Chapter 3 Print ISBN: 978-81-934224-1-0, eBook ISBN: 978-93-89246-18-6 There is No Standard Model of ZFC and ZFC2 Jaykov Foukzon1 ∗ and Elena Men'kova2 DOI:10.9734/bpi/amacs/v1 ABSTRACT In this Chapter we obtain a contradictions in formal set theories under assumption that these theories have omega-models or nonstandard model with standard part. An posible generalization of Lob's theorem is considered. Main results are: (i) ¬Con(ZFC + ∃MZFCst ), (ii) ¬Con(NF + ∃MNFst ), (iii) ¬Con(ZFC2), (iv) let k be an inaccessible cardinal then ¬Con(ZFC + ∃κ), (v) ¬Con(ZFC + (V = L)), (vi) ¬Con(ZF + (V = L)). Keywords: Gödel encoding; Russell's paradox; standard model; Henkin semantics; inaccessible cardinal. 2010 Mathematics Subject Classification: 53C25; 83C05; 57N16. 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 inconsistent, that fact would have been uncovered by now. This much is certain-ZFC and ZFC2 is immune to the classic paradoxes of naive set theory: Russell's paradox, the Burali-Forti paradox, and Cantor's paradox. Remark 1.1.1. The inconsistency of the second order set theory ZFC2082 originally have been uncovered in [2] and officially announced in [3], see also ref. [4], [5], [6]. 1Israel Institute of Technology, Haifa, Israel. 2All-Russian Research Institute for Optical and Physical Measurements, Moscow, Russia. *Corresponding author: E-mail: jaykovfoukzon@list.ru; Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 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 443 for which there is a formula A (z, t1, ..., tk) in ZF such that if AX denotes A with all bound variables restricted toX, then for some ti, i = 1, ..., k. inX, 443 = { z ∈ X | AX ( z, t1, ..., tk )} . (1.1.1) Observe X ′ j 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 conditionA ( 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 ZFC Hs 2 and that Y = X ′ is a single formula A(X,Y ) in ZFCHs2 . (ii)We will denote the set Y of all sets 443 definable relative to a given set X by Y , IHs2 . Definition 1.1.3. Let RHs2 be a set of the all sets definable relative to a given setX by the first order 1-place open wff's and such that ∀x ( x ∈ IHs2 ) [ x ∈ RHs2 ⇐⇒ x /∈ x ] . (1.1.3) Remark 1.1.5.(a) Note that RHs2 ∈ IHs2 since RHs2 is a set definable by the first order 1-place open wff Ψ ( Z,IHs2 ) : Ψ ( Z,IHs2 ) , ∀x ( x ∈ IHs2 ) [x ∈ Z ⇐⇒ x /∈ x] , (1.1.4) Theorem 1.1.1. [6].Set theory ZFCHs2 is inconsistent. Proof. From (1.1.3) and Remark 1.1.2 one obtains RHs2 ∈ RHs2 ⇐⇒ RHs2 /∈ RHs2 . (1.1.5) From (1.1.5) one obtains a contradiction ( RHs2 ∈ RHs2 ) ∧ ( RHs2 /∈ RHs2 ) . (1.1.6) Remark 1.1.6. Note that in paper [6] we dealing by using following definability condition: a set 443 is definable if there is a formula A (z) in ZFC such that 27 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 ∃z [A (z) ∧ y = z] . (1.1.7) Obviously in this case a set Y = RHs2 is a countable set. Definition 1.1.4. Let RHs2 be the countable set of the all sets definable by the first order 1-place open wff's and such that ∀x ( x ∈ IHs2 ) [ x ∈ RHs2 ⇐⇒ x /∈ x ] . (1.1.8) Remark 1.1.7. (a) Note that RHs2 ∈ IHs2 since RHs2 is a ZFC-set definable by the first order 1-place open wff Ψ ( Z,IHs2 ) , ∀x ( x ∈ IHs2 ) [x ∈ Z ⇐⇒ x /∈ x] , (1.1.9) one obtains a contradiction ( RHs2 ∈ RHs2 ) ∧ ( RHs2 /∈ RHs2 ) . In this paper we dealing by using following definability condition. Definition 1.1.5.(i) Let Mst = M ZFC st 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 M ′st, 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], [8], [9], [10], [11], [12], [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: 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⟩ |= ∃XΦ(X) ⇐⇒ ∃S (S ∈ S) [⟨U,S⟩ |= Φ(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 secondorder 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 28 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 Henkin models [7], [8], [9], [10], [11], [12], [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. 4. 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 Z2 such that the semiring N of natural numbers is the only full model of Z2 up to isomorphism. 8. There is a finite second-order axiom system RCF2 such that the field R 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∃ν∀r [r ∈ ν ⇐⇒ ∃s (s ∈ u ∧ (s, r) ∈ X)]] , (1.1.12) whereX 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 Ur logic [13]. Designation 1.1.1. We will denote: (i) by ZFCHs2 set theory ZFC2 with the Henkin semantics, (ii) by ZFCfss2 set theory ZFC2 with the full second-order semantics,(iii) by ZFC Hs 2 set theory ZFCHs2 + ∃M ZFCHs2 st and (iv) by ZFCst set theory ZFC + ∃MZFCst , where MThst is a standard model of the theory Th. Remark 1.1.12. There is no completeness theorem for second-order logic with the full secondorder semantics. Nor do the axioms of ZFCfss2 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 MZFC fss 2 of ZFCfss2 [11]. Let Z be the conjunction of all the axioms of ZFCfss2 . We assume now that: Z is true,i.e. Con ( ZFCfss2 ) . 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 3ba is a strongly inaccessible if and only if (H3ba,∈) is a model of ZFCfss2 . Thus ¬Con(ZFCfss2 ) =⇒ ¬Con(ZFC + ∃κ)). (1.1.13) In this paper we prove that: (i) ZFCst , ZFC + ∃MZFCst (ii) ZFC Hs 2 , ZFCHs2 + ∃M ZFCHs2 st and (iii) ZFC fss 2 is inconsistent, where MThst is a standard model of the theory Th. Axiom ∃MZFC . [8]. There is a set MZFC and a binary relation ε ⊆ MZFC ×MZFC which makes 29 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 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 ZFCHs2 such that ∃MZFC Hs 2 ⇐⇒ Con ( ZFCHs2 ) . We denote this statement throught all this paper by symbol Con ( ZFCHs2 ;M ZFCHs2 ) and there exists a single statement ∃MZ Hs 2 in ZHs2 . We denote this statement throught all this paper by symbol Con ( ZHs2 ;M ZHs2 ) . Axiom ∃MZFCst . [8].There is a set MZFCst such that if R is{ ⟨x, y⟩ |x ∈ y ∧ x ∈ MZFCst ∧ y ∈ MZFCst } then MZFCst is a model for ZFC under the relation R. Definition 1.1.6. [8].The model MZFCst 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 ∃MZFCst ,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(ZFCHs2 ) from Con(ZFC Hs 2 ), (ii)  Con(ZFC) from Con(ZFC),by using Gödel encoding, one needs something more than the consistency of ZFCHs2 , e.g., that ZFC Hs 2 has an omega-model M ZFCHs2 ω or an standard model M ZFCHs2 st i.e., a model in which the integers are the standard integers and the all wff of ZFC Hs 2 , ZFC,etc. represented by standard objects.To put it another way, why should we believe a statement just because there's a ZFCHs2 -proof of it? It's clear that if ZFC Hs 2 is inconsistent, then we won't believe ZFCHs2 -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 ZFCHs2 ; we must also believe that these arithmetical theorems are asserting something about the standard naturals. It is "conceivable" that ZFCHs2 might be consistent but that the only nonstandard models M ZFCHs2 Nst it has are those in which the integers are nonstandard, in which case we might not "believe" an arithmetical statement such as "ZFCHs2 is inconsistent" even if there is a ZFC Hs 2 -proof of it. Remark 1.1.17. Note that assumption ∃MZFC Hs 2 st is not necessary if nonstandard model M ZFCHs2 Nst 30 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 is a transtive or has an standard part M ZHs2 st ⊂ M ZHs2 Nst ,see [14], [15]. Remark 1.1.18. Remind that if M is a transitive model, then 3c9M is the standard 3c9. 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 M ZHs2 Nst of the second-order arithmetic Z Hs 2 the terms 0, S0 = 1,SS0 = 2, . . . comprise the initial segment isomorphic to M ZHs2 st ⊂ M ZHs2 Nst . This initial segment is called the standard cut of the M ZHs2 Nst . The order type of any nonstandard model of M ZHs2 Nst is equal to N+A× Z,see ref. [16], for some linear order A Thus one can to choose Gödel encoding inside the standard model M ZHs2 st . 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 Z fss 2 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 3c9-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 3c9-model. The unique full omega-model M Z fss 2 ω , 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. 1.2 Remarks on the Tarski's Undefinability Theorem Theorem 1.2.1.(Tarski's undefinability theorem) LetTh 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 MThst and Con (Th,st) where Th,st , Th + ∃MThst . (1.2.1) Let T ∗be the set of Gödel numbers of -sentences true in MThst . 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]MThst (1.2.2) where the abbraviation [A] M Th st means that A holds in standard model MThst ,i.e. [A]MThst ⇐⇒ |= M Th st A.Thus Con (Th,st) implies that ¬∃True(x) ( True(g(A)) ⇐⇒ [A] M Th st ) (1.2.3) 31 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 Thus Tarski's undefinability theorem reads Con (Th,st) =⇒ ¬∃True(x) ( True(g(A)) ⇐⇒ [A] M Th st ) . (1.2.4) Remark 1.2.2. By the other hand the Theorem 1.2.1 says that given some 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. Remark 1.2.3. In this paper under the following assumption Con ( ZFC + ∃MZFCst ) (1.2.5) in particular we prove that there exists countable Rassel's set Rω such that the following statement holds: ZFC + ∃MZFCst ⊢ ∃Rω ( Rω ∈ MZFCst ) ∧ (card (Rω) = א0) ∧ [ |=MZFCst ∀x (x ∈ Rω ⇐⇒ x /∈ x) ] . (1.2.6) From (1.2.6) immediately follows a contradiction |=MZFCst (Rω ∈ Rω) ∧ (Rω /∈ Rω) . (1.2.7) From (1.2.5) and (1.2.7) by reductio ad absurdum it follows ¬Con ( ZFC + ∃MZFCst ) . (1.2.8) Remark 1.2.4. It follows from (1.2.8) that Tarski's undefinability theorem (Theorem 1.2.1) obviously no longer holds. Definition 1.2.1. 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 1.2.2.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 1.2.2. [4], [5]. Assume that:Con (ZFCst) ,where ZFCst , ZFC + ∃MZFCst .Then there exists completion ZFC#st of the theory ZFCst such that the following condtions holds: (i) For every formula A in the language of ZFCst that formula [A]MZFCst or formula [¬A]MZFCst is provable in ZFC#st i.e., for any wff A, always ZFC # st ⊢ [A]MZFCst or ZFC # st ⊢ [¬A]MZFCst . (ii) ZFC#st = ∪m∈NThm,where for any m a theory Thm+1 is finite extension of the theory Thm. (iii) Let Prstm (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 Pr st m (y, x) is expressible in the theory Thm by canonical Gödel encoding and really asserts provability in Thm. (iv) Let Pr#st (y, x) be relation such that: y is a Gödel number of a proof of the wff of the theory ZFC#st and x is a Gödel number of this wff.Then the relation Pr # st (y, x) 32 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 is expressible in the theory ZFC#st by the following formula Pr#st (y, x) ⇐⇒ ∃mPrstm (y, x)) (1.2.9) (v) The predicate Pr#st (y, x) really asserts provability in the set theory ZFC # st . Remark 1.2.5.Note that the relation Prstm (y, x) is expressible in the theory Thm since a theory Thm is an finite extension of the recursively axiomatizable theory ZFCst and therefore the predicate Prstm (y, x) exists since any theory Thm is recursively axiomatizable. Remark 1.2.6. Note that a theory ZFC#st obviously is not recursively axiomatizable nevertheless Gödel encoding holds by Theorem 1.2.2. Theorem 1.2.3. Assume that:Con (ZFCst) ,where ZFCst , ZFC+∃MZFCst .Then truth predicate True(n) is expressible by using first order language by the following formula True(g(A)) ⇐⇒ ∃y (y ∈ N) ∃m (m ∈ N) Prstm (y, g(A)) . (1.2.10) Proof.Assume that: ZFC#st ⊢ [A]MZFCst . (1.2.11) It follows from (1.2.11) there exists m∗ = m∗ (g(A)) such that Thm∗ ⊢ [A]MZFCst and therefore by (1.2.9) we obtain Pr#st (y, g(A)) ⇐⇒ Prstm∗ (y, g(A))) . (1.2.12) From (1.2.12) immediately by definitions one obtains (1.2.10). Remark 1.2.7. Note that Theorem 1.2.3 reads Con (ZFCst) =⇒ ∃True(x) ( True(g(A)) ⇐⇒ [A]MZFCst ) . (1.2.13) Theorem 1.2.4. ¬Con (ZFCst) . Proof.Assume that: Con (ZFCst) .From (1.2.10) and (1.2.13) one obtains a condradiction Con (ZFCst)∧ ¬Con (ZFCst) and therefore by reductio ad absurdum it follows ¬Con (ZFCst) . Theorem 1.2.5. [4], [5]. Let MZFCNst be a nonstandard model of ZFC and let M PA st be a standard model of PA.We assume now that MPAst ⊂ MZFCNst and denote such nonstandard model of the set theory ZFC by MZFCNst = M ZFC Nst [PA] .Let ZFCNst be the theory ZFCNst = ZFC +M ZFC Nst [PA] . Assume that:Con (ZFCNst) ,where ZFCst , ZFC+∃MZFCNst .Then there exists completion ZFC#Nst of the theory ZFCNst such that the following condtions holds: (i) For every formula A in the language of ZFC that formula [A]MZFC Nst or formula [¬A]MZFC Nst is provable in ZFC#Nst i.e., for any wff A, always ZFC # Nst ⊢ [A]MZFC Nst or ZFC#Nst ⊢ [¬A]MZFC Nst . (ii) ZFC#Nst = ∪m∈NThm,where for any m a theory Thm+1 is finite extension of the theory Thm. (iii) Let PrNstm (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 expressible in the theory Thm by canonical Gödel encoding and really asserts provability in Thm. (iv) Let Pr#Nst (y, x) be relation such that: y is a Gödel number of a proof of the wff of the theory 33 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 ZFC#Nst and x is a Gödel number of this wff.Then the relation Pr # Nst (y, x) is expressible in the theory ZFC#Nst by the following formula Pr#Nst (y, x) ⇐⇒ ∃m ( m ∈ MPAst ) PrNstm (y, x)) (1.2.14) (v) The predicate Pr#Nst (y, x) really asserts provability in the set theory ZFC # Nst. Remark 1.2.8.Note that the relation PrNstm (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 PrNstm (y, x) exists since any theory Thm is recursively axiomatizable. Remark 1.2.9. Note that a theory ZFC#Nst obviously is not recursively axiomatizable nevertheless Gödel encoding holds by Theorem 1.2.5. Theorem 1.2.6. Assume that:Con (ZFCNst) ,where ZFCNst , ZFC + ∃MZFCNst .Then truth predicate True(n) is expressible by using first order language by the following formula True(g(A)) ⇐⇒ ∃y ( y ∈ MPANst ) ∃m ( m ∈ MPANst ) PrNstm (y, g(A)) . (1.2.15) Proof.Assume that: ZFC#Nst ⊢ [A]MZFC Nst . (1.2.16) It follows from (1.2.14) there exists m∗ = m∗ (g(A)) such that Thm∗ ⊢ [A]MZFC Nst and therefore by (1.2.14) we obtain Pr#Nst (y, g(A)) ⇐⇒ Pr Nst m∗ (y, g(A))) . (1.2.17) From (1.2.17) immediately by definitions one obtains (1.2.15). Remark 1.2.10.Note that Theorem 1.2.6 reads Con (ZFCNst) =⇒ ∃True(x) ( True(g(A)) ⇐⇒ [A]MZFC Nst ) . (1.2.18) Theorem 1.2.7. ¬Con (ZFCNst) . Proof. Assume that: Con (ZFCNst) .From (1.2.15) and (1.2.18) one obtains a condradiction Con (ZFCNst)∧¬Con (ZFCNst) and therefore by reductio ad absurdum it follows ¬Con (ZFCNst) . 2 DERIVATION OF THE INCONSISTENT DEFINABLE SET IN SET THEORY ZFCHS2 AND IN SET THEORY ZFCST 2.1 Derivation of the Inconsistent Definable Set in Set Theory ZFC Hs 2 In this section we obtain a contradiction in set theory ZFC Hs 2 by using a set of the all sets definable by first order 1-place open wff's of the set theory ZFC Hs 2 . 34 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 We start to explain main idea from some simply formal definitions. Definition 2.1.1. Let Mst , MZFC Hs 2 st . Let Hs X ,HsX,Mst be a set of the all first order 1-place open wff's Ψ (X) = ΨMst (X) (wff1) of the set theory ZFC Hs 2 with all bound variables restricted to standard model Mst and that contains free occurrences of the first order individual variable X and quantifiers only over first order individual variables, i.e. HsX is a set of the all first order 1-place open wff's with all bound variables restricted to standard model M ZFCHs2 st . We define now a set ΓHsX , ΓHsX,Mst $ Hs X,Mst by the following (second order) formula ∀ΨMst (X) [ ΨMst (X) ∈ ΓHsX,Mst ⇐⇒( ∃!X ( X ∈ MZFC Hs 2 st ) ΨMst (X) ) ∧ ( ΨMst (X) ∈HsX,Mst )] , (2.1.1) or in the following equivalent form ∀ΨMst (X) [ ΨMst (X) ∈ ΓHsX,Mst ⇐⇒ ∃yFr Hs 1 (y, v) ↘[( g ZFC Hs 2 (ΨMst (X)) = y ) ∧ ( g ZFC Hs 2 (X) = ν )] ∧ ( ∃!X ( X ∈ MZFC Hs 2 st ) ΨMst (X) ) ∧ ( ΨMst (X) ∈HsX,Mst )] , (2.1.1.a) see Remark 2.1.10 (ix) and Eq.(2.1.28). Note that there exist a set ΓHsX,Mst by the second order separaton axiom. Notation 2.1.1. In this subsection we often write for short Ψ (X) ,HsX ,Γ Hs X instead ΨMst (X) , Hs X,Mst ,Γ Hs X,Mst but this should not lead to a confusion. Assumption 2.1.1. We assume now for simplicity but without loss of generality that Hs X,Mst ∈ Mst (2.1.1.b) and therefore by definition of model Mst one obtains Γ Hs X,Mst ∈ Mst. Definition 2.1.2. Let Ψ1 = Ψ1 (X) = Ψ1,Mst (X) and Ψ2 = Ψ2 (X) = Ψ2,Mst (X) be a first order 1-place open wff's of the set theory ZFC Hs 2 and with all bound variables restricted to standard model Mst . (i) We define now the equivalence relation (* ∼X *) , (* ∼X,Mst *) ⊂ ΓHsX,Mst × Γ Hs X,Mst by the following formula ∀Ψ1∀Ψ2 (Ψ1 ∼X Ψ2) ⇐⇒ ∀Ψ1 (X) ∀Ψ2 (X) {[Ψ1 (X) ∼X Ψ2 (X)] ⇐⇒ ∀X ( X ∈ MZFC Hs 2 st ) [Ψ1 (X) ⇐⇒ Ψ2 (X)] } ⇐⇒ ∀Ψ1 (X) ∀Ψ2 (X) {[Ψ1 (X) ∼X Ψ2 (X)] ⇐⇒[ ∀X ( X ∈ MZFC Hs 2 st ) Ψ1 (X) ⇐⇒ ∀X ( X ∈ MZFC Hs 2 st ) Ψ2 (X) ]} . (2.1.2) or in the following equivalent form 35 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 ∀Ψ1∀Ψ2 (Ψ1 ∼X Ψ2) ⇐⇒ ∀Ψ1 (X) ∀Ψ2 (X) {[Ψ1 (X) ∼X Ψ2 (X)] ⇐⇒ ∃y1FrHs1 (y1, v)∃y2Fr Hs 1 (y2, v)∀X ( X ∈ M ZFCHs2 st ) [Ψ1 (X) ⇐⇒ Ψ2 (X)] } ⇐⇒ ∀Ψ1 (X) ∀Ψ2 (X) {[Ψ1 (X) ∼X Ψ2 (X)] ⇐⇒ ∃y1FrHs1 (y1, v)∃y2Fr Hs 1 (y2, v) ↘[( g ZFCHs2 (Ψ1 (X)) = y1 ) ∧ ( g ZFCHs2 (Ψ2 (X)) = y2 ) ∧ ( g ZFCHs2 (X) = ν )] ∧[ ∀X ( X ∈ M ZFCHs2 st ) Ψ1 (X) ⇐⇒ ∀X ( X ∈ M ZFCHs2 st ) Ψ2 (X) ]} . (2.1.2.a) (ii) Note that the equivalence relation (* ∼X *) well defined as a set of ordered pair Z1,2 such that Z1,2 = { (Ψ1,Ψ2) | [ (Ψ1,Ψ2) ∈ ΓHsX × ΓHsX ] ∧ [Θ (Ψ1,Ψ2)] } , Θ(Ψ1,Ψ2) , |= M ZFCHs2 st ∀X [Ψ1 (X) ⇐⇒ Ψ2 (X)] , (2.1.3) or in the following equivalent form Z1,2 = { (Ψ1,Ψ2) | [ (Ψ1,Ψ2) ∈ ΓHsX × ΓHsX ] ∧ ∃y1FrHs1 (y1, v)∃y2FrHs1 (y2, v) ↘[( g ZFC Hs 2 (Ψ1 (X)) = y1 ) ∧ ( g ZFC Hs 2 (Ψ2 (X)) = y2 ) ∧ ( g ZFC Hs 2 (X) = ν )] ∧ [Θ (Ψ1,Ψ2)]} , Θ(Ψ1,Ψ2) , |= M ZFCHs2 st ∀X [Ψ1 (X) ⇐⇒ Ψ2 (X)] , (2.1.3.a) i.e. (Ψ1,Ψ2) ∈ Z1,2 if and only if the sentence ∀X [Ψ1 (X) ⇐⇒ Ψ2 (X)] holds in standard model M ZFCHs2 st .Note that the relation |= M ZFCHs2 st ∀X [Ψ1 (X) ⇐⇒ Ψ2 (X)] is expressible in ZFCHs2 by a single formula Θ (Ψ1,Ψ2) of the set theory ZFC Hs 2 , since there exists a single statement Con ( ZFCHs2 ,M ZFCHs2 ) in ZFCHs2 such that Con ( ZFCHs2 ;M ZFCHs2 ) ⇐⇒ ∃MZFC Hs 2 ⇐⇒ ⇐⇒ Con ( ZFCHs2 ) .see Remark 1.1.4(ii). (iii) It follows from the statement (ii) and Axiom schema of separation that Z1,2 is a set in the sense of the set theory ZFCHs2 . (iv) A subset ΛHsX of Γ Hs X such that Ψ1 (X) ∼X Ψ2 (X) holds for all Ψ1 (X) and Ψ2 (X) in ΛHsX , and never for Ψ1 (X) in Λ Hs X and Ψ2 (X) outside Λ Hs X , is called an equivalence class of Γ Hs X by ∼X . (v) A set of the all possible equivalence classes of a set ΓHsX devided by  X , will by denoted by ΓHsX / ∼X ΓHsX / ∼X, { [Ψ (X)]Hs |Ψ(X) ∈ Γ Hs X } , (2.1.4) is the quotient set of a set ΓHsX devided by the equivalence relation ∼X . (vi) For any Ψ (X) ∈ ΓHsX by symbol [Ψ (X)]Hs , { Φ(X) ∈ ΓHsX |Ψ(X) ∼X Φ(X) } we denote the equivalence class to which Ψ (X) belongs. All elements of ΓHsX that equivalent to each other are also elements of the same equivalence class. Definition 2.1.3. We define now the operations join ∨,meet ∧,and complementation, denoted [Φ (X)]′on ΓHsX / ∼X by : (1) [Φ (X)] ∨ [Ψ (X)] = [Φ (X) ∨Ψ(X)] , 36 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 (2) [Φ (X)] ∧ [Ψ (X)] = [Φ (X) ∧Ψ(X)] , (3) [Φ (X)]′ = [¬Φ(X)] . The resulting bulean algebra BX is the Lindenbaum-Tarski algebra of the second order language Hs2 of ZFC Hs 2 and it may be shown that t∈T [Φ (t)] = [ ∃X ( X ∈ MZFC Hs 2 st ) Φ(X) ] , t∈T [Φ (t)] = [ ∀X ( X ∈ MZFC Hs 2 st ) Φ(X) ] , (2.1.5) where t ∈ T = MZFC Hs 2 st and T is the set of all terms in the language Hs 2 . Remark 2.1.1. Note that in bulean notations definition (2.1.2) reads Z1,2 = { (Ψ1,Ψ2) | ( (Ψ1,Ψ2) ∈ ΓHsX × ΓHsX ) ∧ ([Ω (Ψ1,Ψ2)] = 1BX ) } , Ω(Ψ1,Ψ2) , ∀X ( X ∈ MZFC Hs 2 st ) [Ψ1 (X) ⇐⇒ Ψ2 (X)] (2.1.6) Definition 2.1.4. [17]. Let Th be any theory in the recursive language Th ⊃PA, where PA is a language of Peano arithmetic.We say that a number-theoretic relation R (x1, ..., xn) of n arguments is expressible in Th if and only if there is a wff R (x1, ..., xn) of Th with the free variables x1, ..., xn such that,for any natural numbers k1, ..., kn, the following holds: (i) If R (k1, ..., kn) is true, then ⊢Th R ( k1, ..., kn ) . (ii) If R (k1, ..., kn) is false, then ⊢Th ¬R ( k1, ..., kn ) . Remark 2.1.2. Recoll that any recursive language Th except logical connectives and quantifiers contains the following sets of symbols (see for example ref. [17], p.51): (i) a set of symbols ∆0 = {(, ), ,,¬, =⇒ , ∀} and we will identify these symbols with a 1-tuples ∆0 = {{(} , {)} , {, } , {¬} , { =⇒ , {∀}}} by using a one-one function℘∆0 : ℘∆0 ({(}) = (, ℘∆0 ({)}) =), ℘∆0 ({, }) =, , ℘∆0 ({¬}) = ¬, ℘∆0 ({ =⇒ }) = =⇒ , ℘∆0 ({∀}) = ∀, ℘−1∆0 (() = {(} , ℘ −1 ∆0 ()) = {)} , ℘−1∆0 (, ) = {, } , ℘ −1 ∆0 (¬) = {¬} , ℘−1∆0 ( =⇒ ) = { =⇒ } , ℘−1∆0 (∀) = {∀} , (2.1.7) and we will be often abbreviate ( = {(} , ) = {)} , , = {, } , ¬ = {¬} , =⇒ = { =⇒ } , ∀ = {∀} ; (2.1.8) (ii) a set of the first order individual variables: ∆1 = {x1, x2, ..., xn, ...} and we will identify these individual variables with a 1-tuples ∆1 = {{x1} , {x2} , ..., {xn} , ...} by using a one-one function℘∆1 : ℘∆1 ({xn}) = xn, ℘−1∆1 (xn) = {xn} , (2.1.9) and we will be often abbreviate xn = {xn} ; (iii) a set of the second order individual variables: ∆ = {y1, y2, ..., yn, ...} 37 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 (iv) a set of the individual constants: ∆2 = {a1, a2, ..., an, ...} and we will identify these individual constants with a 1-tuples ∆2 = {{a1} , {a2} , ..., {an} , ...} by using a one-one function℘∆2 ({an}) ℘∆2 ({an}) = an, ℘−1∆2 (an) = {an} , (2.1.10) and we will be often abbreviate ân = {an} ; (v) for every integer n ≥ 0 there is a set of n-ary, or n-place, predicate symbols: ∆3 = {Ank}n,k∈N and we will identify these predicate symbols with a 1-tuples ∆3 = {{Ank}}n,k∈N by using a one-one function ℘∆3 : ℘∆3 ({Ank}) = Ank , ℘−1∆3 (A n k ) = {Ank} ; (2.1.11) and we will be often abbreviate Ânk = {A n k} ; (vi) for every integer n ≥ 0 there is a set of n-ary, or n-place, function symbols: ∆4 = {fnk }n,k∈N and we will identify these predicate symbols with a 1-tuples ∆4 = {{fnk }}n,k∈N by using a one-one function ℘∆4 : ℘∆4 ({fnk }) = fnk , ℘−1∆4 (f n k ) = {fnk } , (2.1.12) and we will be often abbreviate fnk = {f n k } ; (vii) A theory Th is said to have a primitive recursive vocabulary (or a recursive vocabulary) if the following predicates are primitive recursive (or recursive) (a) ICHs(x): x is the Godel number of an individual constant of ZFC Hs 2 , (b) FLHs(x): x is the Godel number of a function letter of ZFC Hs 2 , (c) PLHs(x): x is the Godel number of a predicate letter of ZFC Hs 2 . Remark 2.1.3. (i) Note that in fact it was alwais implicitly assumed that these sets ∆1, ∆2,∆3,∆4 are a sets in a sense of ZFC (ZFC-set),see ref.[8],[17]. (ii) we will write for short A is a ZFC-set instead A is a set in a sense of ZFC,etc. Remark 2.1.4.(a) Recoll that the function symbols applied to the variables and individual constants inductively generate a full ZFC-set of the terms [8], [17]: (1) Variables and individual constants are terms. (1.a) First order variables and individual constants are first order terms. (1.b) Second order variables are second order terms. (2) If f nk is a function symbol and t1, t2, ..., tn, are terms, then f n k (t1, t2, ..., tn) is a term. (2.a) If f nk is a function symbol and t1, t2, ..., tn, are first order terms, then f n k (t1, t2, ..., tn) is a first order term. 38 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 (2.b) If f nk is a function symbol and sequence t1, t2, ..., tn, contain at least one second order term, then f nk (t1, t2, ..., tn) is a second order term. (2.b) If f nk is a function symbol and sequence t1, t2, ..., tn, contain at least one second order term, then f nk (t1, t2, ..., tn) is a second order term. (3) An expression is a term only iff it can be shown to be a term on the basis of conditions (1) and (2). (3.a) An expression is a first order term only iff it can be shown to be a term on the basis of conditions (1.a) and (2.a). (3.b) An expression is a second order term only iff it can be shown to be a term on the basis of conditions (1.b) and (2.b). (3.c) We will be identify the first order terms with ordered n+ 3-tuples {{f nk } , {(} , {t1} , {t2} , ..., {tn} , {)}} (2.1.13) by using a one-one function ℘τ : ℘τ ({{f nk } , {(} , {t1} , {t2} , ..., {tn} , {)}}) = f nk (t1, t2, ..., tn) , ℘−1τ (f n k (t1, t2, ..., tn)) = {{f nk } , {(} , {t1} , {t2} , ..., {tn} , {)}} . (2.1.14) It follows from Remark 2.1.2 and Remark 2.1.4 that there is a ZFC-set of the all first order terms Υ1: Υ1 = ∆1 ∪∆2 ∪ {f nk (t1, t2, ..., tn)}n,k∈N . (2.1.15) (3.d) We will be denoted the image ℘−1τ (Υ1) by ℘−1τ (Υ1) = Υ1. (2.1.16) (4) Recoll that the predicate symbols applied to terms yield the atomic formulas; that is, if Akn is a predicate letter and t1, t2, ..., tn, are terms, then A k n (t1, t2, ..., tn) is an atomic formula. (4.a) The predicate symbols applied to the first order terms yield the first order atomic formulas; that is, if Akn is a predicate letter and t1, t2, ..., tn, are first order terms, then A k n (t1, t2, ..., tn) is an first order atomic formula. (4.b) The predicate symbols applied to the second order terms yield the second order atomic formulas; that is, if Akn is a predicate letter and t1, t2, ..., tn, are second order terms, then A k n (t1, t2, ..., tn) is an second order atomic formula. (4.c) We will be identify the first order atomic formulas with ordered n+ 3-tuples {A nk , (, t1, t2, ..., tn, )} by using a one-one function ℘π : ℘π ({A nk , (, t1, t2, ..., tn, )}) = A nk (t1, t2, ..., tn) , ℘−1π (A n k (t1, t2, ..., tn)) = {A nk , (, t1, t2, ..., tn, )} . (2.1.17) It follows from Remark 2.1.2 and Remark 2.1.4 that there is a ZFC-set of the all first order atomic formulas Σ1: Σ1 = {A nk (t1, t2, ..., tn)}n,k∈N . (2.1.18) 39 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 (4.d)We will be denoted the image ℘−1π (Σ) by ℘−1π (Σ1) = Σ1. (2.1.19) (5) We introduce now a one-one function ℘τ,π such that ℘τ,π|Υ1 = ℘τ , ℘τ,π|Σ1 = ℘π. (2.1.20) (6) Recoll that the well-formed formulas (wff's) of quantification theory are defined inductively as follows [8], [17], [18]: (6.a) Every atomic formula is a wff. (6.b) Every first order atomic formula is a first order wff. (6.c) Every second order atomic formula is a second order wff. (6.d) If B and C are wff's and y is a variable, then (¬B) , (B =⇒ C) and ((∀y)B) are wff's. (6.e) An expression is a wff only if it can be shown to be a wff on the basis of conditions (6.a) and (6.d). (7) If B and C are first order wff's and y is a first order variable, then (¬B) , (B =⇒ C) and ((∀y)B) are first order wff's. (7.a) An expression is a first order wff only if it can be shown to be a wff on the basis of conditions (6.b) and (6.d). Remark 2.1.5. It follows from Remark 2.1.1-Remark 2.1.3 that there is a ZFC-set Ξ1 of the all first order wff's and in partcular Υ1 ∪ Σ1 ⊂ Ξ1. (2.1.21) We extend now one-one function℘τ,π up one-one function ℘Ξ1 by natural way,i.e., ℘Ξ |Υ∪Σ = ℘τ,π and we will be denoted the image ℘ −1 Ξ1 (Ξ1) by ℘−1Ξ1 (Ξ1) = Ξ1. (2.1.22) Remark 2.1.6. Recoll that for an arbitrary second-order theory Th, we correlate with each symbol u of Th an odd positive integer g(u), called the Godel number of u, in the following rules [10] [[10]]:1.g (() = 3,2.g ()) = 5,3.g (, ) = 7,4.g (¬) = 9,5.g ( =⇒ ) = 11,6.g (∀) = 13, 7.g (xk) = 13 + 8k,8.g (ak) = 7 + 8k,9.g (f n k ) = 1 + 8 ( 2n3k ) ,10.g (Ank ) = 3 + 8 ( 2n3k ) , 11.g (yk) = 15 + 8k, where k, n ≥ 1. Example 2.1.1.g (x2) = 29; g (a4) = 39; g ( f 21 ) = 97; g ( A12 ) = 147. Remark 2.1.7. Note that g is a bijection and therefore there exist a functon g−1such that 1.g−1 (3) = (;2.g−1 (5) =);3.g−1 (7) =, ;4.g−1 (9) = ¬;5.g−1 (11) = =⇒ ,6.g−1 (13) = ∀; 7.g (13 + 8k) = xk;8.g (7 + 8k) = ak;9.g ( 1 + 8 ( 2n3k )) = fnk ;10.g ( 3 + 8 ( 2n3k )) = Ank ,where k, n ≥ 1. 40 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 Example 2.1.2.g−1 (29) = x2; g −1 (39) = a4; g −1 (97) = f 21 ; g −1 (147) = A12. Remark 2.1.8. Note that g ◦ g−1 (x) = x. Remark 2.1.9. Given an expression u0u1..uj ...ur,where each uj is a symbol of Th, i.e., each uj ∈ ∆0 ∪ ∆1 ∪ ∆2 ∪ ∆3 ∪ ∆4 (∆0 = {(, ), , ,¬, =⇒ , ∀}) we define its Godel number g ZFC Hs 2 (u0u1..uj ...ur) , ğ (u0u1..uj ...ur) by the formula: g , ğ (u0u1..uj ...ur) = ğ (u0) * ğ (u1) * ... * ğ (uj) * ... * ğ (ur) = = 2g(u0) * 3g(u1) * ... * pg(uj)j * ... * p g(ur) r , (2.1.23) where ğ (uj) = p g(uj) j and where pj denotes the j-th prime number and we assume that p0 = 2. Example 2.1.3. g ( A12 (x1, x2) ) = 2g(A 1 2) * 3g(() * 5g(x1) * 7g(,) * 11g(x2) * 13g()) = 299 * 33 * 521* *77 * 1129 * 135. Definition 2.1.5. Given any natural number k ∈ N wich has representation of the form k = 2g(u0) * 3g(u1) * ... * pg(uj)j * ... * p g(ur) r for some sequance of a symbols u0, u1, .., uj , ..., ur, where each uj is a symbol of Th, we define a function ğ −1 : N → Ξ1 by the following formula ğ−1 (k) = ğ−1 ( 2g(u0) ) * ... * ğ−1 ( p g(uj) j ) * ... * ğ−1 ( p g(ur) r ) = = u0u1..uj ...ur ∈ Ξ1, (2.1.24) where ğ−1 ( p g(uj) j ) = g−1 (g (uj)) = uj ∈ ∆0 ∪∆1 ∪∆2 ∪∆3 ∪∆4. Definition 2.1.6. [10] Thus g is one-one function from the set S# = ∪n∈NSn,where S = ∆0 ∪ ∆1 ∪∆2 ∪∆3 ∪∆4,of symbols of Th, first order expressions of Th and finite sequences of first order expressions of Th into the set of positive integers. The following conditions are to be satisfied by the function g : (1) g is effectively computable, (2) there is an effective procedure that determines whether any given positive integer m is in the range of g and, if m is in the range of g, the procedure: finds the object z ∈ ∪n∈NΞn1 = Ω1 such that g(z) = m. We extend now one-one function℘Ξ1 up one-one function ℘Ω1 by natural way,i.e., ℘Ω1 |Ξ1 = ℘Ξ1 , ℘Ω1 |Ξn1 = ℘Ξ1×...×} {{ } n ℘Ξ1 , (2.1.25) n ∈ N, and we will be denoted the image ℘−1Ω1 (Ω1) by ℘−1Ω1 (Ω1) = Ω1. (2.1.26) Proposition 2.1.1. Let Th be a theory with a primitive recursive (or recursive) vocabulary. Then withe following relations and functions (1-11) are primitive recursive (or recursive). In each case, we give first the notation and intuitive definition for the relation or function, and then an equivalent formula from which its primitive recursiveness (or recursiveness) 41 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 can be deduced. (1) EVbl1(x) : x is the Godel number of an expression consisting of a first order variable, (∃z)z<x(1 ≤ z ∧ x = 213+8z). By [17], Proposition 3.18, this is primitive recursive. EVbl2(x) : x is the Godel number of an expression consisting of a second order variable, (∃z)z<x(1 ≤ z ∧ x = 215+8z). By [17], Proposition 3.18, this is primitive recursive. EVbl1∨2(x) : x is the Godel number of an expression consisting of a variable, (∃z)z<x(1 ≤ z ∧ ( x = 213+8z ) ∨ ( x = 215+8z ) ). By [17], Proposition 3.18, this is primitive recursive. EIC(x) : x is the Godel number of an expression consisting of an individual constant, (∃y)y<x(IC(y)∧ x = 2y) [17], Proposition 3.18). EFL(x) : x is the Godel number of an expression consisting of a function letter, (∃y)y<x(FL(y) ∧ x = 2y) [17], Proposition 3.18. EPL(x) : x is the Godel number of an expression consisting of a predicate letter, (∃y)y<x(PL(y) ∧ x = 2y) [17], Proposition 3.18. (2) ArgT(x) = (qt(8, x * −1))0 : If x is the Godel*number of a function letter f nj , then ArgT(x) = n. ArgT(x) is primitive recursive [17], Proposition 3.18. ArgP(x) = (qt(8, x * − 3))0 : If x is the Godel number of a predicate letter A nj , then ArgP(x) = n.ArgP(x) is primitive recursive [17], Proposition 3.18. (3) Gd1(x) : x is the Godel number of an first order expression of Th, EVbl1(x) ∨EIC(x) ∨EFL(x) ∨ EPL(x) ∨ x = 23 ∨ x = 25 ∨ x = 27 ∨ x = 29 ∨ x = 211 ∨ x = 213 ∨ (∃u)u<x(∃v)v<x(x = u ∗ v ∧Gd1(u) ∧Gd1(v)). Gd1∨2(x) : x is the Godel number of an expression of Th, EVbl1∨2(x) ∨EIC(x) ∨EFL(x) ∨ EPL(x) ∨ x = 23 ∨ x = 25 ∨ x = 27 ∨ x = 29 ∨ x = 211 ∨ x = 213 ∨ x = 215 ∨ (∃u)u<x(∃v)v<x(x = u ∗ v ∧Gd1∨2(u) ∧Gd1∨2(v)). (4) MP1(x, y, z) : The first order expression with Godel number z is a direct consequence of the first order expressions with Godel numbers x and y by modus ponens, y = 23 ∗ x ∗ 211 ∗ z ∗ 25 ∧ Gd1(x) ∧Gd1(z). MP1∨2(x, y, z) : The expression with Godel number z is a direct consequence of the expressions with Godel numbers x and y by modus ponens, y = 23 ∗ x ∗ 211 ∗ z ∗ 25 ∧Gd1∨2(x) ∧Gd1∨2(z). (5)Gen1(x, y) : The first order expression with Godel number y comes from the first order expression with Godel number x by the generalization rule: (∃v)v<y(EVbl1(v) ∧ y = 23 ∗ 23 ∗ 213 ∗ v ∗ 25 ∗ x ∗ 25 ∧Gd1(x)). 42 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 Gen1∨2(x, y) : The expression with Godel number y comes from the an expression with Godel number x by the generalization rule: (∃v)v<y(EVbl1∨2(v) ∧ y = 23 ∗ 23 ∗ 213 ∗ v ∗ 25 ∗ x ∗ 25 ∧Gd1∨2(x)). (6) Trm1(x) : x is the Godel number of an first order term of Th. Trm1(x) is equivalent to the following relation: ' EVbl1(x) ∨EIC(x) ∨ (∃y)y<(px!)x [x = (y)lh(y) *−1∧ lh(y) = ArgT((x)0) + 1 ∧ FL(((y)0)0) ∧ ((y)0)1 = 3∧ lh((y)0) = 2 ∧ (∀u) u<lh(y) * −1 (∃v)v<x((y)u+1 = (y)u ∗ v ∗ 27 ∧Trm1(v))∧ (∃v)v<x((y) lh(y) * −1 = (y) lh(y) * −2 ∗ v ∗ 25 ∧Trm1(v))]. (7) Atfml1(x) : x is the Godel number of an atomic first order wff of Th. Atfml1(x) is equivalent to the following: (∃y)y<(px!)x [x = (y)lh(y) *−1 ∧ lh(y) = ArgP((x)0) + 1∧ PL(((y)0)0) ∧ ((y)0)1 = 3 ∧ lh((y)0) = 2 ∧ (∀u) u<lh(y) * −2 (∃v)v<x((y)u+1 = (y)u ∗ v ∗ 27 ∧Trm1(v))∧ (∃v)v<x((y) lh(y) * −1 = (y) u<lh(y) * −2 ∗ v ∗ 25 ∧Trm1(v))]. (8)Fml1(y) : y is the Godel number of an first order formula of Th: Atfml1(y) ∨ (∃z)z<y[(Fml1(z)∧ = 23 ∗ 29 ∗ z ∗ 25)∨ (Fml1((z)0) ∧ Fml1((z)1 ∧ y = 23 ∗ (z)0 ∗ 211 ∗ (z)1 ∗ 25)∨ (Fml1((z)0) ∧EVbl1((z)1) ∧ y = 23 ∗ 23 ∗ 213 ∗ ((z)1 ∗ 2 5 ∗ (z)0 ∗ 25)]. (9) Subst1(x, y, u, v) : x is the Godel number of the result of substituting in the first order expression with Godel number y the first order term with Godel number u for all free occurrences of the variable with Godel number v. (10) Sub1(y, u, v) : the Godel number of the result of substituting the first order term with Godel number u for all free occurrences in the first order expression with Godel number y of the variable with Godel number v : Sub1(y, u, v) = μxx<(puy !)uySubst1(u, y, u, v). (11) Fr1(y, v) : y is the Godel number of the first order wff or the first order term of Th that contains free occurrences of the variable with Godel number v : (Fml1(y) ∨Trm1(y)) ∧EVbl1(2v) ∧ ¬Subst1(y, y, 213+8v, v). Remark 2.1.10.Note that in order to obtain completely formal definitions of the first order predicates EVblHs1 (x),EIC Hs 1 (x),EFL Hs 1 (x), ...,Fr Hs 1 (y, v) one needs the following second order predicates: (i) EVblHs1 (x, α) : x is the Godel number of the first order expression α ∈ ∆1 consisting of a first order variable; (ii) EICHs1 (x, β) : x is the Godel number of the first order expression β ∈ ∆2 consisting of individual 43 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 constant; (iii) EFLHs1 (x, γ) : x is the Godel number of the first order expression γ ∈ ∆3 consisting of function letter; (iv) EPLHs1 (x, δ) : x is the Godel number of the first order expression δ ∈ ∆4 consisting of predicate letter; (v) GdHs1 (x, ζ) : x is the Godel number of the first order expression ζ ∈ Sn, n ∈ N of the ZFC Hs 2 ; (vi) TrmHs1 (x, τ) : x is the Godel number of the first order term τ ∈ Υ1 of the ZFC Hs 2 . (vii) AtfmlHs1 (x, π) : x is the Godel number of the first order atomic wff π ∈ Σ1 of the ZFC Hs 2 . (viii) FmlHs1 (y, φ) : y is the Godel number of the the first order wff formula φ ∈ Ξ1 of the ZFC Hs 2 . (ix) FrHs1 (y, v,π) : y is the Godel number of the the first order wff π or the the first order term π ∈ Ξ1 of the ZFC Hs 2 that contains free occurrences of the variable with Godel number v. Thus finally we obtain: EVblHs1 (x) ⇐⇒ ∃α (α ∈ ∆1)EVbl Hs 1 (x, α) ⇐⇒ (∃z)z<x(1 ≤ z ∧ x = 2 13+8z), EICHs1 (x) ⇐⇒ ∃β (β ∈ ∆2)EIC Hs 1 (x, β) ⇐⇒ (∃y)y<x(IC Hs(y) ∧ x = 2y), EFLHs1 (x) ⇐⇒ ∃γ (γ ∈ ∆3)EFL Hs 1 (x, γ) ⇐⇒ (∃y)y<x(FL Hs(y) ∧ x = 2y), EPLHs1 (x) ⇐⇒ ∃δ (δ ∈ ∆4)EPL Hs 1 (x, δ) ⇐⇒ (∃y)y<x(PL Hs(y) ∧ x = 2y), GdHs1 (x) ⇐⇒ ∃ζ∃n (ζ ∈ S n)GdHs1 (x, ζ) ⇐⇒ EVblHs1 (x) ∨ EIC Hs 1 (x) ∨ EFL Hs 1 (x) ∨ EPL Hs 1 (x)∨ x = 27 ∨ x = 29 ∨ x = 211 ∨ x = 213∨ (∃u)u<x(∃v)v<x(x = u ∗ v ∧ GdHs2 (u) ∧ Gd Hs 2 (v)). TrmHs1 (x) ⇐⇒ ∃τ (τ ∈ Υ)Trm Hs 1 (x, τ) ⇐⇒ EVbl(x) ∨ EIC(x) ∨ (∃y)y<(px!)x [x = (y) lh(y) * −1 ∧ lh(y) = ArgT((x)0) + 1 ∧ FL(((y)0)0) ∧ ((y)0)1 = 3∧ lh((y)0) = 2 ∧ (∀u) u<lh(y) * −1 (∃v)v<x((y)u+1 = (y)u ∗ v ∗ 27 ∧ TrmHs1 (v))∧ (∃v)v<x((y) lh(y) * −1 = (y) lh(y) * −2 ∗ v ∗ 25 ∧ TrmHs1 (v))]. AtfmlHs1 (x) ⇐⇒ ∃π (π ∈ Σ1)Atfml Hs 1 (x, π) ⇐⇒ (∃y)y<(px!)x [x = (y) lh(y) * −1 ∧ lh(y) = ArgP ((x)0) + 1∧ PLHs1 (((y)0)0) ∧ ((y)0)1 = 3 ∧ lh((y)0) = 2∧ (∀u) u<lh(y) * −2 (∃v)v<x((y)u+1 = (y)u ∗ v ∗ 27 ∧ TrmHs1 (v))∧ (∃v)v<x((y) lh(y) * −1 = (y) u<lh(y) * −2 ∗ v ∗ 25 ∧ TrmHs1 (v))]. FmlHs1 (y) ⇐⇒ ∃ (φ ∈ Ξ1)Fml Hs 1 (y, φ) ⇐⇒ AtfmlHs1 (y) ∨ (∃z)z<y [(Fml Hs 1 (z)∧ = 2 3 ∗ 29 ∗ z ∗ 25)∨ (FmlHs1 ((z)0) ∧ Fml Hs 1 ((z)1 ∧ y = 2 3 ∗ (z)0 ∗ 211 ∗ (z)1 ∗ 25)∨ (FmlHs1 ((z)0) ∧ EVbl Hs 1 ((z)1) ∧ y = 2 3 ∗ 23 ∗ 213 ∗ ((z)1 ∗ 2 5 ∗ (z)0 ∗ 25)]. FrHs1 (y, v) ⇐⇒ ∃π [(π ∈ Ξ1) ∨ (π ∈ Υ1)]Fr Hs 1 (y, v,π) ⇐⇒ (FmlHs1 (y) ∨ Trm Hs 1 (y)) ∧ EVbl Hs 1 (2 v) ∧ ¬SubstHs1 (y, y, 2 13+8v, v). (2.1.27) Designation 2.1.2. (i) Let gZFCHs2 (u) be a Gödel number of an given expression u of the set theory ZFC Hs 2 , ZFCHs2 + ∃M ZFCHs2 st . (ii) Let FrHs1 (y, v) be the relation : y is the Gödel number of a first order wff of the set theory ZFC Hs 2 that contains free occurrences of the variableX with Gödel number v,see Remark 2.1.10(ix). (iii) Note that the relation FrHs1 (y, v) is recursive and thus an equivalent from which it recursiveness can be deduced,i.e. the relation FrHs1 (y, v) is expressible in ZFC Hs 2 by a wff Fr Hs 1 (y, v) : FrHs1 (y, v) ≡ ( FmlHs1 (y) ∨TrmHs1 (y) ) EvblHs1 (2 ν) ∧ ¬SubstHs1 ( y, y, 213+8ν , ν ) . (2.1.28) 44 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 (iv) Note that for any y, v ∈ N by the definition of the relation FrHs1 (y, v) follows that FrHs1 (y, v) ⇐⇒ ∃!Ψ (X) [( g ZFC Hs 2 (Ψ (X)) = y ) ∧ ( g ZFC Hs 2 (X) = ν )] , (2.1.29) where Ψ (X) is a unique wff of ZFC Hs 2 which contains free occurrences of the variable X with Gödel number v.We will be often denote the unique wff Ψ (X) defined by using equivalence (2.1.29) by the symbol Ψy,ν (X) ,i.e. FrHs1 (y, v) ⇐⇒ ∃!Ψy,ν (X) [( g ZFC Hs 2 (Ψy,ν (X)) = y ) ∧ ( g ZFC Hs 2 (X) = ν )] . (2.1.30) Remark 2.1.11. (i) Note that a function g ZFC Hs 2 (Ψy,ν (X)) = y is expressible in set theory ZFC Hs 2 by a wff of the set theory ZFC Hs 2 (see Remark 2.1.13) that contains free occurrences of the variable y ∈ N.Note that formula Ψy,ν (X) is given by an expression u0u1..uj ...ur,i.e.Ψy,ν (X) = u0u1..uj ...ur, where each uj is a symbol of ZFC Hs 2 . (ii) Note that in order to obtain Gödel encoding (2.1.23) rigorously without any refference to non formal notion of the expression u0u1..uj ...ur and by using only notion of ZFC-set Ξ1(see Remark 2.1.5) we remind that Ψy,ν (X) = u0u1..uj ...ur ∈ Ξ1 and therefore Ψy,ν (X) = û0û1..ûj ...ûr ∈ Ξ1. (iii) In order to obtain Gödel encoding as required above in Remark 2.1.11(ii) we introduce now a countale secuence of the functions [Ψy,ν (X) ; j] : Ξ1 × N → ∆0 ∪∆1 ∪∆2 ∪∆3 ∪∆4, j = 0, 1, ... (2.1.31) which are defined by the following formulas [Ψy,ν (X) ; j] = uj , j = 0, 1, ..., (2.1.32) and we revrite now the expression u0u1..uj ...ur ∈ Ξ1 in the following equivalent form [Ψy,ν (X) ; 0] [Ψy,ν (X) ; 1] ... [Ψy,ν (X) ; j] ... [Ψy,ν (X) ; r] . (2.1.33) By definitions are given above (see Remark 2.1.11(i)-(ii)) we obtain that g ZFC Hs 2 (Ψy,ν (X)) = y ⇐⇒ y = 2g([Ψy,ν(X);0]) * 3g([Ψy,ν(X);1]) * ... * pg([Ψy,ν(X);j])j * ... * p g([Ψy,ν(X);r]) r . (2.1.34) Let us denote by (y)j (see ref.[10] [[10]]) the exponent g ([Ψy,ν (X) ; j]) in this factorization: y = 2g([Ψy,ν(X);0]) * 3g([Ψy,ν(X);1]) * ... * pg([Ψy,ν(X);j])j * ... * p g([Ψy,ν(X);r]) r . (2.1.35) Recoll that every positive integer y has a unique factorization into prime powers: y = pa00 * p a1 1 * ... * p aj j * ... * p ar r (2.1.36) Let us denote by (y)j the exponent aj in this factorization (2.1.29).If y = 1, (y)j = 1 for all j. If y = 0, we arbitrarily let (y)j = 0 for all j. Then the functions (y)j , j = 0, 1, ... is primitive recursive, since (y)j = μz<y ( pzj |y ∧ ¬pz+1j |y ) ,see [17], p.181. Remark 2.1.12. Thus the functions (y)j = g ([Ψ (X) ; j]) , j = 0, 1, ... are expressible in set theory ZFC Hs 2 by the formulas denoted below for a short by the symbol λj (y, g ([Ψ (X) ; j])) . 45 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 For y > 0, let lh(y) be the number of non-zero exponents in the factorization of y into powers of primes, or, equivalently, the number of distinct primes that divide y. Let lh(0) = 0, then lh(y) is primitive recursive. Remark 2.1.13. (i) Note that a function ( g ZFC Hs 2 (Ψ (X)) = y ) ∧ ( g ZFC Hs 2 (X) = ν ) is expressible in set theory ZFC Hs 2 by the following formula Ξ1 (Ψ (X) , y, ν) : Ξ1 (Ψ (X) , y, ν) ⇐⇒ (y ∈ N) ∧ (ν ∈ N) FrHs1 (y, v) ∧ [ j≤lh(y)λj (y, g ([Ψ (X) ; j])) ] , (2.1.37) where Ψ (X) is 1-open first order wff of the set theory ZFC Hs 2 . (ii) Note that the length of the formula (2.1.37) depend on numerals y, ν but nevertheless Ξ (Ψ (X) , y, ν) is a single 3-open wiff of ZFC Hs 2 . (iii) Note that g ZFC Hs 2 (Ψy,ν (X)) = y ⇐⇒ Ξ ( ğ−1 (y) , y, ν ) . (2.1.38) Definition 2.1.7.LetHsX,ν be a set of the all 1-place open wff's Ψ (X) of the set theory ZFC Hs 2 that contains free occurrences of the individual variable X with Gödel number v and quantifiers only over individual variables.We define now a set ΓHsX,ν $HsX,ν by the following formula ∀Ψ(X) [ Ψ(X) ∈ ΓHsX,ν ⇐⇒ ( ∃!X ( X ∈ MZFC Hs 2 st ) Ψ(X) ) ∧ ( Ψ(X) ∈HsX,ν )] . (2.1.39) Remark 2.1.14. Let g ZFC Hs 2 (X) = ν.We define now a set ΓHsν $ N by the following formula ΓHsν = { y ∈ N| ( ⟨y, ν⟩ ∈ FrHs1 (y, v) ) ∧ ğ−1 (y) ∈ ΓHsX,ν } , (2.1.40) or in the following equivalent form: ∀y (y ∈ N) [ y ∈ ΓHsν ⇐⇒ (y ∈ N) ∧ FrHs1 (y, v) ∧ ğ−1 (y) ∈ ΓHsX,ν ] . (2.1.41) Definition 2.1.8. Let Ξ1,X be a ZFC-set of the all first order 1-open wiff's of the set theory ZFC Hs 2 ,see Remark 2.1.4, then we abbreviate Ξ1,X , Wff1,X [ ZFC Hs 2 ] . Remark 2.1.15.(a) Note that a ZFC-setWff1 [ ZFC Hs 2 ] in canonical handbooks always considered as an standard set in the sense of the set theory ZFC,see ref. [8]. See for example the proof of the Gödel Completness Theorem, ref. [8] Theorem 2, sect.4,p.13. (b) Note that from statement (a) (see also Remark 2.1.4) and from the axiom of separation it follows directly that ΓHsν is a standard set in the sense of the set theory ZFC Hs 2 , (441) note that the collections ΓHsX and Γ Hs X,ν in fact can be considered as a standard set directly 46 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 without any reference to Gödel number, since a countable collection of the all first order wff's of the set theory ZFC Hs 2 is a set in the sense of the set theory ZFC. Definition 2.1.9.(i) We define now the equivalence relation (* ∼ν *) ⊂ ΓHsν × ΓHsν (2.1.42) in the sense of the set theory ZFC Hs 2 by the following formula ∀y1∀y1 [ y1 ∼ν y2 ⇐⇒ ( ∀X ( X ∈ MZFC Hs 2 st ) [Ψy1,ν (X) ⇐⇒ Ψy2,ν (X)] )] . (2.1.43) Remark 2.1.16. Note that (2.1.43) by using second order lenguage of the set theory ZFC Hs 2 can be written in the following equivalent form y1 ∼ν y2 ⇐⇒ FrHs1 (y1, v) ∧ FrHs1 (y2, v) ∧ ∃Ψy1,ν (X) ( g ZFC Hs 2 (Ψy1,ν (X)) = y1 ) ∧ ∃Ψy2,ν (X) ( g ZFC Hs 2 (Ψy2,ν (X)) = y2 ) ∧ ( g ZFC Hs 2 (X) = ν ) ∧[ ∀X ( X ∈ MZFC Hs 2 st ) [Ψy1,ν (X) ⇐⇒ Ψy2,ν (X)] ] . (2.1.44) Remark 2.1.17. Note that from the axiom of separation it follows directly that the equivalence relation (* ∼ν *) is a relation in the sense of the set theory ZFC Hs 2 . (i) A subset ΛHsν of Γ Hs ν such that y1 ∼ν y2 holds for all y1 and y1 in ΛHsν , and never for y1 in ΛHsν and y2 outside Λ Hs ν , is an equivalence class of Γ Hs ν . (iii) For any y ∈ ΓHsν by symbol [y]Hs , { z ∈ ΓHsν |y ∼ν z } we denote the equivalence class to which y belongs. All elements of ΓHsν equivalent to each other are also elements of the same equivalence class. (iii)The collection of all possible equivalence classes of ΓHsν by  ν , denoted by symbol ΓHsν / ∼ν : ΓHsν / ∼ν, { [y]Hs |y ∈ Γ Hs ν } . (2.1.45) Remark 2.1.18. Note that from the axiom of separation it follows directly that ΓHsν / ∼ν is a set in the sense of the set theory ZFC Hs 2 . Definition 2.1.10. Let IHs2 be the countable ZFC-set of the all sets definable by the first order 1-place open wff's of the set theory ZFC Hs 2 , i.e. ∀Y { Y ∈ IHs2 ⇐⇒ ∃Ψ(X) [( [Ψ (X)]Hs ∈ Γ Hs X / ∼X ) ∧ [∃!X [Ψ (X) ∧ Y = X]] ]} . (2.1.46) Definition 2.1.11. We rewrite now (2.1.46) in the following equivalent form ∀Y { Y ∈ IHs2 ⇐⇒ ∃Ψ(X) [( [Ψ (X)]Hs ∈ Γ ∗Hs X / ∼X ) ∧ (Y = X) ]} , (2.1.47) where the countable set Γ∗HsX / ∼X is defined by the following formula ∀Ψ(X) { [Ψ (X)] ∈ Γ∗HsX / ∼X ⇐⇒ [( [Ψ (X)] ∈ ΓHsX / ∼X ) ∧ ∃!XΨ(X) ]} (2.1.48) Definition 2.1.12. Let RHs2 be the countable set of the all sets definable by the first order 1-place 47 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 open wff's and such that ∀X {( X ∈ Mst ∧X ∈ IHs2 ) [ X ∈ RHs2 ⇐⇒ X /∈ X ]} . (2.1.49) Remark 2.1.19.(a) Note that RHs2 ∈ IHs2 since RHs2 is a ZFC-set definable by the first order 1-place open wff ΨMst ( Z,IHs2 ) , Ψ ( Z,IHs2 ) , ∀X {( X ∈ Mst ∧X ∈ IHs2 ) [X ∈ Z ⇐⇒ X /∈ X] } , (2.1.50) and obviously ΨMst ( Z,IHs2 ) ∈ Wff1,Z [ ZFC Hs 2 ] . From (2.1.47)-(2.1.50) one obtains RHs2 ∈ RHs2 ⇐⇒ RHs2 /∈ RHs2 . (2.1.51) But (2.1.51) immediately gives a contradiction( RHs2 ∈ RHs2 ) ∧ ( RHs2 /∈ RHs2 ) . (2.1.52) (b) Note that the contradiction (2.1.52) that is a contradiction inside ZFC Hs 2 for the reason that the countable set IHs2 is a standard set in a sense of the set theory ZFC Hs 2 , see Remark 2.1.15 (a)-(c) and Remark 2.1.4. Theorem 2.1.1.Let ZFC Hs 2 be a theory ZFC Hs 2 , ZFCHs2 + ∃M ZFCHs2 st and Wff1 [ ZFC Hs 2 ] ∈ ∈ MZFC Hs 2 st .Then set theory ZFC Hs 2 is inconsistent. Proof. Immediately from (2.1.52). Remark 2.1.20. In order to obtain a contradiction inside ZFC Hs 2 , in more general case,i.e., without any refference to Assumption 2.1.1 we introduce the following definitions. Definition 2.1.13. We define now the countable set Γ∗Hsν / ∼ν by the following formula ∀y { [y]Hs ∈ Γ ∗Hs ν / ∼ν ⇐⇒ ( [y]Hs ∈ Γ Hs ν / ∼ν ) ∧ FrHs2 (y, v) ∧ [∃!XΨy,ν (X)] } . (2.1.53) Remark 2.1.21. Note that from the axiom of separation it follows directly that Γ∗ν/ is a set in the sense of the set theory ZFC Hs 2 . Definition 2.1.14. We define now the countable set I∗Hs2 by the following formula ∀Y { Y ∈ I∗Hs2 ⇐⇒ ∃y [( [y] ∈ Γ∗Hsν / ∼ν )]} . (2.1.54) Note that from the axiom schema of replacement (1.1.1) it follows directly that I∗Hs2 is a set in a sense of the set theory ZFC Hs 2 . Definition 2.1.15. We define now the countable set R∗Hs2 by formula ∀X {[( X ∈ I∗Hs2 ) ∧ (X ∈ Mst) ] ∧ [ X ∈ R∗Hs2 ⇐⇒ X /∈ X ]} . (2.1.55) 48 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 Note that from the axiom schema of separation it follows directly that R∗Hs2 is a set in the sense of the set theory ZFC Hs 2 . Remark 2.1.22. Note that R∗Hs2 ∈ I∗Hs2 since R∗Hs2 is definable by the following formula Ψ∗Mst ( Z,I∗Hs2 ) , Ψ∗ ( Z,I∗Hs2 ) , ∀X {( X ∈ I∗Hs2 ∧X ∈ Mst ) [X ∈ Z ⇐⇒ X /∈ X] } , (2.1.56) where obviously Ψ∗ ( Z,I∗Hs2 ) ∈ Wff1,X [ ZFC Hs 2 ] . Theorem 2.1.2.Set theory ZFC Hs 2 is inconsistent. Proof. From (2.1.55) and Remark 2.1.22 we obtain R∗Hs2 ∈ R∗Hs2 ⇐⇒ R∗Hs2 /∈ R∗Hs2 . (2.1.57) From (2.1.57) one obtains a contradiction( R∗Hs2 ∈ R∗Hs2 ) ∧ ( R∗Hs2 /∈ R∗Hs2 ) . (2.1.58) Definition 2.1.16. Let ZFC Hs 2 be a set theory ZFC Hs 2 , ZFCHs2 + ∃M ZFCHs2 Nst . We assume now that:∃MZFC Hs 2 st such that M ZFCHs2 st ⊂ M ZFCHs2 Nst .Then we will say that M ZFCHs2 st is a standard part of M ZFCHs2 Nst . Theorem 2.1.3. Set theory ZFC Hs 2 is inconsistent. Proof. Similarly to proof of the Theorem 2.1.2 but with quantifiers bounded on standard part M ZFCHs2 st of M ZFCHs2 Nst . Definition 2.1.17. Let ∆ be an standard set in the sense of the set theory ZFC.We will say that: (i) a set ∆ is admissible relative to model M ZFCHs2 Nst iff Con ( ZFC Hs 2 ) =⇒ Con ( ZFC Hs 2 + ( ∆ ∈ MZFC Hs 2 Nst )) . (2.1.59) (ii) a set ∆ is not admissible relative to model M ZFCHs2 Nst iff Con ( ZFC Hs 2 ) =⇒ ¬Con ( ZFC Hs 2 + ( ∆ ∈ MZFC Hs 2 Nst )) . (2.1.60) (iii) a set ∆ is absolute not admissible iff ∆ is not admissible relative to any model M ZFCHs2 Nst . Definition 2.1.18. Let Ξ1,X by a ZFC-set of the all the first order 1-place open wff's of the set theory ZFC Hs 2 , then we abbreviate Ξ1,X , Wff1,X [ ZFC Hs 2 ] . Theorem 2.1.4.(1) Set theory ZFC Hs 2 + ( Wff1,X [ ZFC Hs 2 ] ∈ MZFC Hs 2 Nst ) is inconsistent. (2) A set Wff1,X [ ZFC Hs 2 ] absolute is not admissible. 49 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 Proof. Similarly to proof of the Theorem 2.1.2 since canonical Gödel encoding holds by property N ∈ MZFC Hs 2 Nst . Proof. (2) Immediately from (1) and Definition 2.1.17. 2.2 Derivation of the Inconsistent Definable Set in Set Theory ZFCst In this section we obtain a contradiction in the set theory ZFCst , ZFC+∃MZFCst . by using a set of the all sets definable by 1-place open wff's of the set theory ZFCst. Definition 2.2.1. Let stX be a set of the all 1-place open wff's Ψ (X) of the set theory ZFCst with all bound variables restricted to standard model Mst that contains free occurrences of the individual variable X with Godel number v and we will be denoted these wff's by Ψ (X) = ΨMst (X) ,ΨX = ΨX,Mst ,Ψy,ν (X) = Ψy,ν,Mst (X) , y, ν ∈ N.We define now a set ΓstX $stX by the following second order formula ∀Ψ(X) [ Ψ(X) ∈ ΓstX ⇐⇒ ( ∃!X ( X ∈ MZFCst ) Ψ(X) ) ∧ ( Ψ(X) ∈stX )] . (2.2.1) or in the following equivalent form ∀Ψ(X) [ Ψ(X) ∈ ΓstX ⇐⇒ ∃yFrZFC(y, v) ↘ [(gZFC (Ψ (X)) = y) ∧ (gZFC (X) = ν)] ∧ ( ∃!X ( X ∈ MZFCst ) Ψ(X) ) ∧ ( Ψ(X) ∈stX )] , (2.2.1.a) see Remark 2.2.2 (ix) and Eq.(2.2.). Note that there exist a set ΓstX by the second order separaton axiom of ZFCHs2 . Notation 2.2.1. In this subsection we often write for short Ψ (X) ,ΨX ,Ψy,ν (X) instead ΨMst (X) ,ΨX,Mst ,Ψy,ν,Mst (X) but this should not lead to a confusion. Assumption 2.2.1. We assume now for simplicity but without loss of generality that st X ∈ Mst (2.2.1.b) and therefore by definition of model Mst one obtains Γ st X ∈ Mst. Definition 2.2.2. Let Ξ1,X be a ZFC-set of the all 1-open wiff's of the set theory ZFCst,then we abbreviate Ξ1,X , Wff1,X [ZFCst] . Definition 2.2.3.LetstX be a setstX = ℘−1Ξ1 (stX) ,and Ψ (X) , ΨX = ℘−1Ξ1 (Ψ (X)) where one-one function ℘−1Ξ1 defined in sec.2.1,see Remark 2.1.5 and Eq.(2.1.22). Remark 2.2.1.(i)We define now a set ΓstX = ℘ −1 Ξ1 ( ΓstX ) , ΓstX $stX by the following first order formula with quantifiers over first order individual varables ΨX and X : ∀ΨX [ ΨX ∈ ΓstX ⇐⇒ ∃!X ( X ∈ MZFCst ) ΨX ∧ ( ΨX ∈stX)] , (2.2.2) (where we write ΨX instead Ψ (X)) or in the following equivalent form 50 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 ∀ΨX [ ΨX ∈ ΓstX ⇐⇒ ∃yFrZFC(y, v) ↘[( ĝZFC ( ΨX ) = y ) ∧ ( ĝZFC ( X ) = ν )] ∧ ( ∃!X ( X ∈ MZFCst ) ΨX ) ∧ ( ΨX ∈stX)] , (2.2.2.a) where one-one function ĝZFC ( ΨX ) = y is defined below by Eq.(2.2.4), see Remark 2.2.3. Note that there exist a set ΓstX by the (first order) separaton axiom of ZFC. (ii) Note that second order definition (2.2.1) and first order definition (2.2.2) are equivalent. We abbreviate now: (a) ICZFC(x): x is the Godel number of an individual constant of ZFC, (b) FLZFC(x): x is the Godel number of a function letter of ZFC, (c) PLZFC(x): x is the Godel number of a predicate letter of ZFC. Remark 2.2.2. Note that in order to obtain by using only first order logic the formal definitions of the first order predicates EVblZFC(x),EICZFC(x),EFLZFC(x), ...,FrZFC(y, v) from the first order predicates ICZFC(x),FLZFC(x),PLZFC(x) one needs the following first order predicates: (i) EVblZFC(x, α) : x is the Godel number of the 1-tuple α ∈ ∆1, ∆1 = ℘−1∆1 (∆1) correspondinging to the individual variable α ∈ ∆1, α = ℘∆1 (α) ,see Remark 2.1.2 (i). (ii) EICZFC(x, β) : x is the Godel number of the 1-tuple β ∈ ∆2, ∆2 = ℘−1∆2 (∆2) correspondinging to the individual constant β ∈ ∆2, β = ℘∆2 ( β ) ,see Remark 2.1.2 (ii). (iii) EFLZFC(x, γ) : x is the Godel number of the 1-tuple γ ∈ ∆3, ∆3 = ℘−1∆3 (∆3) correspondinging to the function letter γ ∈ ∆3, γ = ℘∆3 (γ) ,see Remark 2.1.2 (iii). (iv) EPLZFC(x, δ) : x is the Godel number of the 1-tuple δ ∈ ∆4, ∆4 = ℘−1∆4 (∆4) correspondinging to the predicate letter δ ∈ ∆4, δ = ℘∆4 ( δ ) ,see Remark 2.1.2 (iv). (v) GdZFC(x, ζ) : x is the Godel number of the element ζ ∈ Ŝ# of the set Ŝ# = ∪n∈NŜn, Ŝ = ∆1 ∪ ∆2 ∪ ∆3 ∪ ∆4, correspondinging to the expression ζ ∈ S#, of ZFC,where S# = ∪n∈NSn,S = ∆1 ∪∆2 ∪∆3 ∪∆4,see Definition 2.1.6. (vi) TrmZFC(x, τ) : x is the Godel number of the element τ ∈ Υ1 of the set Υ = ℘−1τ (Υ1) ,corres-pondinging to the term τ = ℘τ (τ) of ZFC. (vii) AtfmlZFC(x, π) : x is the Godel number of the element π ∈ Σ1 of the set Σ1 = ℘−1π (Σ1) ,corres-ponding-ing to atomic wff π = ℘π (π) of ZFC. 51 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 (viii) FmlZFC(y, φ) : y is the Godel number of the element φ ∈ Ξ1 of the set Ξ1 = ℘−1Ξ1 (Ξ1) ,corres-pondinging to the wff formula φ = ℘Ξ1 (φ) of ZFC. (ix) FrZFC(y, v, π) : y is the Godel number of the element π ∈ Ξ1,ν of the set Ξ1,ν = ℘−1Ξ1,ν (Ξ1,ν) ,correspondinging to the wff formula or term π = ℘Ξ1,ν (π) of ZFC that contains free occurrences of the variable with Godel number v. Thus finally we obtain: EVblZFC(x) ⇐⇒ ∃α ( α ∈ ∆1 ) EVblZFC(x, α) ⇐⇒ (∃z)z<x(1 ≤ z ∧ x = 213+8z), EICZFC(x) ⇐⇒ ∃β ( β ∈ ∆2 ) EICZFC(x, β) ⇐⇒ (∃y)y<x(ICZFC(y) ∧ x = 2y), EFLZFC(x) ⇐⇒ ∃γ ( γ ∈ ∆3 ) EFLZFC(x, γ) ⇐⇒ (∃y)y<x(FLZFC(y) ∧ x = 2y), EPLZFC(x) ⇐⇒ ∃δ ( δ ∈ ∆4 ) EPLZFC(x, δ) ⇐⇒ (∃y)y<x(PLZFC(y) ∧ x = 2y), GdZFC(x) ⇐⇒ ∃ζ∃n ( ζ ∈ Ŝn ) GdZFC(x, ζ) ⇐⇒ EVblZFC(x) ∨EICZFC(x) ∨EFLZFC(x) ∨EPLZFC(x)∨ x = 27 ∨ x = 29 ∨ x = 211 ∨ x = 213∨ (∃u)u<x(∃v)v<x(x = u ∗ v ∧GdZFC(u) ∧GdZFC(v)). TrmZFC(x) ⇐⇒ ∃τ ( τ ∈ Υ ) TrmZFC(x, τ) ⇐⇒ EVblZFC(x) ∨EICZFC(x) ∨ (∃y)y<(px!)x [x = (y)lh(y) *−1∧ lh(y) = ArgT((x)0) + 1 ∧ FLZFC(((y)0)0) ∧ ((y)0)1 = 3∧ lh((y)0) = 2 ∧ (∀u) u<lh(y) * −1 (∃v)v<x((y)u+1 = (y)u ∗ v ∗ 27 ∧TrmZFC(v))∧ (∃v)v<x((y) lh(y) * −1 = (y) lh(y) * −2 ∗ v ∗ 25 ∧TrmZFC(v))]. AtfmlZFC(x) ⇐⇒ ∃π ( π ∈ Σ ) AtfmlZFC(x, π) ⇐⇒ (∃y)y<(px!)x [x = (y)lh(y) *−1 ∧ lh(y) = ArgP ((x)0) + 1∧ PLZFC(((y)0)0) ∧ ((y)0)1 = 3 ∧ lh((y)0) = 2∧ (∀u) u<lh(y) * −2 (∃v)v<x((y)u+1 = (y)u ∗ v ∗ 27 ∧TrmZFC(v))∧ (∃v)v<x((y) lh(y) * −1 = (y) u<lh(y) * −2 ∗ v ∗ 25 ∧TrmZFC(v))]. FmlZFC(y) ⇐⇒ ∃φ ( φ ∈ Ξ ) FmlZFC(y, φ) ⇐⇒ AtfmlZFC(y) ∨ (∃z)z<y[(FmlZFC(z)∧ = 23 ∗ 29 ∗ z ∗ 25)∨ (FmlZFC((z)0) ∧ FmlZFC((z)1 ∧ y = 23 ∗ (z)0 ∗ 211 ∗ (z)1 ∗ 25)∨ (FmlZFC((z)0) ∧EVblZFC((z)1) ∧ y = 23 ∗ 23 ∗ 213 ∗ ((z)1 ∗ 2 5 ∗ (z)0 ∗ 25)]. FrZFC(y, v) ⇐⇒ ∃π [ (π ∈ Ξ1,ν) ∨ ( π ∈ Υ1 )] FrZFC(y, v,π) ⇐⇒ (FmlZFC(y) ∨TrmZFC(y)) ∧EVblZFC(2v) ∧ ¬SubstZFC(y, y, 213+8v, v). (2.2.3) Remark 2.2.3. (i)LetgZFCst (u) be a Gödel number of given an expression u ∈ Ω of the language of the set theory ZFCst , ZFC + ∃MZFCst . Recall that ℘−1Ω1 (Ω1) = Ω1 see Definition 2.1.6. We set now ĝZFCst (u) = gZFCst (u) . (2.2.4) 52 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 (ii) Let FrZFC(y, v) be the relation : y is the Gödel number of a wff of the set theoryZFCst that contains free occurrences of the variable X with Gödel number v,see Eq.(2.2.3). (iii) Note that the relation FrZFC(y, v) is expressible in ZFCst by a wff Fr ZFC(y, v) (iv) Note that for any y, v ∈ N by definition of the relation FrZFC(y, v) follows that FrZFC(y, v) ⇐⇒ ∃!ΨX [(gZFCst (ΨX) = y) ∧ (gZFCst (X) = ν)] ⇐⇒ ∃!ΨX [( ĝZFCst ( ΨX ) = y ) ∧ ( ĝZFCst ( X ) = ν )] , (2.2.5) where ΨX = Ψ(X) is a unique wff of ZFCst which contains free occurrences of the variable X with Gödel number v.We denote such unique wff Ψ (X) defined by equivalence (2.2.5) by symbol Ψy,ν (X) ,i.e. FrZFC(y, v) ⇐⇒ ∃!Ψy,ν (X) [(gZFCst (Ψy,ν (X)) = y) ∧ (gZFCst (X) = ν)] ⇐⇒ ∃! Ψy,ν (X) [( gZFCst ( Ψy,ν (X) ) = y ) ∧ ( gZFCst ( X ) = ν )] . (2.2.6) Remark 2.2.4. Note that a function gZFCst (Ψy,ν (X)) = y is expressible in set theory ZFCst by a wff of the set theory ZFCst that contains free occurrences of the variable y ∈ N. Note that any formula Ψy,ν (X) is given by an expression u0u1..uj ...ur,i.e. Ψy,ν (X) =: u0u1..uj ...ur, where each uj is a symbol of ZFCst.We introduce now a functions [Ψy,ν (X) ; j] : Ψy,ν (X) → uj , j = 0, 1, ...,i.e. [Ψy,ν (X) ; j] =: uj and revrite expression u0u1..uj ...ur in the following equivalent form [Ψy,ν (X) ; 0] [Ψy,ν (X) ; 1] ... [Ψy,ν (X) ; j] ... [Ψy,ν (X) ; r] . (2.2.7) By definitions we obtain that gZFCst (Ψy,ν (X)) = y ⇐⇒ y = 2g([Ψy,ν(X);0]) * 3g([Ψy,ν(X);1]) * ... * pg([Ψy,ν(X);j])j * ... * p g([Ψy,ν(X);r]) r . (2.2.8) and ĝZFCst ( Ψy,ν (X) ) = y ⇐⇒ y = 2ĝ ([ Ψy,ν(X);0 ]) * 3ĝ ([ Ψy,ν(X);1 ]) * ... * p ĝ ([ Ψy,ν(X);j ]) j * ... * p ĝ ([ Ψy,ν(X);r ]) r . (2.2.9) correspondingly. Let us denote by (y)j the exponent g ([Ψy,ν (X) ; j]) in this factorization y = 2g([Ψy,ν(X);0]) * 3g([Ψy,ν(X);1]) * ... * pg([Ψy,ν(X);j])j * ... * p g([Ψy,ν(X);r]) r . (2.2.10) If y = 1, (y)j = 1 for all j. If x = 0, we arbitrarily let (y)j = 0 for all j. Then the functions (y)j , j = 0, 1, ... is primitive recursive, since (y)j = μz<y ( pzj |y ∧ ¬pz+1j |y ) ,is primitive recursive. Thus the function (y)j is expressible in set theory ZFCst by formula denoted below by λj (y, g ([Ψy,ν (X) ; j])) . For y > 0, let lh(y) be the number of non-zero exponents in the factorization of y into powers of primes, or, equivalently, the number of distinct primes that divide y. (i) Let lh(0) = 0, then lh(y) is primitive recursive. Thus function gZFCst (Ψy,ν (X)) = y is expressible in set theory ZFCst by the following formula Ξ (Ψy,ν (X) , y) Ξ (Ψy,ν (X) , y) ⇐⇒ j≤lh(y)λj (y, g ([Ψy,ν (X) ; j])) . (2.2.11) 53 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 (ii) function gZFCst (Ψy,ν (X)) = y is expressible in set theory ZFCst by the following formula Ξ (Ψy,ν (X) , y) Ξ ( Ψy,ν (X), y ) ⇐⇒ j≤lh(y)λj ( y, ĝ ([ Ψy,ν (X); j ])) . (2.2.12) Definition 2.2.4. Let stν be a set of the all Gödel numbers of the 1-place open wff's of the set theory ZFCst that contains free occurrences of the variable X with Gödel number v,i.e. st ν = { y ∈ N| ⟨y, ν⟩ ∈ FrZFC(y, v) } , (2.2.13) or in the following equivalent form: ∀y (y ∈ N) [ y ∈stν ⇐⇒ (y ∈ N) ∧ FrZFC(y, v) ] . (2.2.14) We define now a set Γstν $Hsν by the following first order formula ∀y (y ∈ N) { y ∈ Γstν ⇐⇒ ( y ∈stν ) ∧∃ΨX [( ĝZFCst ( ΨX ) = y ) ∧ ( ĝ ZFC Hs 2 ( X ) = ν ) ( ∃!X ( X ∈ MZFCst ) ΨX )]} (2.2.15) where ΨX = Ψ(X) is a unique wff of ZFCst which contains free occurrences of the variable X with Gödel number v. or in the following equivalent form ∀y (y ∈ N) [ y ∈ Γstν ⇐⇒ ( y ∈stν ) ∧ ∃yFrZFC(y, v) ↘ ∧∃ΨX [( ĝZFC ( ΨX ) = y ) ∧ ( ĝ ZFC Hs 2 ( X ) = ν )] ∧ ( ∃!X ( X ∈ MZFCst ) ΨX )] , (2.2.16) Remark 2.2.5. Note that from the axiom of separation it follows directly that Γstν is a set in the sense of the set theory ZFCst. Definition 2.2.5. Let Ψ1 = Ψ1 (X) and Ψ2 = Ψ2 (X) be 1-place open wff's of the set theory ZFC. (i) We define now the equivalence relation ( * ∼X * ) ⊂ ΓstX × ΓstX by Ψ1 (X) ∼X Ψ2 (X) ⇐⇒ Ψ1 (X) ∼X Ψ2 (X) ⇐⇒ ( ∀X ( X ∈ MZFCst ) [Ψ1 (X) ⇐⇒ Ψ2 (X)] ) (2.2.17) or more precisely ∀Ψ1∀Ψ2 ( Ψ1 ∼X Ψ2 ) ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) {[ Ψ1 (X) ∼X Ψ2 (X) ] ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) [ ∀X ( X ∈ MZFC Hs 2 st ) [Ψ1 (X) ⇐⇒ Ψ2 (X)] ]} ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) {[ Ψ1 (X) ∼X Ψ2 (X) ] ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) [ ∀X ( X ∈ MZFC Hs 2 st ) Ψ1 (X) ⇐⇒ ∀X ( X ∈ MZFC Hs 2 st ) Ψ2 (X) ]} . (2.2.18) or in the following equivalent form ∀Ψ1∀Ψ2 ( Ψ1 ∼X Ψ2 ) ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) [ Ψ1 (X) ∼X Ψ2 (X) ] ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) {[ Ψ1 (X) ∼X Ψ2 (X) ] ⇐⇒ ∃y1FrHs1 (y1, v)∃y2FrHs1 (y2, v) ↘[( ĝZFC ( Ψ1 (X) ) = y1 ) ∧ ( ĝZFC ( Ψ2 (X) ) = y2 ) ∧ ( ĝZFC ( X ) = ν )] ∧[ ∀X ( X ∈ MZFCst ) Ψ1 (X) ⇐⇒ ∀X ( X ∈ MZFCst ) Ψ2 (X) ]} . (2.2.19) 54 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 (ii) A subset ΛstX of Γ st X such that Ψ1 (X) ∼X Ψ2 (X) holds for all Ψ1 (X) and Ψ2 (X) in Λ st X , and never for Ψ1 (X) in Λ st X and Ψ2 (X) outside Λ st X , is an equivalence class of Γ st X . (iii) For any Ψ (X) ∈ ΓstX let [ Ψ (X) ] st , { Φ (X) ∈ ΓstX ∣∣∣Ψ (X) ∼X Φ (X)} denote the equivalence class to which Ψ (X) belongs. All elements of ΓstX equivalent to each other are also elements of the same equivalence class. (iv) The set of all possible equivalence classes of ΓstX by  X , denoted Γ st X/ ∼X ΓstX/ ∼X, {[ Ψ (X) ] st ∣∣∣Ψ (X) ∈ ΓstX } . (2.2.20) Definition 2.2.6.(i)We define now the equivalence relation (* ∼ν *) ⊂ Γstν × Γstν in the sense of the set theory ZFCst by y1 ∼ν y2 ⇐⇒ [ Ψy1,ν (X) ∼X Ψy2,ν (X) ] (2.2.21) Note that from the axiom of separation it follows directly that the equivalence relation (* ∼ν *) is a relation in the sense of the set theory ZFCst. (ii) A subset Λstν of Γ st ν such that y1 ∼ν y2 holds for all y1 and y1 in Λstν ,and never for y1 in Λstν and y2 outside Λ st ν , is an equivalence class of Γ st ν . (iii) For any y ∈ Γstν let [y]st , { z ∈ Γstν |y ∼ν z } denote the equivalence class to which y belongs. All elements of Γstν equivalent to each other are also elements of the same equivalence class. (iv)The set of all possible equivalence classes of Γstν by  ν , denoted Γ st ν / ∼ν Γstν / ∼ν, { [y]st |y ∈ Γ st ν } . (2.2.22) Remark 2.2.6. Note that from the axiom of separation it follows directly that Γstν / ∼ν is a set in the sense of the set theory ZFCst. Definition 2.2.7. Let Ist be the countable set of the all sets definable by 1-place open wff of the set theory ZFCst, i.e. by using second order lenguage correspondng definition reads ∀Y { Y ∈ Ist ⇐⇒ ∃Ψ(X) [( [Ψ (X)]st ∈ Γ st X/ ∼X ) ∧ [∃!X [Ψ (X) ∧ Y = X]] ]} . (2.2.23) We rewrite now (2.2.23) by using first order lenguage of the set theory ZFCst in the following equvalent form ∀Y { Y ∈ Ist ⇐⇒ ∃Ψ (X) [([ Ψ (X) ] st ∈ ΓstX/ ∼X ) ∧ [∃!X [Ψ (X) ∧ Y = X]] ]} . (2.2.24) Remark 2.2.7. Note that from the axiom of replacement it follows directly that Γstν / ∼ν is a set in the sense of the set theory ZFCst. Definition 2.2.8. We rewrite now (2.2.24) in the following equivalent form ∀Y { Y ∈ Ist ⇐⇒ ∃Ψ (X) [([ Ψ (X) ] st ∈ Γ∗stX / ∼X ) ∧ (Y = X) ]} , (2.2.25) 55 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 where the countable set Γ∗stX / ∼X is defined by ∀Ψ (X) {[ Ψ (X) ] st ∈ Γ∗stX / ∼X ⇐⇒ [([ Ψ (X) ] st ∈ ΓstX / ∼X ) ∧ ∃!XΨ(X) ]} (2.2.26) Definition 2.2.9. Let Rst be the countable set of the all sets such that ∀X (X ∈ Ist) [X ∈ Rst ⇐⇒ X /∈ X] . (2.2.27) Remark 2.2.8. Note that Rst ∈ Ist since Rst is a set definable by 1-place open wff Ψ (Z,Ist) , ∀X (X ∈ Ist) [X ∈ Z ⇐⇒ X /∈ X] . (2.2.28) From (2.2.27) and Remark 2.2.8 one obtains directly Rst ∈ Rst ⇐⇒ Rst /∈ Rst. (2.2.29) But (2.2.29) immediately gives a contradiction (Rst ∈ Rst) ∧ (Rst /∈ Rst) . (2.2.30) The contradiction (2.2.30) it is a true contradiction inside ZFCst for the reason that the countable set Ist is a set in the sense of the set theory ZFCst. Definition 2.2.10. Let Ξ1,X be a ZFC-set of the all first order 1-open wiff's of the set theory ZFCst,then we abbreviate Ξ1,X , Wff1,X [ZFCst] . Theorem 2.2.1.Let ZFC∗st be a theory ZFC ∗ st , ZFC + ∃MZFCst and Wff1,X [ZFCst] ∈ MZFCst . Then set theory ZFC∗st is inconsistent. Proof. Immediately from (2.2.29). Remark 2.2.9. In order to obtain a contradiction inside ZFC Hs 2 , in more general case,i.e., without any refference to Assumption 2.2.1 we introduce the following definitions. Definition 2.2.11.We define now countable set Γ∗stν / ∼νby the following formula ∀y { [y]st ∈ Γ ∗st ν / ∼ν ⇐⇒ ( [y]st ∈ Γ st ν / ∼ν ) ∧ Frst(y, v) ∧ [∃!XΨy,ν (X)] } . (2.2.31) Remark 2.2.10. Note that from the axiom of separation it follows directly that Γ∗stν / ∼ν is a set in the sense of the set theory ZFCst. Definition 2.2.12. We define now the countable set I∗st by the following formula 56 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 ∀Y { Y ∈ I∗st ⇐⇒ ∃y [( [y]st ∈ Γ ∗st ν / ∼ν ) ∧ (ĝZFCst (X) = ν) ∧ Y = X ]} . (2.2.32) Note that from the axiom schema of replacement it follows directly that I∗st is a set in the sense of the set theory ZFCst. Definition 2.2.13. We define now the countable set R∗st by the following formula ∀X (X ∈ I∗st) [X ∈ R∗st ⇐⇒ X /∈ X] . (2.2.33) Note that from the axiom schema of separation it follows directly that R∗st is a set in the sense of the set theory ZFCst. Remark 2.2.11.Note that R∗st ∈ I∗st since R∗st is a definable by the following formula Ψ∗ (Z) , ∀X (X ∈ I∗st) [X ∈ Z ⇐⇒ X /∈ X] . (2.2.34) Theorem 2.2.2. Set theory ZFCst is inconsistent. Proof. From (2.2.34) and Remark 2.2.11 we obtain R∗st ∈ R∗st ⇐⇒ R∗st /∈ R∗st . (2.2.35) From (2.2.34) immediately one obtains a contradiction (R∗st ∈ R∗st) ∧ (R∗st /∈ R∗st) . 2.3 Derivation of the Inconsistent Definable Set in ZFCNst Definition 2.3.1. Let PA be a first order theory which contain usual postulates of Peano arithmetic [17] and recursive defining equations for every primitive recursive function as desired.So for any (n+1)-place function f defined by primitive recursion over any n-place base function g and (n+ 2)-place iteration function h there would be the defining equations: (i) f (0, y1, ..., yn) = g (y1, ..., yn) ,(ii) f (x+ 1, y1, ..., yn) = h (x, f (x, y1, ..., yn) , y1, ..., yn) . Designation 2.3.1.(i) Let MZFCNst be a nonstandard model of ZFC and let M PA st be a standard model of PA.We assume now that MPAst ⊂ MZFCNst and denote such nonstandard model of the set theory ZFC by MZFCNst [ PA ] .(ii) Let ZFCNst be the theory ZFCNst = ZFC +M ZFC Nst [ PA ] . (2.3.1) Designation 2.3.2.(i) Let gZFCNst (u) be a Gödel number of given an expression u of the set theory ZFCNst , ZFC + ∃MZFCNst [ PA ] . (ii) Let FrNst(y, v) be the relation : y is the Gödel number of a wff of the set theory ZFCNst that contains free occurrences of the variable X with Gödel number v,see Remark 2.3.2. (iii) Note that the relation FrNst(y, v) is expressible in ZFCNst by a wff FrNst(y, v) (iv) Note that for any y, v ∈ N by definition of the relation FrNst(y, v) follows that FrNst(y, v) ⇐⇒ ∃!Ψ (X) [(gZFCNst (Ψ (X)) = y) ∧ (gZFCNst (X) = ν)] , (2.3.2) 57 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 where Ψ (X) is a unique wff of ZFCst which contains free occurrences of the variable X with Gödel number v.We denote a unique wff Ψ (X) defined by using equivalence (2.3.2) by symbol Ψy,ν (X) ,i.e. FrNst(y, v) ⇐⇒ ∃!Ψy,ν (X) [(gZFCNst (Ψy,ν (X)) = y) ∧ (gZFCNst (X) = ν)] , (2.3.3) Definition 2.3.2. Let NstX be a set of the all 1-place open wff's Ψ (X) (with all bound variables restricted to nonstandard model MZFCNst of the set theory ZFC) that contains free occurrences of the individual variable X with Godel number v and we will be denoted these wff's by Ψ (X) ,ΨX ,Ψy,ν (X) , y, ν ∈ N. We define now a set ΓNstX $NstX by the following second order formula ∀Ψ(X) [ Ψ(X) ∈ ΓNstX ⇐⇒ ( ∃!X ( X ∈ MZFCNst ) Ψ(X) ) ∧ ( Ψ(X) ∈stX )] . (2.3.4) or in the following equivalent form ∀Ψ(X) [ Ψ(X) ∈ ΓNstX ⇐⇒ ( ∃y ∈ MPAst ) FrZFCNst(y, v) ↘ [(gZFCNst (Ψ (X)) = y) ∧ (gZFCNst (X) = ν)] ∧ ( ∃!X ( X ⊂ MZFCNst ) Ψ(X) ) ∧ ( Ψ(X) ∈NstX )] , (2.3.4.a) Note that there exist a set ΓNstX by the second order separaton axiom of ZFC Hs 2 . Assumption 2.3.1. We assume now for simplicity but without loss of generality that Nst X ∈ MNst (2.2.1.b) and therefore by definition of model MZFCNst one obtains Γ Nst X ∈ MZFCNst . Definition 2.3.3. Let Ξ1,X be a ZFC-set of the all 1-open wiff's of the set theory ZFCNst, then we abbreviate Ξ1,X , Wff1,X [ZFCNst] . Definition 2.3.4.LetNstX be a setNstX = ℘−1Ξ1 (NstX ) ,and Ψ (X) , ΨX = ℘−1Ξ1 (Ψ (X)) where one-one function ℘−1Ξ1 defined in sec.2.1,see Remark 2.1.5 and Eq.(2.1.22). Remark 2.3.1. (i)We define now a set ΓNstX = ℘ −1 Ξ1 ( ΓNstX ) , ΓNstX $NstX by the following first order formula with quantifiers over first order individual varables ΨX and X : ∀ΨX [ ΨX ∈ ΓNstX ⇐⇒ ∃!X ( X ∈ MZFCNst ) ΨX ∧ ( ΨX ∈NstX )] , (2.3.5) (where we write ΨX instead Ψ (X)) or in the following equivalent form ∀ΨX [ ΨX ∈ ΓstX ⇐⇒ ( ∃y ∈ MPAst ) FrZFCNst(y, v) ↘[( ĝZFCNst ( ΨX ) = y ) ∧ ( ĝZFCNst ( X ) = ν )] ∧ ( ∃!X ( X ∈ MZFCNst ) ΨX ) ∧ ( ΨX ∈NstX )] , (2.3.5.a) where one-one function ĝ where one-one function ĝZFCNst ( ΨX ) = y is defined below by Eq.(2.3.), see Remark 2.3.. Note that there exist a set ΓNstX by the (first order) separaton axiom of ZFC. (ii) Note that second order definition (2.3.4) and first order definition (2.3.5) are equivalent. 58 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 We abbreviate now: (a) ICZFCNst(x): x is the Godel number of an individual constant of ZFCNst, (b) FLZFCNst(x): x is the Godel number of a function letter of ZFCNst, (c) PLZFCNst(x): x is the Godel number of a predicate letter of ZFCNst. Remark 2.3.2. Note that in order to obtain by using only first order logic the formal definitions of the first order predicates EVblZFCNst(x),EICZFCNst(x),EFLZFCNst(x), ..., FrZFCNst(y, v) from the first order predicates ICZFCNst(x),FLZFCNst(x),PLZFCNst(x) one needs the following first order predicates: (i)EVblZFCNst(x, α) : x is the Godel number of the 1-tuple α ∈ ∆1, ∆1 = ℘−1∆1 (∆1) correspondinging to the individual variable α ∈ ∆1, α = ℘∆1 (α) ,see Remark 2.1.2 (i). (ii)EICZFCNst(x, β) : x is the Godel number of the 1-tuple β ∈ ∆2, ∆2 = ℘−1∆2 (∆2) correspondinging to the individual constant β ∈ ∆2, β = ℘∆2 ( β ) ,see Remark 2.1.2 (ii). (iii)EFLZFCNst(x, γ) : x is the Godel number of the 1-tuple γ ∈ ∆3, ∆3 = ℘−1∆3 (∆3) correspondinging to the function letter γ ∈ ∆3, γ = ℘∆3 (γ) ,see Remark 2.1.2 (iii). (iv)EPLZFCNst(x, δ) : x is the Godel number of the 1-tuple δ ∈ ∆4, ∆4 = ℘−1∆4 (∆4) correspondinging to the predicate letter δ ∈ ∆4, δ = ℘∆4 ( δ ) ,see Remark 2.1.2 (iv). (v) GdZFCNst(x, ζ) : x is the Godel number of the element ζ ∈ Ŝ# of the set Ŝ# = ∪n∈NŜn, Ŝ = ∆1 ∪ ∆2 ∪ ∆3 ∪ ∆4, correspondinging to the expression ζ ∈ S#, of ZFCNst,where S# = ∪n∈NSn,S = ∆1 ∪∆2 ∪∆3 ∪∆4,see Definition 2.1.6. (vi) TrmZFCNst(x, τ) : x is the Godel number of the element τ ∈ Υ1 of the set Υ = ℘−1τ (Υ1) ,correspondinging to the term τ = ℘τ (τ) of ZFCNst. (vii) AtfmlZFCNst(x, π) : x is the Godel number of the element π ∈ Σ1 of the set Σ1 = ℘−1π (Σ1) ,correspondinging to atomic wff π = ℘π (π) of ZFCNst. (viii) FmlZFCNst(y, φ) : y is the Godel number of the element φ ∈ Ξ1 of the set Ξ1 = ℘−1Ξ1 (Ξ1) ,correspondinging to the wff formula φ = ℘Ξ1 (φ) of ZFCNst. (ix) FrZFCNst(y, v, π) : y is the Godel number of the element π ∈ Ξ1,ν of the set Ξ1,ν = ℘−1Ξ1,ν (Ξ1,ν) ,correspondinging to the wff formula or term π = ℘Ξ1,ν (π) of ZFCNst that contains free occurrences of the variable with Godel number v. Thus finally we obtain: 59 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 EVblZFCNst (x) ⇐⇒ ∃α ( α ∈ ∆1 ) EVblZFCNst (x, α) ⇐⇒ (∃z ∈ MPAst )z<x(1 ≤ z ∧ x = 2 13+8z), EICZFCNst (x) ⇐⇒ ∃β ( β ∈ ∆2 ) EICZFCNst (x, β) ⇐⇒ (∃y ∈ MPAst )y<x(IC ZFCNst (y) ∧ x = 2y), EFLZFCNst (x) ⇐⇒ ∃γ ( γ ∈ ∆3 ) EFLZFCNst (x, γ) ⇐⇒ (∃y ∈ MPAst )y<x(FL ZFCNst (y) ∧ x = 2y), EPLZFCNst (x) ⇐⇒ ∃δ ( δ ∈ ∆4 ) EPLZFCNst (x, δ) ⇐⇒ (∃y ∈ MPAst )y<x(PL ZFCNst (y) ∧ x = 2y), GdZFCNst (x) ⇐⇒ ∃ζ∃n ( n ∈ MPAst ∧ ζ ∈ Ŝ n ) GdZFCNst (x, ζ) ⇐⇒ EVblZFCNst (x) ∨ EICZFCNst (x) ∨ EFLZFCNst (x) ∨ EPLZFCNst (x)∨ x = 27 ∨ x = 29 ∨ x = 211 ∨ x = 213∨ (∃u ∈ MPAst )u<x(∃v ∈ M PA st )v<x(x = u ∗ v ∧ Gd ZFCNst (u) ∧ GdZFCNst (v)). TrmZFCNst (x) ⇐⇒ ∃τ ( τ ∈ Υ ) TrmZFCNst (x, τ) ⇐⇒ EVblZFCNst (x) ∨ EICZFCNst (x) ∨ (∃y ∈ MPAst )y<(px!)x [x = (y) lh(y) * −1 ∧ lh(y) = ArgT((x)0) + 1 ∧ FL ZFCNst (((y)0)0) ∧ ((y)0)1 = 3∧ lh((y)0) = 2 ∧ (∀u) u<lh(y) * −1 (∃v)v<x((y)u+1 = (y)u ∗ v ∗ 27 ∧ TrmZFCNst (v))∧ (∃v)v<x((y) lh(y) * −1 = (y) lh(y) * −2 ∗ v ∗ 25 ∧ TrmZFCNst (v))]. AtfmlZFCNst (x) ⇐⇒ ∃π ( π ∈ Σ ) AtfmlZFCNst (x, π) ⇐⇒ (∃y)y<(px!)x [x = (y) lh(y) * −1 ∧ lh(y) = ArgP ((x)0) + 1∧ PLZFCNst (((y)0)0) ∧ ((y)0)1 = 3 ∧ lh((y)0) = 2∧ (∀u) u<lh(y) * −2 (∃v)v<x((y)u+1 = (y)u ∗ v ∗ 27 ∧ TrmZFCNst (v))∧ (∃v)v<x((y) lh(y) * −1 = (y) u<lh(y) * −2 ∗ v ∗ 25 ∧ TrmZFCNst (v))]. FmlZFCNst (y) ⇐⇒ ∃φ ( φ ∈ Ξ ) FmlZFCNst (y, φ) ⇐⇒ AtfmlZFCNst (y) ∨ (∃z ∈ MPAst )z<y [(Fml ZFCNst (z)∧ = 23 ∗ 29 ∗ z ∗ 25)∨ (FmlZFCNst ((z)0) ∧ FmlZFCNst ((z)1 ∧ y = 23 ∗ (z)0 ∗ 211 ∗ (z)1 ∗ 25)∨ (FmlZFCNst ((z)0)∧ ∧EVblZFCNst ((z)1) ∧ y = 23 ∗ 23 ∗ 213 ∗ ((z)1 ∗ 2 5 ∗ (z)0 ∗ 25)]. FrZFCNst (y, v) ⇐⇒ ∃π [( π ∈ Ξ1,ν ) ∨ ( π ∈ Υ1 )] FrZFCNst (y, v,π) ⇐⇒ (FmlZFCNst (y) ∨ TrmZFCNst (y)) ∧ EVblZFCNst (2v) ∧ ¬SubstZFCNst (y, y, 213+8v , v). (2.3.6) Remark 2.3.3. Let gZFCNst (u) be a Gödel number of given an expression u ∈ Ω of the language of the set theory ZFCNst , ZFC + ∃MZFCNst . Recall that ℘−1Ω1 (Ω1) = Ω1 see Definition 2.1.6. We set now ĝZFCNst (u) = gZFCNst (u) (2.3.7) (ii) Let FrZFCNst(y, v) be the relation : y is the Gödel number of a wff of the set theoryZFCNst that contains free occurrences of the variable X with Gödel number v,see Eq.(2.2.3) (iii) Note that the relation FrZFCNst(y, v) is expressible in ZFCNst by a wff FrZFCNst(y, v) (iv) Note that for any y, v ∈ N by definition of the relation FrZFCNst(y, v) follows that FrZFCNst(y, v) ⇐⇒ ∃!ΨX [(gZFCst (ΨX) = y) ∧ (gZFCNst (X) = ν)] ⇐⇒ ∃!ΨX [( ĝZFC Nst ( ΨX ) = y ) ∧ ( ĝZFCNst ( X ) = ν )] , (2.3.8) where ΨX = Ψ(X) is a unique wff of ZFCNst which contains free occurrences of the variable X with Gödel number v.We denote such unique wff Ψ (X) defined by equivalence (2.3.8) by symbol Ψy,ν (X) ,i.e. FrZFCNst(y, v) ⇐⇒ ∃!Ψy,ν (X) [(gZFCNst (Ψy,ν (X)) = y) ∧ (gZFCNst (X) = ν)] ⇐⇒ ∃! Ψy,ν (X) [( ĝZFCNst ( Ψy,ν (X) ) = y ) ∧ ( ĝZFCNst ( X ) = ν )] . (2.3.9) where ΨX = Ψ(X) is a unique wff of ZFCNst which contains free occurrences of the variable X with Gödel number v.We denote such unique wff Ψ (X) defined by equivalence (2.3.9) by symbol 60 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 Ψy,ν (X) ,i.e. FrZFCNst (y, v) ⇐⇒ ∃!Ψy,ν (X) [( gZFCNst ( Ψy,ν (X) ) = y ) ∧ ( gZFCNst (X) = ν )] ⇐⇒ ∃! Ψy,ν (X) [( ĝZFCNst ( Ψy,ν (X) ) = y ) ∧ ( ĝZFCNst ( X ) = ν )] . (2.3.10) Remark 2.3.4. Note that a function gZFCNst (Ψy,ν (X)) = y is expressible in set theory ZFCNst by a wff of the set theory ZFCNst that contains free occurrences of the variable y ∈ N. Note that any formula Ψy,ν (X) is given by an expression u0u1..uj ...ur,i.e. Ψy,ν (X) =: u0u1..uj ...ur, where each uj is a symbol of ZFCNst. We introduce now a functions [Ψy,ν (X) ; j] : Ψy,ν (X) → uj , j = 0, 1, ...,i.e. [Ψy,ν (X) ; j] =: uj and revrite expression u0u1..uj ...ur in the following equivalent form [Ψy,ν (X) ; 0] [Ψy,ν (X) ; 1] ... [Ψy,ν (X) ; j] ... [Ψy,ν (X) ; r] . (2.3.11) By definitions we obtain that gZFCNst (Ψy,ν (X)) = y ⇐⇒ y = 2g([Ψy,ν(X);0]) * 3g([Ψy,ν(X);1]) * ... * pg([Ψy,ν(X);j])j * ... * p g([Ψy,ν(X);r]) r . (2.3.12) and ĝZFCNst ( Ψy,ν (X) ) = y ⇐⇒ y = 2ĝ ([ Ψy,ν(X);0 ]) * 3ĝ ([ Ψy,ν(X);1 ]) * ... * p ĝ ([ Ψy,ν(X);j ]) j * ... * p ĝ ([ Ψy,ν(X);r ]) r . (2.3.13) correspondingly.Let us denote by (y)j the exponent g ([Ψy,ν (X) ; j]) in this factorization y = 2g([Ψy,ν(X);0]) * 3g([Ψy,ν(X);1]) * ... * pg([Ψy,ν(X);j])j * ... * p g([Ψy,ν(X);r]) r . (2.3.14) If y = 1, (y)j = 1 for all j. If x = 0, we arbitrarily let (y)j = 0 for all j. Then the functions (y)j , j = 0, 1, ... is primitive recursive, since (y)j = μz<y ( pzj |y ∧ ¬pz+1j |y ) ,is primitive recursive. Thus the function (y)j is expressible in set theory ZFCNst by formula denoted below by λj (y, g ([Ψy,ν (X) ; j])) . For y > 0, let lh(y) be the number of non-zero exponents in the factorization of y into powers of primes, or, equivalently, the number of distinct primes that divide y. Let lh(0) = 0, then lh(y) is primitive recursive. Thus (i) function gZFCNst (Ψy,ν (X)) = y is expressible in set theory ZFCNst by the following formula Ξ (Ψy,ν (X) , y) Ξ (Ψy,ν (X) , y) ⇐⇒ j≤lh(y)λj (y, g ([Ψy,ν (X) ; j])) . (2.3.15) (ii) function gZFCNst (Ψy,ν (X)) = y is expressible in set theory ZFCNst by the following formula Ξ (Ψy,ν (X) , y) Ξ ( Ψy,ν (X), y ) ⇐⇒ j≤lh(y)λj ( y, ĝ ([ Ψy,ν (X); j ])) . (2.3.16) Definition 2.3.5. Let gZFCNst (X) = ν.Let Γ Nst ν be a set of the all Gödel numbers of the 1-place open wff's of the set theory ZFCNst that contains free occurrences of the variable X with Gödel number v,i.e. ΓNstν = { y ∈ N| ⟨y, ν⟩ ∈ FrZFCNst(y, v) } , (2.3.17) or in the following equivalent form: 61 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 ∀y (y ∈ N) [ y ∈ ΓNstν ⇐⇒ (y ∈ N) ∧ FrZFCNst(y, v) ] . (2.3.18) Remark 2.3.5. Note that from the axiom of separation it follows directly that ΓNstν is a set in the sense of the set theory ZFCNst. We define now a set ΓNstν $Nstν by the following first order formula ∀y (y ∈ N) { y ∈ ΓNstν ⇐⇒ ( y ∈Nstν ) ∧∃ΨX [( ĝZFCNst ( ΨX ) = y ) ∧ ( ĝZFCNst ( X ) = ν ) ( ∃!X ( X ∈ MZFCNst ) ΨX )]} (2.3.19) where ΨX = Ψ(X) is a unique wff of ZFCNst which contains free occurrences of the variable X with Gödel number v. or in the following equivalent form ∀y (y ∈ N) [ y ∈ ΓNstν ⇐⇒ ( y ∈Nstν ) ∧ FrZFCNst(y, v) ↘ ∧∃ΨX [( ĝZFCNst ( ΨX ) = y ) ∧ ( ĝZFCNst ( X ) = ν )] ∧ ( ∃!X ( X ∈ MZFCNst ) ΨX )] , (2.3.20) Remark 2.3.6. Note that from the axiom of separation it follows directly that ΓNstν is a set in the sense of the set theory ZFCNst. Definition 2.3.6. Let Ψ1 = Ψ1 (X) and Ψ2 = Ψ2 (X) be 1-place open wff's of the set theory ZFCNst. (i) We define now the equivalence relation ( * ∼X * ) ⊂ ΓNstX × ΓNstX by Ψ1 (X) ∼X Ψ2 (X) ⇐⇒ Ψ1 (X) ∼X Ψ2 (X) ⇐⇒ ( ∀X ( X ∈ MZFCNst ) [Ψ1 (X) ⇐⇒ Ψ2 (X)] ) (2.3.21) or more precisely ∀Ψ1∀Ψ2 ( Ψ1 ∼X Ψ2 ) ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) {[ Ψ1 (X) ∼X Ψ2 (X) ] ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) [ ∀X ( X ∈ MZFCNst ) [Ψ1 (X) ⇐⇒ Ψ2 (X)] ]} ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) {[ Ψ1 (X) ∼X Ψ2 (X) ] ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) [ ∀X ( X ∈ MZFCNst ) Ψ1 (X) ⇐⇒ ∀X ( X ∈ MZFCNst ) Ψ2 (X) ]} . (2.3.22) or in the following equivalent form ∀Ψ1∀Ψ2 ( Ψ1 ∼X Ψ2 ) ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) [ Ψ1 (X) ∼X Ψ2 (X) ] ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) {[ Ψ1 (X) ∼X Ψ2 (X) ] ⇐⇒( ∃y1 ∈ MPAst ) FrZFCNst(y1, v) ( ∃y2 ∈ MPAst ) FrZFCNst(y2, v) ↘[( ĝZFCNst ( Ψ1 (X) ) = y1 ) ∧ ( ĝZFCNst ( Ψ2 (X) ) = y2 ) ∧ ( ĝZFCNst ( X ) = ν )] ∧[ ∀X ( X ∈ MZFCNst ) Ψ1 (X) ⇐⇒ ∀X ( X ∈ MZFCNst ) Ψ2 (X) ]} . (2.3.23) or in the following equivalent form ∀Ψ1∀Ψ2 ( Ψ1 ∼X Ψ2 ) ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) [ Ψ1 (X) ∼X Ψ2 (X) ] ⇐⇒ ∀Ψ1 (X)∀Ψ2 (X) {[ Ψ1 (X) ∼X Ψ2 (X) ] ⇐⇒ ∃y1 FrZFCNst(y1, v)∃y2 FrZFCNst(y2, v) ↘[( ĝZFCNst ( Ψ1 (X) ) = y1 ) ∧ ( ĝZFCNst ( Ψ2 (X) ) = y2 ) ∧ ( ĝZFCNst ( X ) = ν )] ∧[ ∀X ( X ∈ MZFCNst ) Ψ1 (X) ⇐⇒ ∀X ( X ∈ MZFCNst ) Ψ2 (X) ]} . (2.3.24) (ii) A subset ΛNstX of Γ Nst X such that Ψ1 (X) ∼X Ψ2 (X) holds for all Ψ1 (X) and Ψ2 (X) in Λ Nst X , 62 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 and never for Ψ1 (X) in Λ Nst X and Ψ2 (X) outside Λ Nst X , is an equivalence class of Γ Nst X . (iii) For any Ψ (X) ∈ ΓNstX let [ Ψ (X) ] Nst , { Φ (X) ∈ ΓNstX ∣∣∣Ψ (X) ∼X Φ (X)} denote the equivalence class to which Ψ (X) belongs. All elements of ΓNstX equivalent to each other are also elements of the same equivalence class. (iv) The set of all possible equivalence classes of ΓNstX by  X , denoted by Γ Nst X / ∼X : ΓNstX / ∼X, {[ Ψ (X) ] Nst ∣∣∣Ψ (X) ∈ ΓNstX } . (2.3.25) Definition 2.3.7.(i)We define now the equivalence relation (* ∼ν *) ⊂ ΓNstν × ΓNstν in the sense of the set theory ZFCNst by y1 ∼ν y2 ⇐⇒ [ Ψy1,ν (X) ∼X Ψy2,ν (X) ] (2.3.26) Note that from the axiom of separation it follows directly that the equivalence relation (* ∼ν *) is a relation in the sense of the set theory ZFCNst. (ii) A subset ΛNstν of Γ Nst ν such that y1 ∼ν y2 holds for all y1 and y1 in ΛNstν , and never for y1 in ΛNstν and y2 outside Λ Nst ν , is an equivalence class of Γ Nst ν . (iii) For any y ∈ ΓNstν let [y]Nst , { z ∈ ΓNstν |y ∼ν z } denote the equivalence class to which y belongs. All elements of ΓNstν equivalent to each other are also elements of the same equivalence class. (iv)The set of all possible equivalence classes of ΓNstν by  ν , denoted by Γ Nst ν / ∼ν : ΓNstν / ∼ν, { [y]Nst |y ∈ Γ Nst ν } . (2.3.27) Remark 2.3.7. Note that from the axiom of separation it follows directly that ΓNstν / ∼ν is a set in the sense of the set theory ZFCNst. Definition 2.3.8.Let INst be a countable set of the all sets definable by 1-place open wff of the set theory ZFCNst, i.e. by using second order lenguage correspondng definition reads ∀Y {Y ∈ INst ⇐⇒ ∃Ψ(X) [( [Ψ (X)]Nst ∈ Γ st X/ ∼X ) ∧ [ ∃!X ( X ∈ MZFCNst ) [Ψ (X) ∧ Y = X] ]]} . (2.3.28) We rewrite now (2.3.28) by using first order lenguage of the set theory ZFCNst in the following equvalent form ∀ {Y ∈ INst ⇐⇒ ∃Ψ (X) [([ Ψ (X) ] Nst ∈ ΓNstX / ∼X ) ∧ [ ∃!X ( X ∈ MZFCNst ) [Ψ (X) ∧ Y = X] ]]} . (2.3.29) Remark 2.3.8. Note that from the axiom of replacement it follows directly that Γstν / ∼ν is a set in the sense of the set theory ZFCNst. Definition 2.3.9.We rewrite now (2.3.29) in the following equivalent form ∀Y { Y ∈ Ist ⇐⇒ ∃Ψ (X) [([ Ψ (X) ] Nst ∈ Γ∗NstX / ∼X ) ∧ (Y = X) ]} , (2.3.30) where the countable set Γ∗NstX / ∼X is defined by 63 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 ∀Ψ (X) {[ Ψ (X) ] Nst ∈ Γ∗NstX / ∼X ⇐⇒[([ Ψ (X) ] Nst ∈ ΓstX / ∼X ) ∧ ∃!X ( X ∈ MZFCNst ) Ψ(X) ]} . (2.3.31) Definition 2.3.10. Let RNst be the countable set of the all sets such that ∀X (X ∈ INst) [X ∈ RNst ⇐⇒ X /∈ X] . (2.3.32) Remark 2.3.9. Note that Rst ∈ INst since RNst is a set definable by 1-place open wff Ψ (Z,INst) , ∀X (X ∈ INst) [X ∈ Z ⇐⇒ X /∈ X] . (2.3.33) From (2.3.32) and Remark 2.3.9 one obtains directly RNst ∈ RNst ⇐⇒ RNst /∈ RNst. (2.3.34) But (2.3.34) immediately gives a contradiction (RNst ∈ RNst) ∧ (RNst /∈ RNst) . (2.3.35) The contradiction (2.3.35) it is a true contradiction inside ZFCNst for the reason that the countable set INst is a set in the sense of the set theory ZFCNst. Definition 2.3.11. Let Ξ1,X be a ZFC-set of the all first order 1-open wiff's of the set theory ZFCNst,then we abbreviate Ξ1,X , Wff1,X [ZFCNst] . Theorem 2.3.1. Let ZFC∗Nst be a theory ZFC ∗ Nst , ZFC + ∃MZFCNst and Wff1,X [ZFCNst] ∈ MZFCNst . Then set theory ZFC∗Nst is inconsistent. Proof. Immediately from (2.3.33). Remark 2.3.10. In order to obtain a contradiction inside ZFCNst in more general case,i.e., without any refference to Assumption 2.3.1 we introduce the following definitions. Definition 2.3.12. We define now countable set Γ∗Nstν / ∼νby the following formula ∀y { [y]st ∈ Γ ∗Nst ν / ∼ν ⇐⇒( [y]Nst ∈ Γ Nst ν / ∼ν ) ∧ FrZFCNst(y, v) ∧ [ ∃!X ( X ∈ MZFCNst ) Ψy,ν (X) ]} . (2.3.36) Remark 2.3.11. Note that from the axiom of separation it follows directly that Γ∗stν / ∼ν is a set in the sense of the set theory ZFCNst. Definition 2.3.13. We define now the countable set I∗st by the following formula ∀Y { Y ∈ I∗Nst ⇐⇒ ∃y [( [y]Nst ∈ Γ ∗Nst ν / ∼ν ) ∧ (ĝZFCNst (X) = ν) ∧ Y = X ]} . (2.3.37) Note that from the axiom schema of replacement it follows directly that I∗Nst is a set in the sense of the set theory ZFCNst. Definition 2.3.14. We define now the countable set R∗Nst by the following formula ∀X (X ∈ I∗Nst) [X ∈ R∗Nst ⇐⇒ X /∈ X] . (2.3.38) Note that from the axiom schema of separation it follows directly that R∗Nst is a set in the sense of the set theory ZFCNst. Remark 2.3.12. Note that R∗Nst ∈ I∗Nst since R∗Nst is a definable by the following formula 64 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 Ψ∗ (Z) , ∀X (X ∈ I∗Nst) [X ∈ Z ⇐⇒ X /∈ X] . (2.3.39) Theorem 2.3.2.Set theory ZFCNst is inconsistent. Proof. From (2.3.39) and Remark 2.3.12 we obtain R∗Nst ∈ R∗Nst ⇐⇒ R∗Nst /∈ R∗Nst . (2.3.40) From (2.3.40) immediately one obtains a contradiction (R∗Nst ∈ R∗Nst) ∧ (R∗Nst /∈ R∗Nst) . 3 AVOIDING THE CONTRADICTIONS FROM SET THEORY ZFCHS2 AND SET THEORY ZFCST USING QUINEAN APPROACH In order to avoid difficulties mentioned above we use well known Quinean approach [19]. 3.1 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 Φ(xn) is a formula, then the set {xn | Φ(xn)}n+1 exists i.e., given any formula Φ(xn), the formula ∃An+1∀xn[xn ∈ An+1 ↔ Φ(xn)] (3.1.1) is an axiom where An+1 represents the set {xn | Φ(xn)}n+1 and is not free in Φ(xn). Quinean set theory [19] (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 3.1.1. In New Foundations (NF ) and related set theories, a formula Φ in the language of first-order logic with equality and membership is said to be stratified if and only if there is a function f which sends each variable appearing in Φ [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 Φ satisfies f (y) = f (x) + 1 and any atomic formula x = y appearing in Φ satisfies f (x) = f (y) Quinean set theory. 65 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 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. Axiom schema of comprehension: {x | Φs} exists for each stratified formula Φs. 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, [20] 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. 3.2 Set Theory ZFC Hs 2 , 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 Φs (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 Φ s. Then ∀A∀w1∀w2...∀wn [∀x (x ∈ A =⇒ ∃!yΦs (x, y, w1, w2, . . . , wn)) =⇒ =⇒ ∃B∀x (x ∈ A =⇒ ∃y (y ∈ B ∧ Φs (x, y, w1, w2, . . . , wn)))] , (3.2.1) i.e., iftherelationΦs (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 Φs (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 Φ s. Then ∀w1∀w2...∀wn∀A∃B∀x [x ∈ B ⇐⇒ (x ∈ A ∧ Φs (x,w1, w2, . . . , wn))] , (3.2.2) Remark 3.2.1. Notice that the stratified axiom schema of separation follows from the stratified axiom schema of replacement together with the axiom of empty set. Remark 3.2.2. 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 R∗Hs2 ,R∗st and R∗Nst impossible, because x /∈ x cannot be stratified. 66 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 4 SECOND-ORDER SET THEORY ZFC2 WITH THE FULL SECOND-ORDER SEMANTICS 4.1 Second-order Set Theory ZFC2 with Urlogic Remind that urlogic has the following characteristics [13]. 1. Sentences of urlogic are finite strings of symbols. That a string of symbols is a sentence of urlogic, is a non-mathematical judgement. 2. Some sentences are accepted as axioms. That a sentence is an axiom, is a non-mathematical judgement. 3. Derivations are made from axioms. The derivations obey certain rules of proof. That a derivation obeys the rules of proof, is a non-mathematical judgement. 4. Derived sentences can be asserted as facts. Remark 4.1.1. Let ZFCUl2 be second order set theory ZFC2 with Ur logic. Note that in ZFC Ul 2 by using the rules of DED2 we dealing without any reference to semantics, i.e. satisfiability in some standard model, validity etc. Definition 4.1.1. Let ΓUlX be the countable set of the all first order 1-place open wff's of the set theory ZFCUl2 that contains free occurrences of the variable X. Let Ψ1 (X) ,Ψ2 (X) be a first order 1-place open wff's of the set theory ZFC Ul 2 . We define now the equivalence relation (* ∼X *) ⊂ ΓUlX × ΓUlX by Ψ1 (X) ∼X Ψ2 (X) ⇐⇒ ∀X [Ψ1 (X) ⇐⇒ Ψ2 (X)] (4.1.1) For any Ψ (X) ∈ ΓUlX let [Ψ (X)]Ul , { Φ(X) ∈ ΓUlX |Ψ(X) ∼ Φ(X) } denote the equivalence class to which Ψ (X) belongs. All elements of ΓUlX equivalent to each other are also elements of the same equivalence class. The set of the all possible equivalence classes of ΓUlX by  X , denoted by Γ Ul X / ∼X ΓUlX / ∼X, { [Ψ (X)]Ul |Ψ(X) ∈ Γ Ul X } . (4.1.2) Let FrUl1 (y, v) be the relation : y is the Gödel number of a first order 1-open wff of the set theoryZFCUl2 that contains free occurrences of the variable X with Gödel number v [15]. Note that the relation FrUl1 (y, v) is expressible in ZFC Ul 2 by a wff Fr Ul 1 (y, v). Note that for any y, v ∈ N by definition of the relation FrUl1 (y, v) follows that Fr Ul 1 (y, v) ⇐⇒ ∃!Ψ (X) [( gZFCUl2 (Ψ (X)) = y ) ∧ ( gZFCUl2 (X) = ν )] , (4.1.3) where Ψ (X) is a unique wff of ZFCUl2 which contains free occurrences of the first order variable X with Gödel number v. We denote a unique wff Ψ (X) defined by using equivalence (4.1.3) by symbol ΨUly,ν (X) , i.e. Fr Ul 1 (y, v) ⇐⇒ ∃!ΨUly,ν (X) [( gZFCUl2 ( ΨUly,ν (X) ) = y ) ∧ ( gZFCUl2 (X) = ν )] . (4.1.4) Definition 4.1.2. Let gZFCUl2 (X) = ν. Let ΓUlν be a set of the all Gödel numbers of the first order 1-place open wff's of the set theory ZFCUl2 that contains free occurrences of the variable X with Gödel number v, i.e. ΓUlν = { y ∈ N| ⟨y, ν⟩ ∈ FrUl1 (y, v) } , (4.1.5) 67 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 or in the following equivalent form: ∀y (y ∈ N) [ y ∈ ΓUlν ⇐⇒ (y ∈ N) ∧ Fr Ul 1 (y, v) ] . (4.1.6) Remark 4.1.2. Note that from the axiom of separation it follows directly that ΓUlν is a set in the sense of the set theory ZFCUl2 . Definition 4.1.3. (i) We define now the equivalence relation (* ∼ν *) ⊂ ΓUlν × ΓUlν (4.1.7) inthesenseofthesettheoryZFCUl2 by y1 ∼ν y2 ⇐⇒ ( ∀X [ ΨUly1,ν (X) ⇐⇒ Ψ Ul y2,ν (X) ]) . (4.1.8) For any y1 ∈ ΓUlv let [y1]Ul , { y ∈ ΓUlX |y1 ∼ν y2 } denote the equivalence class to which y1 belongs. The set of the all possible equivalence classes of ΓUlν by  ν , denoted Γ Ul ν / ∼ν ΓUlν / ∼ν, { [y]Ul |y ∈ Γ Ul ν } . (4.1.9) Remark 4.1.3. Note that from the axiom of separation it follows directly that ΓHsν / ∼ν is a set in the sense of the set theory ZFCUl2 . Definition 4.1.4. Let IUl2 be the countable set of the all sets definable by a first order 1-place open wff of the set theory ZFCUl2 , i.e. ∀Y { Y ∈ IUl2 ⇐⇒ ∃Ψ(X) [( [Ψ (X)]Ul ∈ Γ Ul X / ∼X ) ∧ [∃!X [Ψ (X) ∧ Y = X]] ]} . (4.1.10) Definition 4.1.5. We rewrite now (4.1.10) in the following equivalent form ∀Y { Y ∈ IUl2 ⇐⇒ ∃Ψ(X) [( [Ψ (X)]Ul ∈ Γ ∗Ul X / ∼X ) ∧ (Y = X) ]} , (4.1.11) where the countable set Γ∗UlX / ∼X is defined by the following formula ∀Ψ(X) { [Ψ (X)]Ul ∈ Γ ∗Ul X / ∼X ⇐⇒ [( [Ψ (X)]Ul ∈ Γ Ul X / ∼X ) ∧ ∃!XΨ(X) ]} . (4.1.12) Definition 4.1.6. Let RUl2 be the countable set of all sets such that ∀X ( X ∈ IUl2 ) [ X ∈ RUl2 ⇐⇒ X /∈ X ] . (4.1.13) Remark 4.1.4. Note that RUl2 ∈ IUl2 since RUl2 is a set definable by first order 1-place open wff Ψ ( Z,IUl2 ) , ∀X ( X ∈ IUl2 ) [X ∈ Z ⇐⇒ X /∈ X] . (4.1.14) From (4.1.13) one obtains RUl2 ∈ RUl2 ⇐⇒ RUl2 /∈ RUl2 . (4.1.15) But (4.1.15) gives a contradiction( RUl2 ∈ RUl2 ) ∧ ( RUl2 /∈ RUl2 ) . (4.1.16) 421ontradiction (4.1.16) is a contradiction inside ZFCUl2 for the reason that the countable set IUl2 is a set in the sense of the set theory ZFCUl2 . In order to obtain a contradiction inside ZFCUl2 in more general case we introduce the following definitions. Definition 4.1.7. We define now the countable set Γ∗Ulν / ∼ν by the following formula ∀y { [y]Ul ∈ Γ ∗Ul ν / ∼ν ⇐⇒ ( [y]Ul ∈ Γ Ul ν / ∼ν ) ∧ Fr Ul 1 (y, v) ∧ [ ∃!XΨUly,ν (X) ]} . (4.1.17) Remark 4.1.5. Note that from the axiom of separation it follows directly that Γ∗Ulν / ∼ν is a set 68 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 in the sense of the set theory ZFCUl2 . Definition 4.1.8.We define now the countable set I∗Ul2 by the following formula ∀Y { Y ∈ I∗Ul2 ⇐⇒ ∃y [( [y]Ul ∈ Γ ∗Ul ν / ∼ν ) ∧ ( gZFCUl2 (X) = ν ) ∧ Y = X ]} . (4.1.18) Note that from the axiom schema of replacement (1.1.1) it follows directly that I∗Hs2 is a set in the sense of the set theory ZFCUl2 . Definition 4.1.9. We define now the countable set R∗Ul2 by formula ∀X ( X ∈ I∗Ul2 ) [ X ∈ R∗Ul2 ⇐⇒ X /∈ X ] . (4.1.19) Note that from the axiom schema of separation it follows directly that R∗Ul2 is a set in the sense of the set theory ZFCUl2 . Remark 4.1.6. Note that R∗Ul2 ∈ I∗Ul2 since R∗Ul2 is definable by the following formula Ψ∗ (Z) , ∀X ( X ∈ I∗Ul2 ) [X ∈ Z ⇐⇒ X /∈ X] . (4.1.20) Theorem 4.1.1. Set theory ZFCUl2 is inconsistent. Proof. From (4.1.19) and Remark 4.1.6 we obtain R∗Ul2 ∈ R∗Ul2 ⇐⇒ R∗Ul2 /∈ R∗Ul2 from which immediately one obtains a contradiction( R∗Ul2 ∈ R∗Ul2 ) ∧ ( R∗Ul2 /∈ R∗Ul2 ) . (4.1.21) 4.2 Second-order Set Theory ZFC2 with the Full Se-condorder Semantics Remind that the canonical approach of second order logic with full second-order semantics to the foundations of mathematics is that mathematical propositions have the form U |= Φ (4.2.1) where U is a mathematical structure, such as integers, reals etc., and is a mathematical statement written in second order logic. If A is one of the structures, such as (N,+,×, <) or (R,+,×, <), for which there is a second order sentence ΞU such that ∀W (W |= ΞU ⇐⇒ W ∼= U) , (4.2.2) then (4.2.2) can be expressed as a second order semantic logical truth |= ΞU =⇒ Φ. (4.2.3) Remark 4.2.1. Let ZFCfss2 be second order set theory ZFC2 with the full second-order semantics. (1) There is no completeness theorem for second-order logic. (2) Nor do the axioms of second-order ZFCfss2 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. Remark 4.2.2. Thus there may be sentences of the language of second-order set theory ZFCfss2 : (i) that are true but unsatisfiable, or (ii) sentences that are valid, but false. Remark 4.2.3. For example let Z be the conjunction of all the axioms of second-order ZFCfss2 . Z is surely true. But the existence of a model for Z requires the existence of strongly inaccessible 69 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 cardinals. The axioms of ZFCfss2 don't entail the existence of strongly inaccessible cardinals, and hence the satisfiability of Z is independent of ZFCfss2 . Thus, Z is true but its unsatisfiability is consistent with ZFCfss2 . Definition 4.2.1. We will say that Ψ is a well formed first order formula Ψ of ZFCfss2 (wff1) if Ψ contain only first-order variables and first-order quantifiers. Let Γ♯fssX be the countable set of the all first order 1-place open wff1's of the set theory ZFC fss 2 that contains free occurrences of the first-order variable X. Let Ψ1 (X) ,Ψ2 (X) be 1-place open wff1's of the set theory ZFC fss 2 .We define now the equivalence relation (* ∼X *) ⊂ Γ♯fssX × Γ ♯fss X by Ψ1 (X) ∼X Ψ2 (X) ⇐⇒ ∀X [Ψ1 (X) ⇐⇒ Ψ2 (X)] (4.2.4) For any Ψ (X) ∈ Γ♯fssX let [Ψ (X)]♯fss , { Φ(X) ∈ Γ♯fssX |Ψ(X) ∼ Φ(X) } be the equivalence class to which Ψ (X) belongs. All elements of Γ♯fssX equivalent to each other are also elements of the same equivalence class. The collection of all possible equivalence classes of Γ♯fssX by  X , denoted Γ♯fssX / ∼X Γ♯fssX / ∼X, { [Ψ (X)]♯fss |Ψ(X) ∈ Γ ♯fss X } . (4.2.5) Let Fr♯fss2 (y, v) be the relation : y is the Gödel number of a wff of the set theory ZFC ♯fss 2 that contains free occurrences of the first-order variable X with Gödel number v [17]. Note that the relation Fr♯fss1 (y, v) is expressible in ZFC fss 2 by a wff Fr ♯fss 1 (y, v). Note that for any y, v ∈ N by definition of the relation Fr♯fss1 (y, v) follows that Fr ♯fss 1 (y, v) ⇐⇒ ∃!Ψ (X) [( g ZFC fss 2 (Ψ (X)) = y ) ∧ ( g ZFC fss 2 (X) = ν )] , (4.2.6) where Ψ (X) is a unique wff1 of ZFC fss 2 which contains free occurrences of the variable X with Gödel number v. We denote a unique wff1 Ψ(X) defined by using equivalence (4.2.6) by symbol Ψ♯fssy,ν (X) , i.e. Fr ♯fss 1 (y, v) ⇐⇒ ∃!Ψ♯fssy,ν (X) [( g ZFC fss 2 ( Ψ♯y,ν (X) ) = y ) ∧ ( g ZFC fss 2 (X) = ν )] . (4.2.7) Remark 4.2.4. In order to avoid difficulties mentioned above,see Remark 4.2.1-Remark 4.2.3 we dealing with the countable set Γ♯fssX of the all first order 1-place open wff1's of the set theory ZFCfss2 . Definition 4.2.2. Let g ZFC fss 2 (X) = ν.Let Γ♯fssν be a set of all Gödel numbers of the all first order 1-place open wff1's of the set theory ZFC fss 2 that contains free occurrences of the first-order variable X with Gödel number v, i.e. Γ♯fssν = { y ∈ N| ⟨y, ν⟩ ∈ Fr♯fss1 (y, v) } , (4.2.8) or in the following equivalent form ∀y (y ∈ N) [ y ∈ Γ♯fssν ⇐⇒ (y ∈ N) ∧ Fr ♯fss 1 (y, v) ] . (4.2.9) Remark 4.2.5. Note that from the axiom of separation it follows directly that Γ♯fssν is a set in the sense of the set theory ZFCfss2 . Definition 4.2.3. (i) We define now the equivalence relation 70 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 (* ∼ν *) ⊂ Γ♯fssν × Γ♯fssν (4.2.10) in the sense of the set theory ZFCfss2 by y1 ∼ν y2 ⇐⇒ ( ∀X [ Ψ♯fssy1,ν (X) ⇐⇒ Ψ ♯fss y2,ν (X) ]) . (4.2.11) The collection of all possible equivalence classes of Γ♯fssν by  ν , denoted Γ ♯fss ν / ∼ν Γ♯fssv / ∼ν, { [y]♯fss |y ∈ Γ ♯fss ν } . (4.2.12) Remark 4.2.6. Note that from the axiom of separation it follows directly that Γ♯fssν / ∼ν is a set in the sense of the set theory ZFCfss2 . Definition 4.2.4. Let I♯fss2 be the countable collection of the all sets definable by a first order 1-place open wff1 of the set theory ZFC fss 2 , i.e. ∀Y { Y ∈ I♯fss2 ⇐⇒ ∃Ψ(X) [( [Ψ (X)]♯fss ∈ Γ ♯fss X / ∼X ) ∧ [∃!X [Ψ (X) ∧ Y = X]] ]} . (4.2.13) Definition 4.2.5. We rewrite now (4.2.13) in the following equivalent form ∀Y { Y ∈ I∗♯fss2 ⇐⇒ ∃Ψ(X) [( [Ψ (X)]♯fss ∈ Γ ∗♯fss X / ∼X ) ∧ (Y = X) ]} , (4.2.14) where the countable collection Γ∗♯fssX / ∼X is defined by the following formula ∀Ψ(X) { [Ψ (X)]♯fss ∈ Γ ∗♯fss X / ∼X ⇐⇒ [( [Ψ (X)]♯fss ∈ Γ ♯fss X / ∼X ) ∧ ∃!XΨ(X) ]} . (4.2.15) Definition 4.2.6. Let R∗♯fss2 be the countable collection of all sets such that ∀X ( X ∈ I♯fss2 ) [ X ∈ R∗♯fss2 ⇐⇒ X /∈ X ] . (4.2.16) Remark 4.2.7. Note that R∗♯fss2 ∈ I ∗♯fss 2 since R ∗♯fss 2 is a collection definable by 1-place open wff1 Ψ ( Z,I∗♯fss2 ) , ∀X ( X ∈ I∗♯fss2 ) [X ∈ Z ⇐⇒ X /∈ X] . (4.2.17) From (4.2.16) and Remark 4.2.7 one obtains R∗♯fss2 ∈ R ∗♯fss 2 ⇐⇒ R ∗♯fss 2 /∈ R ∗♯fss 2 . (4.2.18) But (4.2.18) gives a contradiction( R∗♯fss2 ∈ R ∗♯fss 2 ) ∧ ( R∗♯fss2 /∈ R ∗♯fss 2 ) . (4.2.19) The contradiction (4.2.19) it a contradiction inside ZFCfss2 for the reason that the countable collection I∗♯fss2 is a set in the sense of the set theory ZFC fss 2 . In order to obtain a contradiction inside ZFCfss2 in more general case we introduce the following definitions. Definition 4.2.7. We define now the countable set Γ∗♯fssν / ∼ν by the following formula ∀y { [y]Ul ∈ Γ ∗♯fss ν / ∼ν ⇐⇒ ( [y]♯fss ∈ Γ ∗♯fss ν / ∼ν ) ∧ Fr ∗♯fss 2 (y, v) ∧ [ ∃!XΨ♯fssy,ν (X) ]} . (4.2.20) Remark 4.2.8. Note that from the axiom of separation it follows directly that Γ∗Ulν / ∼ν is a set in the sense of the set theory ZFCfss2 . 71 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 Definition 4.2.8. We define now the countable set I∗♯fss2 by formula ∀Y { Y ∈ I∗♯fss2 ⇐⇒ ∃y [( [y]♯fss ∈ Γ ∗♯fss ν / ∼ν ) ∧ ( g ZFC fss 2 (X) = ν ) ∧ Y = X ]} . (4.2.21) Note that from the axiom schema of replacement (1.1.1) it follows directly that I∗♯fss2 is a set in the sense of the set theory ZFCfss2 . Definition 4.2.9. We define now the countable set R∗♯fss2 by the following formula ∀X ( X ∈ I∗♯fss2 ) [ X ∈ R∗♯fss2 ⇐⇒ X /∈ X ] . (4.2.22) Note that from the axiom schema of separation it follows directly that R∗♯fss2 is a set in the sense of the set theory ZFCfss2 . Remark 4.2.9. Note that R∗♯fss2 ∈ I∗Ul2 since R∗Ul2 is definable by the following formula Ψ∗ (Z) , ∀X ( X ∈ I∗♯fss2 ) [X ∈ Z ⇐⇒ X /∈ X] . (4.2.23) Theorem 4.2.1. Set theory ZFCfss2 is inconsistent. Proof. From (4.2.22) and Remark 4.1.6 we obtain R∗♯fss2 ∈ R ∗♯fss 2 ⇐⇒ R ∗♯fss 2 /∈ R∗Ul2 from which immediately one obtains a contradiction( R∗♯fss2 ∈ R ∗♯fss 2 ) ∧ ( R∗♯fss2 /∈ R ∗♯fss 2 ) . (4.2.24) 5 CONCLUSIONS a In this Chapter we have proved that set theory ZFC + ∃MZFCst is inconsistent in particular ¬Con(ZF + V = L).. b This result originally was obtained in [2], [4]. [5] by using essentially another approach. ACKNOWLEDGMENT The reviewers provided important clarifications. COMPETING INTERESTS Authors have declared that no competing interests exist. REFERENCES [1] Nelson E. Warning signs of a possible collapse of contemporary mathematics. In Infinity: New Research Frontiers, by Michael Heller (Editor), W. Hugh Woodin P. 75–85. Published February 14th 2013 by Cambridge University Press, Hardcover. 2011;311. ISBN: 1107003873 (ISBN13: 9781107003873). Available:https://web.math.princeton.edu/  nelson/papers/warn.pdf [2] Foukzon J. Generalized Lob's Theotem. Strong Reflection Pronciples and Large Cardinal Axioms. Consistency Results in Topology; 2013. 72 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 arXiv: 1301.5340 {math.GM] Available:https://arxiv.org/abs/1301.5340 [3] Lemhoff R. The bulletin of symbolic logic. 2016;23(2):213-266. COLLOQUIUM'16, Leeds, UK, July 31-August 6, 2016. Foulzon, J., Inconsistent countable set in second order ZFC and unexistence of the stringly inaccessible cardinals. Available:https://www.jstor.org/stable/44259451?seq=1#page scan tab contents. [4] Foukzon J, Men'kova ER. Generalized Löb's theorem. Strong reflection principles and large cardinal axioms. Advances in Pure Mathematics. 2013;3(3):368-373. Available:http://dx.doi.org/10.4236/apm.2013.33053 DOI: 10.4236/apm. 2013.33053 [5] Foukzon J. Inconsistent countable set in second order zfc and nonexistence of the strongly inaccessible cardinals. British Journal of Mathematics & Computer Science. 2015;9(5). ISSN: 2231-0851 DOI : 10.9734/BJMCS/2015/16849 Available:http://www.sciencedomain.org/abstract/9622 [6] Foukzon J, Men'kova ER. There is no standard model of zfc and zfc2Journal of Advances in Mathematics and Computer Science. 2018;26(2):1-20. Published jan. 30, 2018. DOI: 10.9734/JAMCS/2018/38773 [7] Henkin L. Completeness in the theory of types. Journal of Symbolic Logic. 1950;15(2):81-91. DOI:10.2307/2266967. JSTOR 2266967. [8] Cohen P. Set theory and the continuum hypothesis. Reprint of the W. A. Benjamin, Inc., New York, 1966 edition; 1966. ISBN-13: 978-0486469218 [9] Godel K. Consistency of the continuum hypothesis (AM-3). Series: Annals of Mathematics Studies Copyright Date: Published by: Princeton University Press. 1968;69. [10] Rossberg M. First-order logic, second-order logic, and completeness. In: V. Hendricks et al., eds. First-order logic revisited, Berlin: Logos-Verlag. P. 2004;303-321. [11] Shapiro S. Foundations without foundationalism: A case for second-order logic. Oxford University Press;1991. ISBN 0-19-825029-0 [12] Rayo A, Uzquiano G. Toward a theory of second-order consequence. Notre Dame Journal of Formal Logic. 1999;40(3):315-325. [13] Vaananen J. Second-order logic and foundations of mathematics. The Bulletin of Symbolic Logic. 2001;7(4):504-520. [14] Friedman H. Countable models of set theories, cambridge summer school in mathematical logic. Lecture Notes in Mathematics, Springer. 1971;337(1973):539-573. [15] Magidor M, Shelah S, Stavi J. On the standard part of nonstandard models of set theory. The Journal of Symbolic Logic. 1983;48(1):33-38. [16] Bovykin A. On oredr-types of models of arithmetic. Ph. D. Thesis pp.109, University of Birmingham. On order-types of models of arithmetic (with R. Kaye), 2001. Contemporary Mathematics Series of the AMS. 2000;302:275-285. [17] Mendelson E. Introduction to mathematical logic; 1997. ISBN-10: 0412808307. ISBN-13: 978-0412808302. [18] Takeuti G. Proof theory: Second edition. Dover Books on Mathematics; 2013 ISBN-13: 978-0486490731; ISBN-10: 0486490734 73 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 [19] Quine WV. New foundations for mathematical logic. The American mathematical Monthly, mathematical Association of America. 193744(2):70-80. DOI:10.2307/2300564, JSTOR 2300564 [20] Hailperin T. A set of axioms for logic. Journal of Symbolic Logic. 1944;9:1-19. Biography of author(s) Jaykov Foukzon Israel Institute of Technology, Haifa, Israel. Jaykov Foukzon (born October 7, 1954). Professor, Doctor Sc. (Phys. and Math.). Nationality: Soviet Union, Education: Moscow State University (1975) Doctoral adviser: Lev Pontryagin. Soviet and Russian mathematician and physicist. Institutions: Russian Space Research Institute, Institute of Control Sciences of Russian Academy of Sciences, Central Aerohydrodynamic Institute (TsAGI) Russia. Currently works at the Center for Mathematical Sciences, Technion Israel Institute of Technology, Haifa, Israel. Skills and Activities: Optimal Control, Nonlinear Dynamics, Quantum Chaos, Control Theory, Advanced Control Theory, Systems Dynamics, Theory of Reference Frames (TR), Waves, Non-Standard Model, Standard Model, Calculations, Quantum Field Theory, Number Theory, Electromagnetic Fields, Partial Differential Equations, Actuarial Mathematics, Geometry, Computational Physics, QuantumMechanics, Theoretical Physics, Foundations of Quantum Mechanics, Mathematical Physics, First-order Logic, Special and General Relativity, General Relativity, Theoretical Particle Physics, Fundamental Physics, Probability Theory, Gravitational Physics, Field Theory, Theoretical High Energy Physics, High Energy Physics Theory, Particle Physics, Foundations of Physics, Supersymmetry, Classical Mechanics, Quantum Cosmology, Conformal Field Theory, Algebraic Quantum Field Theory, Renormalization-group Theory, Quantum Gravity, Electrodynamics, Set Theory, Logic, IT Systems, Algebra, Analysis, Basic, Numbers, Deterministic Quantum Physics, Mathematical Analysis, Differential Geometry. Books Jaykov Foukzon, Elena R. Men'kova, Alexander A. Potapov: EPRB Paradox Resolution. Bell inequalities revisited. 03/2019; LAMBERT. ISBN: 978-613-9-45511-9 Jaykov Foukzon, Alexander Potapov, Elena Men'kova: Schr446dinger's Cat Paradox Resolution. A New Quantum Mechanical Formalism Based on the Probability Representation of Continuous Observables. 12/2017; LAMBERT. ISBN: 978-3-659-93543-5 S. A. PODOSENOV, A. A. POTAPOV, J. FOUKZON, E. R. MEN'KOVA: Fields, Fractals, Singularities and Quantum Control. 10/2016; in press. 74 Advances in Mathematics and Computer Science Vol. 1 There is No Standard Model of ZFC and ZFC2 S. A. PODOSENOV, A. A. POTAPOV, J. FOUKZON, E. R. MEN'KOVA: 41d43543343e43b43e43d43e43c43d44b435,44444043043a44243043b44c43d44b435 438 441432 44f 43743043d43d44b435 44144244044343a44244344044b 432 44043543b44f442438432438441 44244143a438445 44143f 43b43e44843d44b445 441440435434430445, 44d43b 43543a44244043e 43443843d43043c43843a435, 43a43243043d44243e43243e439 43c43544543043d 43843a435 438 43a43e44143c43e43b43e433438438. Publishing house URSS, 03/2016. ISBN 978-5-9710-2456-9 Jaykov Foukzon, Alexander Potapov, Elena Men'kova: Large deviations principles of Non-FreidlinWentzell type. 07/2015; LAMBERT. ISBN: 978-3-659-66379-6. Menkova Elena Romanovna All-Russian Research Institute for Optical and Physical Measurements, Moscow, Russia. Men'kova Elena Romanovna (born July 27, 1955) Ph. D. Senior researcher of All-Russian Research Institute for Optical and Physical Meas-urements, VNIIOFI. Laboratory of Generation and Measurement of Electromagnetic Pulse Parameters. Education: The Moscow Mendeleev Institute of Chemical Technology (1978). The Moscow Institute of Electronic Mechanical Engineering. Automated Management Systems, qualification improvement faculty (1987). Managed the literature searches and physical interpretation of models. Scientific area: radiation and measurement of pulse electromagnetic fields and their interaction with the conducting bodies. The research interest focuses on the works connected with problems of relativistic mechanics and electrodynamics, noninertial frames of reference in special and general relativity. ----------------------------------------------– c⃝ Copyright 2019 The Author(s), Licensee Book Publisher International, This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. DISCLAIMER This chapter is an extended version of the article published by the same authors in the following journal with CC BY license. Journal of Advances in Mathematics and Computer Science, 26(2): 1-20, 2018. Reviewers Information (1) Xiaolan Liu, Sichuan University of Science Engineering, China. (2) Derya Doan Durgun, Manisa Celal Bayar University, Turkey.