 4. 1. 55 and therefore Thi # i  i  i  i. 4. 1. 56 But this is a contradiction. Definition 4.1.13. A Th# -wff  that is: (i) Th-wff  or (ii) well-formed formula  which contains predicate PrTh#  c given by formula (4.1.28).An Th # -wff  (well-formed formula ) is closed i.e.  is a sentence if  has no free variables; a wff is open if it has free variables. Definition 4.1.14.Let   x be one-place open Th# -wff such that the following condition: Th # !xx 4. 1. 57 is satisfied. Remark 4.1.16.We rewrite now the condition (4.1.57) in the following equivalent form using only the lenguage of the theory Th # : Th # !xx  PrTh# !xx c 4. 1. 58 Definition 4.1.15.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 4.1.16. Let  be a set such that : x x   x is a Th# -set . Proposition 4.1.11. A set  is a Th# -set. Proof. Let us consider an one-place open wff x such that condition (4.1.57) 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 4. 1. 59 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. 4. 1. 60 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, . . . . 4. 1. 61 It is easy to prove that any set k  gk , k  1, 2, . . is a Th #-set.This is done by Gödel encoding and by axiom schema of separation. 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 . 4. 1. 62 Let gn,k n k be a family of the sets gn,k n, k  1, 2, . . . . By axiom of choice 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.Thus we can define Th # -set    : xx   x    PrTh# x  x c. 4. 1. 63 Proposition 4.1.12. 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. 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. 4. 1. 64 We define now a set k such that k  k   gk , nn  gn,k  k   gn,k, vk 4. 1. 65 Obviously definitions (4.1.64) and (4.1.65) are equivalent by Proposition 4.1.1. Proposition 4.1.13. (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 4.1.14. Set  is inconsistent. Proof.From the formula (4.1.63) we obtain Th #     PrTh#     c. 4. 1. 66 From (4.1.66) one obtains Th #        4. 1. 67 and therefore Th #       . 4. 1. 68 But this is a contradiction. Remark 4.1.17.Note that a contradictions mentioned above can be avoid using canonical Quinean approach,see subsection 3.6. IV.2.Proof of the inconsistensy of the set theory ZFC2 Hs  MZFC2 Hsusing 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 sets 1,2, . . . ,  and corresponding inconsistent sets 1,2, . . . ,. Remark 4.2.1.Note that a contradiction mentioned above is a strictly stronger then contradictions derived in subsection 4.1, and these contradiction impossible avoid by using Quinean approach,see subsection 3.6. Proposition 4.2.1.(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 holds. Assume that Th Hs has an standard Model MZFC2 Hs . Then there is no -formula Truen such that for every -formula A such that MZFC2 Hs  A, the following equivalence holds MZFC2 Hs  A  TruegA. 4. 2. 1 Proof.The diagonal lemma yields a counterexample to this equivalence, by giving a "Liar" sentence S such that S  TruegS holds. Remark 4.2.2. Above we has been defined the set (see Definition 4.1.63) in fact using generalized truth predicate True#  c such that True#  c  PrTh#  c. 4. 2. 2 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 biconditional (4.2.3) is correct, i.e.,for every first order ZFC2Hs-formula  such that MZFC2 Hs   and the following equivalence holds MZFC2 Hs    PrTh#  c, 4. 2. 3 where PrTh#  c  . Proposition 4.2.2.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 # . 4. 2. 4 Therefore (4.2.2) gives generalized "truth predicate" for set theory Th# .By Proposition 4.2.1 one obtains a contradiction. Remark 4.2.3.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 4.2.5.Under ZFC it can be shown that κ is inaccessible if and only if Vκ, is a model of second order ZFC, [5]. Remark 4.2.6. 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. IV.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 4.3.1.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 4.3.2.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 4.3.3.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 4.3.1. 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 4.3.4.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 4.3.5.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 4.3.2. 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. 4. 3. 1 Theorem 4.3.1.[12].(Löb's Theorem for ZFC2 fss) Let  be any closed formula with code y  c  MZ2 , then Th PrThc implies Th . 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 4.3.3. Let   x be one-place open wff such that: Th !xx 4. 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 4.3.2. Note that yTh  y  x  PrTh!xxc 4. 3. 3 Definition 4.3.4. Let  be a collection such that : x x   x is a Th-set . Proposition 4.3.1. Collection  is a Th-set. Definition 4.3.4. We define now a Th-set c   : xx  c x    PrThx  xc. 4. 3. 4 Proposition 4.3.2. (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 4.3.3. A set c is inconsistent. Proof.From formla (4.3.4) one obtains Th c  c  PrThc  c c. 4. 3. 5 From formula (4.3.4) and definition 4.3.5 one obtains Th c  c  c  c 4. 3. 6 and therefore Th c  c  c  c. 4. 3. 7 But this is a contradiction. Thus finally we obtain: Theorem 4.3.2.[5].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 [12].Thus finally we obtain. Theorem 4.3.3.[5],[6].ConZFC  MstZFCMstZFC  Hk. 5.Discussion. How can we safe the set theory ZFC  MstZFC. 5.1.The set theory ZFCw with a weakened axiom of infinity We remind that a major part of modern mathematical analysis and related areas based not only on set theory ZFC but on strictly stronger set theory: ZFC  MstZFC. In order to avoid difficultnes which arises from ConZFC  MstZFC in this subsection we introduce set theory ZFCw with a weakened axiom of infinity. Without loss of generality we consider second-order arithmetic 2 with an restricted induction schema. Second-order arithmetic 2 includes, but is significantly stronger than, its first-order counterpart Peano arithmetic. Unlike Peano arithmetic, second-order arithmetic allows quantification over sets of natural numbers as well as numbers themselves. Because real numbers can be represented as (infinite) sets of natural numbers in well-known ways, and because second order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in second-order arithmetic. For this reason, second-order arithmetic is sometimes called "analysis". Induction schema of second-order arithmetic 2. If φn is a formula of second-order arithmetic 2 with a free number variable n and possible other free number or set variables (written m and X), the induction axiom for φ is the axiom: mXφ0  nφn  φn  1  nφn. 5. 1. 1 The (full) second-order induction scheme consists of all instances of this axiom, over all second-order formulas. One particularly important instance of the induction scheme is when φ is the formula "n  X " expressing the fact that n is a member of X (X being a free set variable): in this case, the induction axiom for φ is X0  X  nn  X  n  1  X  nn  X. 5. 1. 2 This sentence is called the second-order induction axiom. Comprehension schema of second-order arithmetic 2 If φn is a formula with a free variable n and possibly other free variables, but not the variable Z, the comprehension axiom for φ is the formula Znn  Z φn. 5. 1. 3 This axiom makes it possible to form the set Z  n|φn of natural numbers satisfying φn. There is a technical restriction that the formula φ may not contain the variable Z. Designation 5.1.1.Let Wffk2 be a set of the all k-place open wff's of the second-order arithmetic 2 and let k2 be a set of the all primitive recursve k-place open wff's k of the second-order arithmetic 2.Let k2 be a set of the all k-place open wff's k of the second-order arithmetic 2 such that k  k2  k2  Wffk2. 5. 1. 4 Let Wff1,X be a set of the all sets definable by 1-place open wff's X  Wff1,X2, let 1 be a set of the all sets definable by 1-place open wff's 1X  12 and let 1be a set of the all sets definable by 1-place open wff's 1X  12 Restricted induction schema of second-order arithmetic 2. If φkn  k  2 is a formula of second-order arithmetic 2 with a free number variable n and possible other free number and set variables (written m and X), the induction axiom for φ is the axiom: mX X  1 φk0  nφkn  φkn  1  nφkn. 5. 1. 5 The restricted second-order induction scheme consists of all instances of this axiom, over all second-order formulas. One particularly important instance of the induction scheme is when φk   is the formula n  X  X  1 expressing the fact that n is a member of X and X  1 (X being a free set variable): in this case, the induction axiom for φk is X X  1 0  X  nn  X  n  1  X  nn  X. 5. 1. 6 Restricted comprehension schema of second-order arithmetic 2 k . If φ1n  1 is a formula with a free variable n and possibly other free variables, but not the variable Z, the comprehension axiom for φ1 is the formula Znn  Z φ1n. 5. 1. 7 Remark 5.1.1.Let  2 k be a theory 2 k  Mst 2 k where Mst 2 k is an standard model of 2. We assume now that Con 2 k  Mst 2 k . 5. 1. 8 Definition 5.1.1. Let gx :    be any real analytic function such that: (i) gx   n0  anxn, |x| r, 5. 1. 9 where nan   and where (ii) the sequence an n  Mst2   (in particular an n  Mst 2 1 if the sequence an nis primitive recursive . Then we will call any function given by Eq.(5.1.9) -analytic -function and denoted such functions by g x. In particular we will call any function g 1x constructive -analytic function. Definition 5.1.2. A transcendental number z   is called -transcendental number over field , if there does not exist -analytic -function g x such that g z  0. In particular a transcendental number z   is called #-transcendental number over field , if there does not exist constructive -analytic function g 1x such that gz  0, i.e. for every constructive -analytic function g 1x the inequality g 1z  0 is satisfied. Example 5.1.1. Number  is transcendental but number  is not #-transcendental number over field since (1) function sin x is a -analytic and (2) sin  2  1, i.e. 1   2   3 233!   5 255!   7 277! . . . 12n12n1 22n12n  1! . . . 0. 5. 1. 10 Remark 5.1.5.Note that a sequence an  12n1 22n12n  1! , n  0, 1, 2. . . . . obviously is primitive recursive and therefore an n  Mst 2 1 , 5. 1. 11 since we assume Con 2 1  Mst 2 1 . Propostion 5.1.1.Let 0  1. For each n  0 choose an rational number n inductively such that 1 k1 n1 kek  n!1 nen 1 k1 n1 kek. 5. 1. 12 The rational number n exists because the rational numbers are dense in . Now the power series fx  1 n1  nen has the radius of convergence  and fe  0.However any sequence n n obviously is not primitive recursive and therefore n n  Mst 2 1 . 5. 1. 13 Theorem 5.1.1.[21] Assume that Con 2 1  Mst 2 1 .Then number e is #-transcendental over the field . Theorem 5.1.2.[21] Number ee is transcendental over the field . Proof.Immediately from Theorem 5.1.2. Theorem 5.1.3.[21] Assume that Con2  Mst2 .Then number e is -transcendental over the field . 5.2.The set theory ZFC# with a nonstandard axiom of infinity We remind that a major part of modern set theory involves the study of different models of ZF and ZFC. It is crucial for the study of such models to know which properties of a set are absolute to different models [8]. It is common to begin with a fixed model of set theory and only consider other transitive models containing the same ordinals as the fixed model. Certain fundamental properties are absolute to all transitive models of set theory, including the following:(i) x is the empty set, (ii) x is an ordinal,(iii) x is a finite ordinal,(iv) x  ω, (v) x is (the graph of) a function. Other properties, such as countability, x  2y are not absolute, see [8]. Remark 5.2.1.Note that for nontransitive models the properties (ii)-(v) no longer holds. Let M, be a non standard model of ZFC. It follows from consideration above that any such model M, is substantially non standard model of ZFC, i.e., there does not exist an standard model Mst, of ZFC such that Mst M where  |Mst  |Mst. 5. 2. 1 and   M, . 5. 2. 2 Theorem 5.2.1.[9].Let M, be a non standard model of ZF.A necessary and sufficient condition for M, to be isomorfic to a standard model M, is that there does not exist a countable sequence xn n of elements in M such that xn1 xn. Definition 5.2.1.Let MNst  M,  be a non standard model of ZFC.We will say that: (i) element zM is a non standard relative to  and abraviate Nstz, if there exist a sequence xn nof elements in M such that xn1 xn and z  x0 ,and (ii) element zMNst is a standard standard relative to  and abraviate stz if there does not exist a countable sequence xn n of elements in M such that xn1 xn and z  x0, i.e., stz  Nstz. Remark 5.2.2.We denote by ZFC set theory which is obtained from set theory ZFC by using wff's of ZFC with quantifiers bounded on a non standard model M,. The firstorder lenguage corresponding to set theory ZFC we denote by . Let WffZFC  be a set of the all wff's of ZFC.Note that stz, Nstz  WffZFC , i.e., predicates stz and Nstz are not well defined in ZFC since   MNst. Definition 5.2.2.In set theory, an ordinal number α is an admissible ordinal if Lα is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, α is admissible when α is a limit ordinal and Lα  Σ0-collection. Definition 5.2.3.Let M, be a non standard model of ZF.Assume that ordinal of M,  have a largest minimal segment isomorfic to some standard ordinal   M,which is called the standard part of M,,see ref.[14]-[15]. We shall assume that  M,and that for  :  |M	  |M	, 5. 2. 3 where M	 is the set of all elements of M with M rank is less then . Which standard ordinal  can be standard part of M,?It well-known that a necessary condition is that  is admissible ordinal. A well-known Friedman theorem (see ref.[14]- [15]) implies that for countable  the admissibility is also sufficient condition. Thus there is no admissible countable ordinal  in any non standard model of ZFC. Remark 5.2.3.We introduce now in consideration an conservative extension of the theory ZFC by adding to lenguage  the atomic predicate Nstz which satisfies the following condition zNstz  xxz  Nstx. 5. 2. 4 1. Axioms of non standartness (a) There exists at least one non standard set zNstz. 5. 2. 5 (b) There exists at least one non standard transitive set zNstz  TRz, 5. 2. 6 where: TRz  xxz  x  z. 2. Axiom of extensionality xyzzx  zy  x  y. 5. 2. 7 3. Axiom of regularity xx  yyx  zzy  zx. 5. 2. 8 4. Axiom schema of specification Let φst be any formula in the language of ZFC such that (i) formula st free from occurrence of the atomic predicate Nstz, i.e., st can not contain the atomic predicate Nstz and (ii) st is a formula with all free variables among x, z, w1, , wn (y is not free in φst ). Then: zw1wnyxx y  xz  φstx, z, w1, , wn. 5. 2. 9 4. Axiom of empty set xy  y x . 5. 2. 10 We will denote the empty set by . 5. Axiom of pairing xyzxz  yz. 5. 2. 11 6. Axiom of union AYx x Y  Y   x A . 5. 2. 12 7. Axiom schema of replacement The axiom schema of replacement asserts that the image of a set under any definable in ZFC function will also fall inside a set. Let φst be any formula in the language of ZFC such that (i) formula st free from occurrence of the atomic predicate Nstz, i.e., st can not contain the atomic predicate Nstz and (ii) φst is a formula whose free variables are among x, y, A, w1, , wn, so that in particular B is not free in φst. Then: Aw1wn x x A  !yφstA, w1, , wn, x, y  Bx x A  yyB  φstA, w1, , wn, x, y . 5. 2. 13 8. Axiom of infinity Let Sx abbreviate x  x ,where w is some set.Then: I I  xI x  x  I . 5. 2. 14 Such a set as usually called an inductive set. Definition 5.2.4. We will say that x is a non standard set and abraviate xNst iff x contain at least one non standard element,i.e., xNst  x  Nst. 5. 2. 15 Remark 5.2.4.It follows from Axiom schema of specification and Axiom schema of replacement (5.2.13) we can not extract from a non standard set the standard and non standard elements separately,i.e. for any non standard set xNst there is no exist a set y and z such that xNst  y  z, 5. 2. 16 where y contain only standard sets and z contain only standard sets! As it follows from Theorem 5.3.1 any inductive set is a non standard set. Thus Axiom of infinity can be written in the following form 8. Axiom of infinity Let Sx abbreviate x  x ,where w is some set.Then: INst INst  xINst x  x  INst . 5. 2. 17 Such a set as usually called a non standard inductive set. 9. Strong axiom of infinity Let Sx abbreviate x  x ,where w is some set.Then: INst TRINst  INst  xINst x  x  INst . 5. 2. 18 5.3.Extracting the standard and non standard natural numbers from the infinite non standard set INst. Definition 5.3.1. We will say that xNst is inductive if there is an formula x of ZFC that says: 'xNst is -inductive'; i.e. xNst   xNst  y y  xNst  Sy  xNst . 5. 3. 1 Thus we wish to prove the existence of a unique non standard set W Nst such that x xW Nst  INstINst  xINst . 5. 3. 2 (1) For existence, we will use the Axiom of Infinity combined with the Axiom schema of specification. Let INst be an inductive (non standard) set guaranteed by the Axiom of Infinity. Then we use the Axiom Schema of Specification to define our set W Nst  x  INst : JNstΦJNst  xJNst , 5. 3. 3 i.e. W Nst is the set of all elements of INst which happen also to be elements of every other inductive set. This clearly satisfies the hypothesis of (5.3.2), since if xW Nst , then x is in every inductive set, and if x is in every inductive set, it is in particular in INst, so it must also be in W Nst . (2) For uniqueness, first note that any set which satisfies (5.3.2) is itself inductive, since is in all inductive sets, and if an element x is in all inductive sets, then by the inductive property so is its successor. Thus if there were another set W1 Nst which satisfied (5.3.2) we would have that W1 Nst  W Nst since W is inductive, and W Nst  W1 Nst since W1 Nst is inductive. Thus W1 Nst  W Nst . Let ω denote this unique set. (3) For non stardarntness we assume that ω is a standard set, i.e. there is no nonstandard element in ω.Then ω   where  isomorphic to , but this is a contradiction, since   M, . Theorem 5.3.1.There exist unique non stardard set ω such that (5.3.2) holds,i.e. x x  ω  INstINst  xINst . 5. 3. 4 Definition 5.3.2. We will say that a set S is -finite if every surjective -function from S onto itself is one-to-one. Theorem 5.3.2. There exist -finite non standard natural numbers in ω. Proof. Assuming that any non standard natural number is not -finite one obviously obtains a contradiction. Remark 5.3.1.Assuming that ω is standard set then this method mentioned above produce system which satisfy the axioms of second-order arithmetic Z2 fss, since the axiom of power set allows us to quantify over the power set of ω , as in second-order logic. Thus it completely determine isomorphic systems, and since they are isomorphic under the identity map, they must in fact be equal. 6.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  MstZFCMstZFC  Hk.This result also was obtained in [3],[4],[5] essentially another approach. For the first time this result has been declared to AMS in [22],[23]. 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] J. Foukzon,Generalized Lob's Theorem.Strong Reflection Principles and Large Cardinal Axioms. Consistency Results in Topology.2013 arXiv:1301.5340 [math.GM] https://arxiv.org/abs/1301.5340 [3] R.Lemhoff,The Bulletin of Symbolic Logic Vol. 23, No. 2, pp. 213-266 COLLOQUIUM '16, Leeds, UK, July 31-August 6, 2016 J. Foukzon Contributed talks, J. Foukzon, Inconsistent countable set in second order ZFC and unexistence of the strongly inaccessible cardinals. https://www.jstor.org/stable/44259451?seq1#page_scan_tab_contents [4] 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 [5] J. Foukzon,Inconsistent Countable Set in Second Order ZFC and Nonexistence of the Strongly Inaccessible Cardinals, British Journal of Mathematics & Computer Science,Vol.: 9, Issue.: 5,DOI : 10.9734/BJMCS/2015/16849 http://www.sciencedomain.org/abstract/9622 [6] J. Foukzon and E. Men'kova,There is No Standard Model of ZFC and ZFC2 Journal of Advances in Mathematics and Computer Science,2018 Volume 26 Issue 2, pp.1-20.Published Jan 30, 2018. DOI: 10.9734/JAMCS/2018/38773 [7] L. Henkin, "Completeness in the theory of types". Journal of Symbolic Logic15(2): 81-91. doi:10.2307/2266967. JSTOR 2266967 [8] P. Cohen, Set Theory and the continuum hypothesis.Reprint of the W. A. Benjamin, Inc.,New York,1966 edition. ISBN-13: 978-0486469218 [9] K. Gödel,Consistency of the Continuum Hypothesis. (AM-3),Series: Annals of Mathematics Studies Copyright Date: 1968 Publishedby: Princeton University Press Pages: 69. [10] M. Rossberg, "First-Order Logic, Second-Order Logic, and Completeness". In V.Hendricks et al., eds. First-order logic revisited. Berlin: Logos-Verlag. [11] S. Shapiro, Foundations without Foundationalism: A Case for Second-order Logic. Oxford University Press. ISBN 0-19-825029-0 [12] A. Rayo and G. Uzquiano,Toward a Theory of Second-Order Consequence, Notre Dame Journal of Formal Logic Volume 40, Number 3, Summer 1999. [13] J. Vaananen, Second-Order Logic and Foundations of Mathematics, The Bulletin of Symbolic Logic, Vol.7, No. 4 (Dec., 2001), pp. 504-520. [14] H. Friedman, Countable models of set theories, Cambridge Summer School in Mathematical Logic, 1971, Lecture Notes in Mathematics, Springer, 337 (1973), 539–573 [15] 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 [16] 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. [17] E. Mendelson,Introduction to mathematical logic.June1,1997. ISBN-10: 0412808307. ISBN-13: 978-0412808302 [18] G. Takeuti,Proof Theory: Second Edition (Dover Books on Mathematics) 2013 ISBN-13: 978-0486490731; ISBN-10: 0486490734 [19] Quine, W. V. (1937). New Foundations for Mathematical Logic. The American Mathematical Monthly, Mathematical Association of America, 44 (2),70-80. (doi:10.2307/2300564, JSTOR 2300564). [20] Hailperin, T. (1944). A set of axioms for logic. Journal of Symbolic Logic, 9,1-19. [21] J. Foukzon, Non-Archimedean Analysis on the Extended Hyperreal Line d and the Solution of Some Very Old Transcendence Conjectures over the Field . Advances in Pure Mathematics Vol.5 No.10,42 pp. Paper ID 58868, DOI: 10.4236/apm.2015.510056 [22] 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 [23] 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 #