There Is No Standard Model of ZFC and ZFC2 with Henkin Semantics

(1)Advances in Pure Mathematics, 2019, 9, 685-744 https://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368. There Is No Standard Model of ZFC and ZFC2 with Henkin Semantics Jaykov Foukzon1, Elena Men’kova2 Department of Mathematics, Israel Institute of Technology, Haifa, Israel Lomonosov Moscow State University, Moscow, Russia. 1 2. How to cite this paper: Foukzon, J. and Men’kova, E. (2019) There Is No Standard Model of ZFC and ZFC2 with Henkin Semantics. Advances in Pure Mathematics, 9, 685-744. https://doi.org/10.4236/apm.2019.99034. Abstract. Received: February 5, 2019 Accepted: September 13, 2019 Published: September 16, 2019. ¬Con NF + ∃M stNF , 4) ¬Con ( ZFC2 ) , 5) let k be inaccessible cardinal then. Copyright © 2019 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/. In this article we proved so-called strong reflection principles corresponding to formal theories Th which has omega-models or nonstandard model with standard part. A possible generalization of Löb’s theorem is considered. Main results are: 1) ¬Con ( ZFC + ∃M stZFC ) , 2) ¬Con ( ZF + (V = L ) ) , 3). (. ). ¬Con ( ZFC + ∃κ ) .. Keywords Gödel Encoding, Completion of ZFC, Russell’s Paradox, ω-Model, Henkin Semantics, Full Second-Order Semantic, Strongly Inaccessible Cardinal. Open Access. 1. Introduction 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 DOI: 10.4236/apm.2019.99034 Sep. 16, 2019. 685. Advances in Pure Mathematics.

(2) J. Foukzon, E. Men’kova. don’t; it is a matter of faith (or of skepticism)”—E. Nelson wrote in his paper [1]. However, it is deemed unlikely that even ZFC2 which is significantly stronger than ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC and ZFC2 were consistent, that fact would have been uncovered by now. This much is certain—ZFC and ZFC2 are 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]. 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 wrote: “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 constructible 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 just 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 y for which there is a formula A ( z , t1 , , tk ) in ZF such. that if AX denotes A with all bound variables restricted to X, then for some ti , i = 1, , k , in X, y=. { z ∈ X | A ( z, t , , t )} . X. 1. (1). k. Observe X ′  P ( x )  X , X ′ = X if X is infinite (and we assume AC). It. should be clear to the reader that the definition of X ′ , as we have given it, can. be done entirely within ZF and that Y = X ′ is a single formula A ( X , Y ) in ZF. In general, one’s intuition is that all normal definitions can be expressed in ZF, except possibly those which involve discussing the truth or falsity of an infinite sequence of statements. Since this is a very important point we shall give a rigorous proof in a later section that the construction of X ′ is expressible in. ZF.” Remark 1.1.3. We will say that a set y is definable by the formula. A ( z , t1 , , tk ) relative to a given set X.. Remark 1.1.4. Note that a simple generalisation of the notion of the definability which has been by Definition 1.1.1 immediately gives Russell’s paradox in second order set theory ZFC2 with the Henkin semantics [7]. Definition 1.1.2. [6]. i) We will say that a set y is definable relative to a given set X iff there is a formula. A ( z , t1 , , tk ). in ZFC then for some. ti ∈ X , i = 1, , k , in X there exists a set z such that the condition A ( z , t1 , , tk ). is satisfied and y = z or symbolically ∃z  A ( z , t1 , , tk ) ∧ y = z  . DOI: 10.4236/apm.2019.99034. 686. (2) Advances in Pure Mathematics.

(3) J. Foukzon, E. Men’kova. It should be clear to the reader that the definition of X ′ , as we have given it, can be done entirely within second order set theory ZFC2 with the Henkin semantics [7] denoted by ZFC2Hs and that Y = X ′ is a single formula A ( X , Y ) in ZFC2Hs . ii) We will denote the set Y of all sets y definable relative to a given set X by Y  ℑ2Hs . Definition 1.1.3. Let ℜ2Hs be a set of the all sets definable relative to a given set X by the first order 1-place open wff’s and such that. (. ). (3). ∀x x ∈ ℑ2Hs  x ∈ ℜ2Hs ⇔ x ∉ x  .. Remark 1.1.5. (a) Note that ℜ2Hs ∈ ℑ2Hs since ℜ2Hs is a set definable by the first order 1-place open wff Ψ ( Z , ℑ2Hs ) :. (. ). (. Ψ Z , ℑ2Hs  ∀x x ∈ ℑ2Hs. ) [ x ∈ Z ⇔ x ∉ x] ,. (4). Theorem 1.1.1. [6]. Set theory ZFC2Hs is inconsistent. Proof. From (3) and Remark 1.1.2 one obtains ℜ2Hs ∈ ℜ2Hs ⇔ ℜ2Hs ∉ ℜ2Hs .. (5). From (5) one obtains a contradiction. (ℜ. Hs 2. ) (. ). (6). ∈ ℜ2Hs ∧ ℜ2Hs ∉ ℜ2Hs .. Remark 1.1.6. Note that in paper [6] we dealing by using following definabil-. ity condition: a set y is definable if there is a formula A ( z ) in ZFC such that ∃z  A ( z ) ∧ y = z  .. (7). Obviously in this case a set Y = ℜ2Hs is a countable set. Definition 1.1.4. Let ℜ2Hs be the countable set of the all sets definable by the first order 1-place open wff’s and such that. (. ). ∀x x ∈ ℑ2Hs  x ∈ ℜ2Hs ⇔ x ∉ x  .. (8). Remark 1.1.7. (a) Note that ℜ2Hs ∈ ℑ2Hs since ℜ2Hs is a set definable by the. first order 1-place open wff Ψ ( Z , ℑ2Hs ) :. (. ). (. Ψ Z , ℑ2Hs  ∀x x ∈ ℑ2Hs. one obtains a contradiction. (ℜ. Hs 2. ) [ x ∈ Z ⇔ x ∉ x] ,. ) (. (9). ). ∈ ℜ2Hs ∧ ℜ2Hs ∉ ℜ2Hs .. In this paper we dealing by using following definability condition. Definition 1.1.5. i) Let M st = M stZFC be a standard model of ZFC. We will say that a set y is definable relative to a given standard model M st of ZFC if. there is a formula A ( z , t1 , , tk ) in ZFC such that if AM st denotes A with all. bound variables restricted to M st , then for some ti ∈ M st , i = 1, , k , in M st. there exists a set z such that the condition AM st ( z , t1 , , tk ) is satisfied and y = z or symbolically. z  . ∃z  AM st ( z , t1 , , tk ) ∧ y =. (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 seDOI: 10.4236/apm.2019.99034. 687. Advances in Pure Mathematics.

(4) J. Foukzon, E. Men’kova. mantics. ii) In this paper we assume for simplicity but without loss of generality that. AM st ( z , t1 , , tk ) = AM st ( z ) .. (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 ZFC2 under the full second-order semantics [8] [9] [10] [11] [12] but also with a proof theory based on formal Urlogic [13]. Remark 1.1.9. Second-order logic essentially differs from the usual first-order predicate calculus in that it has variables and quantifiers not only for individuals but also for subsets of the universe and variables for n-ary relations as well [7]-[13]. The deductive calculus DED2 of second order logic is based on rules and axioms which guarantee that the quantifiers range at least over definable subsets [7]. As to the semantics, there are two types of models: i) Suppose U is an ordinary first-order structure and S is a set of subsets of the domain A of U . The main idea is that the set-variables range over S , i.e. U, S  ∃X Φ ( X ) ⇔ ∃S ( S ∈ S )  U, S  Φ ( S )  .. We call and truth in. U, S U, S. satisfies the axioms of DED2 is preserved by the rules of DED2 . We call this semantics. a Henkin model, if. U, S. 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 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. Henkin models, namely those. U, S. semantics and second-order logic with the full semantics the full second-order logic. Remark 1.1.10. We emphasize that the following facts are the main features of second-order logic: 1) The Completeness Theorem: A sentence is provable in DED2 if and only if it holds in all Henkin models [7]-[13]. 2) The Löwenheim-Skolem Theorem: A sentence with an infinite Henkin model has a countable Henkin model. 3) The Compactness Theorem: A set of sentences, every finite subset of which has a Henkin model, has itself a Henkin model. 4) The Incompleteness Theorem: Neither DED2 nor any other effectively given deductive calculus is complete for full models, that is, there are always sentences which are true in all full models but which are unprovable. 5) Failure of the Compactness Theorem for full models. 6) Failure of the Löwenheim-Skolem Theorem for full models. 7) There is a finite second-order axiom system  2 such that the semiring.  of natural numbers is the only full model of  2 up to isomorphism.. 8) There is a finite second-order axiom system RCF2 such that the field  of. the real numbers is the only full model of RCF2 up to isomorphism. DOI: 10.4236/apm.2019.99034. 688. Advances in Pure Mathematics.

(5) J. Foukzon, E. Men’kova. 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 )   ,. (12). where X is a second-order variable, and where Func ( X ) abbreviates “X is a functional relation”, see [12]. Thus we interpret the wff’s of ZFC2 language with the full second-order semantics as required in [12] [13] but also with a proof theory based on formal urlogic [13]. Designation 1.1.1. We will denote: i) by ZFC2Hs set theory ZFC2 with the Henkin semantics, ii) by ZFC2fss set theory ZFC2 with the full second-order semantics, Hs iii) by ZFC 2Hs set theory ZFC2Hs + ∃M stZFC2 and iv) by ZFCst set theory ZFC + ∃M stZFC , where M stTh is a standard model of the theory Th . Remark 1.1.12. There is no completeness theorem for second-order logic with the full second-order semantics. Nor do the axioms of ZFC2fss imply a reflection principle which ensures that if a sentence Z of second-order set theory ZFC fss. is true, then it is true in some model M 2 of ZFC2fss [11]. Let Z be the conjunction of all the axioms of ZFC2fss . We assume now that: Z. is true, i.e. Con ( ZFC2fss ) . It is known that the existence of a model for Z requires the existence of strongly inaccessible cardinals, i.e. under ZFC it can be. shown that κ is a strongly inaccessible if and only if ZFC2fss . Thus. (. ( Hκ ,∈). is a model of. ). ¬Con ZFC2fss ⇒ ¬Con ( ZFC + ∃κ ) .. (13). In this paper we prove that: Hs. i) ZFCst  ZFC + ∃M stZFC ii) ZFC 2Hs  ZFC2Hs + ∃M stZFC2 and iii) ZFC2fss is inconsistent, where M stTh is a standard model of the theory Th . Axiom. ε ⊆M. ZFC. ∃M ZFC. ×M. ZFC. [8]. There is a set which makes M. ZFC. M ZFC. and a binary relation. a model for ZFC.. Remark 1.1.13. i) We emphasize that it is well known that axiom ∃M ZFC a single statement in ZFC see [8], Ch. II, Section 7. We denote this statement. thought all this paper by symbol Con ( ZFC ; M ZFC ) . The completeness theorem says that ∃M ZFC ⇔ Con ( ZFC ) . ii) Obviously there exists a single statement in ZFC2Hs such that Hs ∃M ZFC2 ⇔ Con ZFC2Hs .. (. ). We denote this statement through all this paper by symbol. (. Hs. ). Hs. and there exists a single statement ∃M Z 2 in Z 2Hs . We Hs denote this statement through all this paper by symbol Con Z 2Hs ; M Z 2 . Con ZFC2Hs ; M ZFC2. (. ). Axiom ∃M stZFC [8]. There is a set M stZFC such that if R is { x, y | x ∈ y ∧ x ∈ M stZFC ∧ y ∈ M stZFC } then M stZFC is a model for ZFC under the relation R.. DOI: 10.4236/apm.2019.99034. 689. Advances in Pure Mathematics.

(6) J. Foukzon, E. Men’kova. Definition 1.1.6. [8]. The model M stZFC is called a standard model since the relation ∈ used is merely the standard ∈ -relation.. Remark 1.1.14. Note that axiom ∃M ZFC doesn’t imply axiom ∃M stZFC , see ref. [8]. Remark 1.1.15. We remind that in Henkin semantics, each sort of second-order variable has a particular domain of its own to range over, which may be a proper subset of all sets or functions of that sort. Leon Henkin (1950) defined these semantics and proved that Gödel’s completeness theorem and compactness theorem, which hold for first-order logic, carry over to second-order logic with Henkin semantics. This is because Henkin semantics are almost identical to many-sorted first-order semantics, where additional sorts of variables are added to simulate the new variables of second-order logic. Second-order logic with Henkin semantics is not more expressive than first-order logic. Henkin semantics are commonly used in the study of second-order arithmetic. Väänänen [13] argued that the choice between Henkin models and full models for second-order logic is analogous to the choice between ZFC and V ( V is von Neumann universe), as a basis for set theory: “As with second-order logic, we cannot really choose whether we axiomatize mathematics using V or ZFC. The result is the same in both cases, as ZFC is the best attempt so far to use V as an axiomatization of mathematics”.. Remark 1.1.16. Note that in order to deduce: i) ~ Con ( ZFC2Hs ) from Con ( ZFC2Hs ) ,. ii) ~ Con ( ZFC ) from Con ( ZFC ) , by using Gödel encoding, one needs something more than the consistency of ZFC2Hs , e.g., that ZFC2Hs has an Hs Hs omega-model M ωZFC2 or an standard model M stZFC2 i.e., a model in which. the integers are the standard integers and the all wff of ZFC2Hs , ZFC, etc. represented by standard objects. To put it another way, why should we believe a. statement just because there’s a ZFC2Hs -proof of it? It’s clear that if ZFC2Hs is inconsistent, then we won’t believe ZFC2Hs -proofs. What’s slightly more subtle is that the mere consistency of ZFC2 isn’t quite enough to get us to believe arithmetical theorems of ZFC2Hs ; we must also believe that these arithmetical theorems are asserting something about the standard naturals. It is “conceivable” that ZFC2Hs might be consistent but that the only nonstandard models ZFC2Hs it has are those in which the integers are nonstandard, in which case we M Nst might not “believe” an arithmetical statement such as “ ZFC2Hs is inconsistent” even if there is a ZFC2Hs -proof of it. Hs Remark 1.1.17. Note that assumption ∃M stZFC2 is not necessary if nonstanHs Z 2Hs ZFC2Hs dard model M Nst is a transitive or has a standard part M stZ 2 ⊂ M Nst , see [14] [15]. Remark 1.1.18. Remind that if M is a transitive model, then ω M is the. standard ω . This implies that the natural numbers, integers, and rational numbers of the model are also the same as their standard counterparts. Each. real number in a transitive model is a standard real number, although not all DOI: 10.4236/apm.2019.99034. 690. Advances in Pure Mathematics.

(7) J. Foukzon, E. Men’kova. standard reals need be included in a particular transitive model. Note that in Hs. Z2 any nonstandard model M Nst of the second-order arithmetic Z 2Hs the terms Hs Z 2Hs = 0, S 0 1,= SS 0 2, comprise the initial segment isomorphic to M stZ 2 ⊂ M Nst . Hs. Z2 This initial segment is called the standard cut of the M Nst . The order type of Hs Z2 any nonstandard model of M Nst is equal to  + A ×  , see ref. [16], for some. linear order A.. Hs. Thus one can choose Gödel encoding inside the standard model M stZ 2 . 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 2fss is categorical. Remark 1.1.20. Note if we view second-order arithmetic Z 2 as a theory in. first-order predicate calculus. Thus a model M Z 2 of the language of second-order arithmetic Z 2 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 power set. of M, the model M Z 2 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. Z 2 , 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, M Z 2 is called an ω-model. In this case we may identify the model with D, its collection of sets of naturals, because this set is enough to completely determine an ω-model. The unique full omega-model fss. Z M ω 2 , which is the usual set of natural numbers with its usual structure and. all its subsets, is called the intended or standard model of second-order arithmetic.. 2. Generalized Löb’s Theorem. Remarks on the Tarski’s Undefinability Theorem 2.1. Remarks on the Tarski’s Undefinability Theorem Remark 2.1.1. In paper [2] under the following assumption. (. Con ZFC + ∃M stZFC. ). (14). it has been proved that there exists countable Russell’s set ℜω such that the following statement is satisfied:. (. ). ZFC + ∃M stZFC  ∃ℜω ℜω ∈ M stZFC ∧ ( card ( ℜω ) = ℵ0 ) ∧  ZFC ∀x ( x ∈ ℜω ⇔ x ∉ x )  .  M st . (15). From (15) it immediately follows a contradiction DOI: 10.4236/apm.2019.99034. 691. Advances in Pure Mathematics.

(8) J. Foukzon, E. Men’kova. M ZFC ( ℜω ∈ ℜω ) ∧ ( ℜω ∉ ℜω ) .. (16). st. From (16) and (14) by reductio and absurdum it follows. (. ¬Con ZFC + ∃M stZFC. ). (17). Theorem 2.1.1. [17] [18] [19]. (Tarski’s undefinability theorem). Let ThL be first order theory with formal language L , which includes negation and has. a Gödel numbering g (  ) such that for every L -formula A ( x ) there is a formula B such that B ↔ A ( g ( B ) ) holds. Assume that ThL has a standard model M stThL and Con ( ThL ,st ) where. ThL , st  ThL + ∃M stThL .. (18). Let T ∗ be the set of Gödel numbers of L -sentences true in M stThL . Then there is no L -formula True ( n ) (truth predicate) which defines T ∗ . That is, there is no L -formula True ( n ) such that for every L -formula A, True ( g ( A ) ) ⇔ [ A]M ThL ,. (19). st. [ A]M. where the abbreviation. i.e.. [ A]M. ThL st. ⇔. Th M st L. ThL st. means that A holds in standard model M stThL ,. A . Therefore Con ( ThL. (. ,st. ). implies that. ¬∃True ( x ) True ( g ( A ) ) ⇔ [ A]M ThL st. ). (20). Thus Tarski’s undefinability theorem reads Con ( ThL. , st. ) ⇒ ¬∃True ( x ) ( True ( g ( A) ) ⇔ [ A]. Th M st L. ).. (21). Remark 2.1.2. i) By the other hand the Theorem 2.1.1 says that given some really consistent formal theory ThL ,st that contains formal arithmetic, the concept of truth in that formal theory ThL ,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 L . To define a truth predicate for the metalanguage would require a still higher metametalanguage, and so on. ii) However if formal theory ThL ,st is inconsistent this is not surprising if. we define a formula True ( n ) = True ( n; ThL , st ) by drawing only on a language L .. iii) Note that if under assumption Con ( ThL ,st ) we define a formula True ( n; ThL , st ) by drawing only on a language L by reductio ad absurdum. it follows ¬Con ( ThL. , st. ).. (22). Remark 2.1.3. i) Let ZFCst be a theory ZFCst  ZFC + ∃M stZFC . In this paper under assumption Con ( ZFCst ) we define a formula True ( n; ZFCst ) by drawing only on a language L ZFCst by using Generalized Löb’s theorem [4] [5]. Thus by reductio ad absurdum it follows DOI: 10.4236/apm.2019.99034. 692. Advances in Pure Mathematics.

(9) J. Foukzon, E. Men’kova. (. ). (23). ¬Con ZFC + ∃M stZFC .. ii) However note that in this case we obtain ¬Con ( ZFCst ) by using approach that completely different in comparison with approach based on derivation of the countable Russell’s set ℜω with conditions (15).. 2.2. Generalized Löb’s Theorem Definition 2.2.1. Let ThL# be first order theory and Con ( Th # ) . A theory ThL# is complete if, for every formula A in the theory’s language L , that formula A or its negation ¬A is provable in ThL# , i.e., for any wff A, always ThL#  A or ThL#  ¬A .. Definition 2.2.2. Let ThL be first order theory and Con ( ThL ) . We will say that a theory ThL# is completion of the theory ThL if i) ThL ⊂ ThL# , ii) a theory ThL# is complete. Theorem 2.2.1. [4] [5]. Assume that: Con ( ZFCst ) , where ZFCst  ZFC + ∃M stZFC . Then there exists completion ZFCst# of the theory. ZFCst such that the following conditions hold: i) For every formula A in the language of ZFC that formula. [¬A]M. ZFC st. is provable in ZFC. # st. [ A]M. ZFC st. or formula. i.e., for any wff A, always ZFCst#  [ A]M ZFC or st. ZFCst#  [ ¬A]M ZFC . st. ii) ZFCst# =  m∈ Th m , where for any m a theory Th m+1 is finite extension of the theory Th m . iii) Let Prmst ( y, x ) be recursive relation such that: y is a Gödel number of a. proof of the wff of the theory Th m and x is a Gödel number of this wff. Then the relation Prmst ( y, x ) is expressible in the theory Th m by canonical Gödel encoding and really asserts provability in Th m . iv) Let Prst# ( y, x ) be relation such that: y is a Gödel number of a proof of the. wff of the theory ZFCst# and x is a Gödel number of this wff. Then the relation Prst# ( y, x ) is expressible in the theory ZFCst# by the following formula. Prst# ( y, x ) ⇔ ∃m ( m ∈  ) Prmst ( y, x ). (24). v) The predicate Prst# ( y, x ) really asserts provability in the set theory ZFCst# . Remark 2.2.1. Note that the relation Prmst ( y, x ) is expressible in the theory. Th m since a theory Th m is a finite extension of the recursively axiomatizable theory ZFC and therefore the predicate Prmst ( y, x ) exists since any theory Th m is recursively axiomatizable. Remark 2.2.2. Note that a theory ZFCst# obviously is not recursively axiomatizable nevertheless Gödel encoding holds by Remark 2.2.1.. Theorem 2.2.2. Assume that: Con ( ZFCst ) , where ZFCst  ZFC + ∃M stZFC . Then truth predicate True ( n ) is expressible by using only first order language. by the following formula True ( g ( A ) ) ⇔ ∃y ( y ∈  ) ∃m ( m ∈  ) Prmst ( y, g ( A ) ) .. (25). Proof. Assume that: DOI: 10.4236/apm.2019.99034. 693. Advances in Pure Mathematics.

(10) J. Foukzon, E. Men’kova. ZFCst#  [ A]M ZFC .. (26). st. It follows from (26) there exists m∗ = m∗ ( g ( A ) ) such that Th m∗  [ A]M ZFC st. and therefore by (24) we obtain Prst# ( y, g ( A ) ) ⇔ Prmst∗ ( y, g ( A ) ) .. (27). From (24) immediately by definitions one obtains (25). Remark 2.2.3. Note that Theorem 2.1.1 in this case reads. (. ). Con ( ZFCst ) ⇒ ¬∃True ( x ) True ( g ( A ) ) ⇔ [ A]M ZFC . st. (28). Theorem 2.2.3. ¬Con ( ZFCst ) .. Proof. Assume that: Con ( ZFCst ) . From (25) and (28) one obtains a contra-. diction Con ( ZFCst ) ∧ ¬Con ( ZFCst ) (see Remark 2.1.3) and therefore by re-. ductio ad absurdum it follows ¬Con ( ZFCst ) .. ZFC Theorem 2.2.4. [4] [5]. Let M Nst be a nonstandard model of ZFC and let. M stPA be a standard model of PA. ZFC We assume now that M stPA ⊂ M Nst and denote such nonstandard model of. ZFC ZFC = M Nst the set theory ZFC by M Nst [ PA] . Let ZFCNst be the theory. ZFC ZFC = ZFC + M Nst [ PA] . Assume that: Con ( ZFCNst ) , where Nst ZFC # of the theory ZFCst  ZFC + ∃M Nst . Then there exists completion ZFC Nst ZFC Nst such that the following conditions hold: i) For every formula A in the language of ZFC that formula [ A]M ZFC or formula Nst. [¬A]M. ZFC Nst. is provable in ZFC. # Nst. # i.e., for any wff A, always ZFCNst  [ A]M ZFC Nst. # or ZFC Nst  [ ¬A]M ZFC . Nst. # ii) ZFC Nst =  m∈ Th m , where for any m a theory Th m+1 is finite extension of the theory Th m .. iii) Let PrmNst ( y, x ) be recursive relation such that: y is a Gödel number of a. proof of the wff of the theory Th m and x is a Gödel number of this wff. Then the relation PrmNst ( y, x ) is expressible in the theory Th m by canonical Gödel. encoding and really asserts provability in Th m .. # iv) Let PrNst ( y, x ) be relation such that: y is a Gödel number of a proof of. # the wff of the theory ZFC Nst and x is a Gödel number of this wff. Then the re-. # # lation PrNst by the following formula ( y, x ) is expressible in the theory ZFCNst. (. ). # PrNst ( y, x ) ⇔ ∃m m ∈ M stPA PrmNst ( y, x ). (29). # v) The predicate PrNst ( y, x ) really asserts provability in the set theory # . ZFC Nst. Remark 2.2.4. Note that the relation PrmNst ( y, x ) is expressible in the theory. Th m since a theory Th m is a finite extension of the recursively axiomatizable theory ZFC and therefore the predicate PrmNst ( y, x ) exists since any theory Th m is recursively axiomatizable. # Remark 2.2.5. Note that a theory ZFC Nst obviously is not recursively axi-. omatizable nevertheless Gödel encoding holds by Remark 2.2.1. DOI: 10.4236/apm.2019.99034. 694. Advances in Pure Mathematics.

(11) J. Foukzon, E. Men’kova ZFC Theorem 2.2.5. Assume that: Con ( ZFCNst ) , where ZFCNst  ZFC + ∃M Nst , PA ZFC M st ⊂ M Nst .. Then truth predicate True ( n ) is expressible by using first order language by. the following formula. (. ) (. ). True ( g ( A ) ) ⇔ ∃y y ∈ M stPA ∃m m ∈ M stPA PrmNst ( y, g ( A ) ) .. (30). Proof. Assume that: # ZFC Nst  [ A]M ZFC .. (31). Nst. It follows from (29) there exists m∗ = m∗ ( g ( A ) ) such that Th m∗  [ A]M ZFC Nst. and therefore by (31) we obtain # PrNst ( y, g ( A) ) ⇔ PrmNst∗ ( y, g ( A) ) .. (32). From (32) immediately by definitions one obtains (30). Remark 2.2.6. Note that Theorem 2.1.1 in this case reads. (. ). Con ( ZFC Nst ) ⇒ ¬∃True ( x ) True ( g ( A ) ) ⇔ [ A]M ZFC . Nst. (33). Theorem 2.2.6. ¬Con ( ZFCNst ) . Proof. Assume that: Con ( ZFC Nst ) . From (30) and (33) one obtains a contra-. diction Con ( ZFCNst ) ∧ ¬Con ( ZFC Nst ) and therefore by reductio ad absurdum it follows ¬Con ( ZFCNst ) . Theorem 2.2.7. Assume that: Con ZFC 2Hs , where. (. Hs. ). ZFC 2Hs  ZFC2Hs + ∃M stZFC2 . Then there exists completion ZFC 2Hs# of the. theory ZFC 2Hs such that the following conditions hold: i) For every first order wff formula A (wff1 A) in the language of ZFC2Hs that formula. [ A]M. ZFC2Hs st. or formula. [¬A]M. ZFC2Hs st. is provable in ZFC 2Hs# i.e., for any. wff1 A, always ZFC 2Hs#  [ A]M ZFC2Hs or ZFC 2Hs#  [ ¬A]M ZFC2Hs . st. st. ii) ZFC 2Hs# =  m∈ Th m , where for any m a theory Th m+1 is finite extension of the theory Th m .. iii) Let Prmst ( y, x ) be recursive relation such that: y is a Gödel number of a. proof of the wff1 of the theory Th m and x is a Gödel number of this wff1. Then the relation Prmst ( y, x ) is expressible in the theory Th m by canonical Gödel encoding and really asserts provability in Th m .. iv) Let Prst# ( y, x ) be relation such that: y is a Gödel number of a proof of the. wff of the set theory ZFC 2Hs# and x is a Gödel number of this wff1. Then the re-. lation Prst# ( y, x ) is expressible in the set theory ZFC 2Hs# by the following formula. Prst# ( y, x ) ⇔ ∃m ( m ∈  ) Prmst ( y, x ). (34). v) The predicate Prst# ( y, x ) really asserts provability in the set theory ZFC 2Hs# .. Remark 2.2.7. Note that the relation Prmst ( y, x ) is expressible in the theory. Th m since a theory Th m is a finite extension of the finite axiomatizable theory DOI: 10.4236/apm.2019.99034. 695. Advances in Pure Mathematics.

(12) J. Foukzon, E. Men’kova Nst ZFC2Hs and therefore the predicate Prm ( y, x ) exists since any theory Th m. is recursively axiomatizable. # Remark 2.2.8. Note that a theory ZFC Nst obviously is not recursively axiomatizable nevertheless Gödel encoding holds by Remark 2.2.1. Theorem 2.2.8. Assume that: Con ZFC 2Hs , where. ). (. Hs 2. ZFC.  ZFC. Hs 2. Hs. + ∃M stZFC2 .. Then truth predicate True ( n ) is expressible by using first order language by the following formula True ( g ( A ) ) ⇔ ∃y ( y ∈  ) ∃m ( m ∈  ) Prmst ( y, g ( A ) ) ,. (35). where A is wff1. Proof. Assume that: ZFC 2Hs#  [ A]M ZFC2Hs .. (36). st. It follows from (34) there exists m∗ = m∗ ( g ( A ) ) such that Th m∗  [ A]M ZFC2Hs st. and therefore by (36) we obtain Prst# ( y, g ( A ) ) ⇔ Prmst∗ ( y, g ( A ) ) .. (37). From (37) immediately by definitions one obtains (35). Remark 2.2.9. Note that in considered case Tarski’s undefinability theorem (2.1.1) reads. (. (. ). ). Con ZFC 2Hs# ⇒ ¬∃True ( x ) True ( g ( A ) ) ⇔ [ A]M ZFC2Hs ,. where A is wff1.. (. st. (38). ). Theorem 2.2.9. ¬Con ZFC 2Hs# .. (. ) . From (35) and (38) one obtains a contradiction Con ( ZFC ) ∧ ¬Con ( ZFC ) and therefore by reductio ad absurdum it follows ¬Con ( ZFC ) . Proof. Assume that: Con ZFC Hs# 2. Hs# 2. Hs# 2. Hs# 2. 3. Derivation of the Inconsistent Provably Definable Set in Set Theory ZFC 2Hs , ZFC st and ZFC Nst 3.1. Derivation of the Inconsistent Provably Definable Set in Set Theory ZFC 2Hs Definition 3.1.1. i) Let Φ be a wff of ZFC 2Hs . We will say that Φ is a first order n-place open wff if Φ contains free occurrences of the first order individual variables X 1 , , X n and quantifiers only over any first order individual variables Y1 , , Ym .  Hs be the countable set of the all first order provable definable sets X, ii) Let ℑ 2. Ψ M st ( X ) is a first ori.e. sets such that ZFC 2Hs  ∃!X Ψ ( X ) , where Ψ ( X ) =. der 1-place open wff that contains only first order variables (we will denote such wff for short by wff1), with all bound variables restricted to standard model Hs. M st = M stZFC2 , i.e. DOI: 10.4236/apm.2019.99034. 696. Advances in Pure Mathematics.

(13) J. Foukzon, E. Men’kova. {. (.  Hs ⇔ ZFC Hs  ∃Ψ ( X )   Ψ ( X )  ∈ Γ Hs / ∼ ∀Y Y ∈ ℑ M st X , M st X 2 2    M st. }. ∧ ∃!X  Ψ M st ( X ) ∧ Y =X    , . ). (39). or in a short notation. {. (.  Hs ⇔ ZFC Hs  ∃Ψ ( X )   Ψ ( X )  ∈ Γ Hs / ∼ ∀Y Y ∈ ℑ X X 2 2   . }. ∧ ∃!X  Ψ ( X ) ∧ Y =X    .. ). (40). Notation 3.1.1. In this subsection we often write for short Ψ ( X ) , FXHs , Γ Hs X. instead Ψ M st ( X ) , FXHs, M st , Γ Hs but this should not lead to a confusion. X , M st Assumption 3.1.1. We assume now for simplicity but without loss of generality that (41). FXHs, M st ∈ M st. Hs. Hs. ZFC2 and therefore by definition of model M st = M stZFC2 one obtains Γ Hs . X , M st ∈ M st Hs Let X ∉ Hs Y be a predicate such that X ∉ Hs Y ↔ ZFC 2  X ∉ Y . Let ZFC 2.  ℜ. Hs 2. ZFC 2. be the countable set of the all sets such that. (. ).  Hs  X ∈ ℜ  Hs ↔ X ∉ ∀X X ∈ ℑ X  . 2 2  Hs  ZFC 2 . (42). From (42) one obtains  Hs ∈ ℜ  Hs ↔ ℜ  Hs ∉ ℜ  2 2 2. Hs ZFC 2.  Hs . ℜ 2. (43). But obviously (43) immediately gives a contradiction. ( ℜ. Hs 2. ).  Hs ∧  ℜ  Hs ∉  Hs  . ∈ℜ ℜ 2 2 2  Hs ZFC 2  . (44). Remark 3.1.1. Note that a contradiction (44) in fact is a contradiction inside ZFC 2Hs for the reason that predicate X ∉. Hs ZFC 2. Y is expressible by first order. language as predicate of ZFC 2Hs (see subsection 1.2, Theorem 1.2.8 (ii)-(iii))  Hs and ℜ  Hs are sets in the sense of the set and therefore countable sets ℑ 2 2 Hs theory ZFC 2 . Remark 3.1.2. Note that by using Gödel encoding the above stated contradiction can be shipped in special completion ZFC 2Hs# of ZFC 2Hs , see subsection 1.2, Theorem 1.2.8. Remark 3.1.3. i) Note that Tarski’s undefinability theorem cannot block the equivalence (43) since this theorem is no longer holds by Proposition 2.2.1. (Generalized Löbs Theorem). ii) In additional note that: since Tarski’s undefinability theorem has been Hs proved under the same assumption ∃M stZFC2 by reductio ad absurdum it follows again ¬Con ( ZFCNst ) , see Theorem 1.2.10. Remark 3.1.4. More formally we can to explain the gist of the contradictions derived in this paper (see Section 4) as follows.  Hs be the set of the all sets of M Let M be Henkin model of ZFC2Hs . Let ℜ 2 DOI: 10.4236/apm.2019.99034. 697. Advances in Pure Mathematics.

(14) J. Foukzon, E. Men’kova.  Hs= { x ∈ ℑ  Hs : ( x ∉ x )} where  A provably definable in ZFC 2Hs , and let ℜ 2 2 means “sentence A derivable in ZFC 2Hs ”, or some appropriate modification. thereof. We replace now formula (39) by the following formula. {. (. }. ).  Hs ↔ ∃Ψ ( X )   Ψ ( X )  ∈ Γ Hs / ∼ ∧ ∃!X  Ψ ( X ) ∧ Y = X   . (45) ∀Y Y ∈ ℑ 2 X X     . and we replace formula (42) by the following formula. (. ).  Hs  X ∈ ℜ  Hs ⇔ ( X ∉ X )  . ∀X X ∈ ℑ 2 2  . (46). Definition 3.1.2. We rewrite now (45) in the following equivalent form. {. (. }. ).  Hs ⇔ ∃Ψ ( X )   Ψ ( X )  ∈ Γ  Hs / ∼ ∧ (Y = X )  , ∀Y Y ∈ ℑ 2 X X  Hs    where the countable set Γ XHs / ∼ X is defined by the following formula. {. (. ). (47). }.  ∀Ψ ( X )  Ψ ( X )  ∈ Γ XHs / ∼ X ⇔   Ψ ( X )  Hs ∈ Γ Hs X / ∼ X ∧ ∃! X Ψ ( X )  (48)   Hs be the countable set of the all sets such that Definition 3.1.3. Let ℜ 2. (. ).  Hs  X ∈ ℜ  Hs ⇔  X ∉ X  . ∀X X ∈ ℑ 2 2  . (49).  Hs is a set definable by the first  Hs ∈ ℑ  Hs since ℜ Remark 3.1.5. Note that ℜ 2 2 2 order 1-place open wff1:. (. ). (. ).  Hs  ∀X X ∈ ℑ  Hs  X ∈ Z ⇔ ( X ∉ X )  . Ψ Z,ℜ 2 2  . (50). From (49) and Remark 3.1.4 one obtains. (. ).  Hs ∈ ℜ  Hs ⇔  ℜ  Hs ∉ ℜ  Hs . ℜ 2 2 2 2. (51). But (51) immediately gives a contradiction. (. ) (. ).  Hs ∈ ℜ  Hs ∧ ℜ  Hs ∉ ℜ  Hs . ZFC 2Hs  ℜ 2 2 2 2. (52). Remark 3.1.6. Note that contradiction (52) is a contradiction inside ZFC 2Hs  Hs is a set in the sense of the set theory for the reason that the countable set ℑ 2. ZFC 2Hs .. In order to obtain a contradiction inside ZFC 2Hs without any reference to Assumption 3.1.1 we introduce the following definitions. Definition 3.1.4. We define now the countable set Γν Hs / ν by the following formula  Hs ( y, v ) ∧ ∃! X Ψ ( X )  [ y ]Hs ∈ Γν Hs / ∼ν ⇔ ([ y ]Hs ∈ ΓνHs / ∼ν ) ∧ Fr y ,ν 2  . (53). Definition 3.1.5. We choose now  A in the following form.  A  Bew. Hs. ZFC 2. ( #A) ∧  BewZFC ( #A) ⇒ A . Hs 2. (54). Here BewZFC Hs ( #A ) is a canonical Gödel formula which says to us that there 2. exists proof in ZFC 2Hs of the formula A with Gödel number #A . Remark 3.1.7. Note that the Definition 3.1.5 holds as definition of predicate really asserting provability of the first order sentence A in ZFC 2Hs . Definition 3.1.6. Using Definition 3.1.5, we replace now formula (48) by the DOI: 10.4236/apm.2019.99034. 698. Advances in Pure Mathematics.

(15) J. Foukzon, E. Men’kova. following formula. {. (. ∀Ψ ( X )  Ψ ( X )  ∈ Γ XHs / ∼ X ⇔ ∃Ψ ( X )  Ψ ( X )  ∈ Γ Hs X / ∼X. (( ( # ( ∃!X Ψ ( X ) ∧ =Y. ). )). ∧  Bew Hs # ∃!X  Ψ ( X ) ∧ Y = X   ZFC 2   ∧  Bew Hs ZFC 2 . (55). }. )). X  ⇒ ∃! X  Ψ ( X ) ∧ = Y X   . . Definition 3.1.7. Using Definition 3.1.5, we replace now formula (49) by the following formula. (. ).  Hs  X ∈ ℜ  Hs ⇔  Bew Hs ( # ( X ∉ X ) )  ∀X X ∈ ℑ 2 2   ZFC 2 . (56). ∧  Bew Hs ( # ( X ∉ X ) ) ⇒ X ∉ X  . ZFC 2  . Definition 3.1.8. Using Proposition 2.1.1 and Remark 2.1.10 [6], we replace now formula (53) by the following formula. {. ∀y [ y ]Hs ∈ Γν Hs / ∼ν ⇔. ([ y ]. Hs. ∈ ΓνHs / ∼ν. ). (. ).  Hs  X   ∧ Fr Hs #∃!X  2 ( y , v ) ∧  Bew  Ψ y ,ν ( X ) ∧ Y = ZFC 2  . (. (57). }. ). ∧  Bew Hs #∃!X  Ψ y ,ν ( X ) ∧ Y= X  ⇒ ∃!X  Ψ y ,ν ( X ) ∧ Y= X   . ZFC 2   Definition 3.1.9. Using Definitions 3.1.4-3.1.6, we define now the countable   Hs by formula set ℑ 2. {. (. ). )}. (.   Hs ⇔ ∃y  [ y ] ∈ Γ  Hs / ∼ ∧ g Hs ( X ) =ν  . ∀Y Y ∈ ℑ 2 ν ν ZFC 2  . (58). Remark 3.1.8. Note that from the second order axiom schema of replacement   Hs is a set in the sense of the set theory ZFC Hs . (12) it follows directly that ℑ 2. 2. Definition 3.1.10. Using Definition 3.1.8 we replace now formula (56) by the following formula. (. ).   Hs  X ∈ ℜ   Hs ⇔  Bew Hs ( # ( X ∉ X ) )  ∀X X ∈ ℑ 2 2 ZFC 2    ∧  Bew Hs ( # ( X ∉ X ) ) ⇒ X ∉ X   . ZFC 2   . (59). Remark 3.1.9. Notice that the expression (60).  Bew Hs ( # ( X ∉ X ) )  ∧  Bew Hs ( # ( X ∉ X ) ) ⇒ X ∉ X      ZFC 2 ZFC 2. (60).   Hs is a set obviously is a well formed formula of ZFC 2Hs and therefore a set ℜ 2 Hs in the sense of ZFC 2 .   Hs is a set definable by   Hs ∈ ℑ   Hs since ℜ Remark 3.1.10. Note that ℜ 2 2 2. 1-place open wff. (. ). (. ).   Hs  ∀X X ∈ ℑ  Hs  X ∈ Z ⇔  Bew Hs ( # ( X ∉ X ) )  Ψ Z,ℜ 2 2    ZFC 2 ∧  Bew Hs ( # ( X ∉ X ) ) ⇒ X ∉ X   . ZFC 2    Hs. Theorem 3.1.1. Set theory ZFC 2Hs  ZFC2Hs + ∃M stZFC2 DOI: 10.4236/apm.2019.99034. 699. (61). is inconsistent.. Advances in Pure Mathematics.

(16) J. Foukzon, E. Men’kova. Proof. From (59) we obtain. (( )). )).   Hs ∈ ℜ   Hs ⇔  Bew Hs # ℜ   Hs ∉ ℜ   Hs  ℜ 2 2 2 2 ZFC 2     Hs ∉ ℜ   Hs ⇒ ℜ   Hs ∉ ℜ   Hs  . ∧  Bew Hs # ℜ 2 2 2 2 ZFC 2  . ((. (62). a) Assume now that:   Hs ∈ ℜ   Hs . ℜ 2 2. Then from (62) we obtain . Hs. ZFC 2. . Hs. ZFC 2. therefore . Hs. ZFC 2. Bew. Hs. ZFC 2. Bew. (63) Hs. ZFC 2. ( # (ℜ.  Hs 2. ( # (ℜ ∉ ℜ )) ) ) ⇒ ℜ ∉ ℜ  Hs 2.   Hs ∉ℜ 2.  Hs 2.  Hs 2.  Hs 2. and ,.   Hs ∉ ℜ   Hs and so ℜ 2 2 . Hs. ZFC 2.   Hs ∈ ℜ   Hs ⇒ ℜ   Hs ∉ ℜ   Hs . ℜ 2 2 2 2. (64). From (63)-(64) we obtain   Hs ∈ ℜ   Hs , ℜ   Hs ∈ ℜ   Hs ⇒ ℜ   Hs ∉ ℜ   Hs  ℜ   Hs ∉ ℜ   Hs ℜ 2 2 2 2 2 2 2 2. (. ) (. ).  Hs ∈ ℜ  Hs ∧ ℜ  Hs ∉ ℜ  Hs . and thus ZFC Hs ℜ 2 2 2 2 2. b) Assume now that. ((. )). ((.    Hs   Hs  ∧  Bew Hs # ℜ   Hs ∉ ℜ   Hs 2 2  BewZFC 2Hs # ℜ2 ∉ ℜ2   ZFC 2. ) ) ⇒ ℜ.  Hs 2.   Hs  . (65) ∉ℜ 2 .   Hs ∉ ℜ   Hs . From (65) and (62) we obtain Then from (65) we obtain  ℜ 2 2  Hs  Hs  Hs  Hs       Hs ∈ ℜ   Hs which immediately  Hs ℜ2 ∈ ℜ2 , so  Hs ℜ2 ∉ ℜ2 , ℜ 2 2 ZFC 2. ZFC 2. gives us a contradiction . Hs. ZFC 2. ( ℜ.  Hs 2. ) (. ).   Hs ∧ ℜ   Hs ∉ ℜ   Hs . ∈ℜ 2 2 2. Definition 3.1.11. We choose now  A in the following form.  A  Bew. Hs. ZFC 2. ( #A) ,. (66). or in the following equivalent form.  A  Bew. Hs. ZFC 2. similar to (46). Here Bew. ( #A) ∧  Bew ZFC ( #A) ⇒ A. (67). Hs 2. Hs. ZFC 2. ( #A). is a Gödel formula which really asserts. provability in ZFC 2Hs of the formula A with Gödel number #A . Remark 3.1.11. Notice that the Definition 3.1.12 with formula (66) holds as definition of predicate really asserting provability in ZFC 2Hs . Definition 3.1.12. Using Definition 3.1.11 with formula (66), we replace now formula (48) by the following formula. {. (. ∀Ψ ( X )  Ψ ( X )  ∈ Γ XHs / ∼ X ⇔ ∃Ψ ( X )  Ψ ( X )  ∈ Γ Hs X / ∼X. ((. )) }. ∧  Bew Hs # ∃! X  Ψ ( X ) ∧ Y = X   . ZFC 2  . ). (68). Definition 3.1.13. Using Definition 3.1.11 with formula (66), we replace now DOI: 10.4236/apm.2019.99034. 700. Advances in Pure Mathematics.

(17) J. Foukzon, E. Men’kova. formula (49) by the following formula. (. ).  Hs  X ∈ ℜ  Hs ⇔  Bew Hs ( # ( X ∉ X ) )   ∀X X ∈ ℑ 2 2  ZFC 2   . (69). Definition 3.1.14. Using Definition 3.1.11 with formula (66), we replace now formula (53) by the following formula. {. ∀y [ y ]Hs ∈ Γν Hs / ∼ν ⇔. ([ y ]. Hs. ).  Hs ( y, v ) ∈ ΓνHs / ∼ν ∧ Fr 2. (70). )}. (. ∧  Bew Hs #∃! X  Ψ y ,ν ( X ) ∧ Y = X   . ZFC 2  . Definition 3.1.15. Using Definitions 3.1.12-3.1.16, we define now the counta  Hs by formula ble set ℑ 2. {. (. )}. (. ).   Hs ⇔ ∃y  [ y ] ∈ Γ  Hs / ∼ ∧ g Hs ( X ) =ν  . ∀Y Y ∈ ℑ 2 ν ν ZFC 2  . (71). Remark 3.1.12. Note that from the axiom schema of replacement (12) it fol  Hs is a set in the sense of the set theory ZFC Hs . lows directly that ℑ 2 2 Definition 3.1.16. Using Definition 3.1.15 we replace now formula (69) by the following formula. (. ).   Hs  X ∈ ℜ   Hs ⇔  Bew Hs ( # ( X ∉ X ) )   . ∀X X ∈ ℑ 2 2   ZFC 2  . (72). Remark 3.1.13. Notice that the expressions (73).  Bew Hs ( # ( X ∉ X ) )  and   ZFC 2  Bew Hs ( # ( X ∉ X ) )  ∧  Bew Hs ( # ( X ∉ X ) ) ⇒ X ∉ X  ZFC 2 ZFC 2    . (73).   Hs obviously are a well formed formula of ZFC 2Hs and therefore collection ℜ 2 is a set in the sense of ZFC 2Hs .   Hs is a set definable by   Hs ∈ ℑ   Hs since ℜ Remark 3.1.14. Note that ℜ 2 2 2 1-place open wff1. (. ). (. ).   Hs  ∀X X ∈ ℑ   Hs  X ∈ Z ⇔ Bew Hs ( # ( X ∉ X ) )  . Ψ Z,ℜ 2 2   ZFC 2 Hs. Theorem 3.1.2. Set theory ZFC 2Hs  ZFC2Hs + ∃M stZFC2 Proof. From (72) we obtain. ((. (74). is inconsistent.. )).   Hs ∈ ℜ   Hs ⇔  Bew Hs # ℜ   Hs ∉ ℜ   Hs  . ℜ 2 2 2 2 ZFC 2  . (75). a) Assume now that:   Hs ∈ ℜ   Hs . ℜ 2 2. Then from (75) we obtain . Hs. ZFC 2. . Hs. ZFC 2. Bew. (76) Hs. ZFC 2. ( # (ℜ.  Hs 2.   Hs ∉ℜ 2. )). and therefore.   Hs ∉ ℜ   Hs , thus we obtain ℜ 2 2 . Hs. ZFC 2.   Hs ∈ ℜ   Hs ⇒ ℜ   Hs ∉ ℜ   Hs . ℜ 2 2 2 2. (77).   Hs ∈ ℜ   Hs ⇒ ℜ   Hs ∉ ℜ   Hs , thus   Hs ∈ ℜ   Hs and ℜ From (76)-(77) we obtain ℜ 2 2 2 2 2 2  Hs ∈ ℜ  Hs ∧ ℜ  Hs ∉ ℜ  Hs .   Hs ∉ ℜ   Hs and finally we obtain  Hs ℜ  Hs ℜ 2 2 2 2 2 2 ZFC 2. DOI: 10.4236/apm.2019.99034. ZFC 2. 701. (. ) (. ). Advances in Pure Mathematics.

(18) J. Foukzon, E. Men’kova. b) Assume now that. ((. )).  Bew Hs # ℜ   Hs ∉ ℜ   Hs  . 2 2 ZFC 2  . (78).   Hs ∉ ℜ   Hs . From (78) and (75) we obtain Then from (78) we obtain ZFC Hs ℜ 2 2 2. . Hs. ZFC 2.   Hs ∈ ℜ   Hs , thus  ℜ 2 2. Hs. ZFC 2.   Hs ∉ ℜ   Hs and  ℜ 2 2. immediately gives us a contradiction . Hs. ZFC 2. ( ℜ. Hs. ZFC 2.  Hs 2.   Hs ∈ ℜ   Hs which ℜ 2 2. ). ) (.   Hs ∧ ℜ   Hs ∉ ℜ   Hs . ∈ℜ 2 2 2. 3.2. Derivation of the Inconsistent Provably Definable Set in Set Theory ZFCst Let ℑst be the countable set of all sets X such that ZFCst  ∃!X Ψ ( X ) , where Ψ ( X ) is a 1-place open wff of ZFC i.e.,. {. (. ∀Y Y ∈ ℑst ⇔ ZFCst  ∃Ψ ( X )   Ψ ( X )  ∈ Γ stX / ∼ X . ). }. ∧∃! X  Ψ ( X ) ∧ Y = X   .. (79). Let X ∉ZFC Y be a predicate such that X ∉ZFC Y ⇔ ZFCst  X ∉ Y . Let st. st. ℜ be the countable set of the all sets such that. (. ). ∀X  X ∈ ℜ st ⇔ ( X ∈ ℑst ) ∧ X ∉ZFC X  . st  . (80). From (80) one obtains. ℜ st ∈ ℜ st ⇔ ℜ st ∉ZFC ℜ st .. (81). st. But (81) immediately gives a contradiction. ( ℜst ∈ ℜst ) ∧ ( ℜst ∉ ℜst ) .. (82). Remark 3.2.1. Note that a contradiction (82) is a contradiction inside ZFCst for the reason that predicate X ∉ZFC Y is expressible by using first order st. language as predicate of ZFCst (see subsection 4.1) and therefore countable sets ℑst and ℜ st are sets in the sense of the set theory ZFCst . Remark 3.2.2. Note that by using Gödel encoding the above stated contradiction can be shipped in special completion ZFCst# of ZFCst , see subsection 1.2, Theorem 1.2.2 (i). Designation 3.2.1. i) Let M stZFC be a standard model of ZFC and ii) let ZFCst be the theory ZFC = ZFC + ∃M stZFC , st iii) let ℑst be the set of the all sets of M stZFC provably definable in ZFCst , and let ℜ= st. { X ∈ ℑst : st ( X ∉ X )} , where. st A means: “sentence A deriva-. ble in ZFCst ”, or some appropriate modification thereof. We replace now (79) by formula. {. }. ∀Y Y ∈ ℑst ↔ st ∃Ψ ( ⋅) ∃!X  Ψ ( X ) ∧ Y = X   ,. (83). and we replace (80) by formula ∀X  X ∈ ℜ st ↔ ( X ∈ ℑst ) ∧ st ( X ∉ X )  . DOI: 10.4236/apm.2019.99034. 702. (84). Advances in Pure Mathematics.

(19) J. Foukzon, E. Men’kova. ZFCst  ℜ st ∈ ℑst . Then, we have that:. Assume that. iff. ℜ st ∈ ℜ st. st ( ℜ st ∉ ℜ st ) , which immediately gives us ℜ st ∈ ℜ st iff ℜ st ∉ ℜ st . But this is. a contradiction, i.e., ZFCst  ( ℜ st ∈ ℜ st ) ∧ ( ℜ st ∉ ℜ st ) . We choose now st A in the following form st A  BewZFCst ( #A ) ∧  BewZFCst ( #A ) ⇒ A .. (85). Here BewZFCst ( #A ) is a canonical Gödel formula which says to us that there exists proof in ZFCst of the formula A with Gödel number #A ∈ M stPA .. Remark 3.2.3. Notice that Definition 3.2.6 holds as definition of predicate really asserting provability in ZFCst . Definition 3.2.1. We rewrite now (83) in the following equivalent form. {. (. }. ).  ⇔ ∃Ψ ( X )   Ψ ( X )  ∈ Γ  st / ∼ ∧ (Y = X )  , ∀Y Y ∈ ℑ st X X  st   . (86). where the countable collection Γ XHs / ∼ X is defined by the following formula. {. (. }. ). ∀Ψ ( X )  Ψ ( X )  st ∈ Γ Xst / ∼ X ⇔   Ψ ( X )  st ∈ Γ stX / ∼ X ∧ st ∃!X Ψ ( X )  (87)    be the countable collection of the all sets such that Definition 3.2.2. Let ℜ st. (. ).  X ∈ℜ  ⇔  ( X ∉ X ) . ∀X X ∈ ℑ st  st st . (88).  Hs ∈ ℑ  Hs since ℜ  Hs is a collection definable by Remark 3.2.4. Note that ℜ 2 2 2 1-place open wff. (. ). (. ).   Ψ Z,ℜ st  ∀X X ∈ ℑst   X ∈ Z ⇔ st ( X ∉ X )  .. (89). Definition 3.2.3. By using formula (85) we rewrite now (86) in the following equivalent form. {. (. }. ).  ⇔ ∃Ψ ( X )   Ψ ( X )  ∈ Γ  st / ∼ ∧ (Y = X )  , ∀Y Y ∈ ℑ st X X  st   . (90). where the countable collection Γ XHs / ∼ X is defined by the following formula. {. ∀Ψ ( X )  Ψ ( X )  st ∈ Γ Xst / ∼ X ⇔. (. ).   Ψ ( X )  ∈ Γ stX / ∼ X ∧ BewZFC ( #∃!X Ψ ( X ) )   st st   . (91). }. ∧  BewZFCst ( #∃!X Ψ ( X ) ) ⇒ ∃!X Ψ ( X ) . Definition 3.2.4. Using formula (85), we replace now formula (88) by the following formula. (. ).  X ∈ℜ  ⇔  Bew    ∀X X ∈ ℑ st  st ZFCst ( # ( X ∉ X ) )  ∧  BewZFCst ( # ( X ∉ X ) )  . (92)  Definition 3.2.5. Using Proposition 2.1.1 and Remark 2.2.2 [6], we replace now formula (89) by the following formula. {. ∀y [ y ]st ∈ Γν st / ∼ν ⇔. ( ( #∃!X Ψ. ([ y ]. st. ).  ( y, v ) ∈ Γνst / ∼ν ∧ Fr st. ) X  ) ⇒ ∃!X  Ψ. X   ∧  BewZFCst #∃!X  Ψ y ,ν ( X ) ∧ Y =   ∧  BewZFCst  DOI: 10.4236/apm.2019.99034. 703. y ,ν. ( X ) ∧ Y=. (93) y ,ν. ( X ) ∧ Y=. }. X   . . Advances in Pure Mathematics.

(20) J. Foukzon, E. Men’kova. Definition 3.2.6. Using Definitions 3.2.3-3.2.5, we define now the countable   by formula set ℑ st. {. )}. ) (. (.   ⇔ ∃y  [ y ] ∈ Γ  st / ∼ ∧ g  . ∀Y Y ∈ ℑ st ν ν ZFCst ( X ) =ν st  . (94). Remark 3.2.5. Note that from the axiom schema of replacement it follows di  is a set in the sense of the set theory ZFC . rectly that ℑ st. st. Definition 3.2.7. Using Definition 3.2.6 we replace now formula (92) by the following formula. (. ).   X ∈ℜ   ⇔  Bew  ∀X X ∈ ℑ st  st ZFCst ( # ( X ∉ X ) )   ∧  BewZFC st ( # ( X ∉ X ) ) ⇒ X ∉ X   . . (95). Remark 3.2.6. Notice that the expression (96)  BewZFCst ( # ( X ∉ X ) )  ∧  BewZFCst ( # ( X ∉ X ) ) ⇒ X ∉ X     . (96).   is a obviously is a well formed formula of ZFCst and therefore collection ℜ st Hs set in the sense of ZFC 2 .   ∈ℑ   since ℜ   is a collection definable by Remark 3.2.7. Note that ℜ st st st 1-place open wff. (. ). (. ).    ∀X X ∈ ℑ    X ∈ Z ⇔  Bew  Ψ Z,ℜ st st  ZFCst ( # ( X ∉ X ) )   ∧  BewZFCst ( # ( X ∉ X ) ) ⇒ X ∉ X   .. (97). Theorem 3.2.1. Set theory ZFCst  ZFC + ∃M stZFC is inconsistent. Proof. From (95) we obtain. (( )). )).   ∈ℜ   ⇔  Bew    ℜ st st ZFCst # ℜ st ∉ ℜ st       ∉ℜ   ∉ℜ   . ∧  BewZFCst # ℜ ⇒ℜ st st st st  . ((. (98). a) Assume now that:   ∈ℜ . ℜ st st. (99). ((.   ∉ℜ  Then from (98) we obtain  BewZFCst # ℜ st st. ((.   ∉ℜ   BewZFCst # ℜ st st. ) ) ⇒ ℜ. )). and.   , therefore  ℜ   ∉ℜ   and so ∉ℜ st st st.  st.   ∈ℜ  ⇒ℜ   ∉ℜ . ZFCst ℜ st st st st. (100).   ∈ℜ   ,ℜ   ∈ℜ  ⇒ℜ   ∉ℜ   ℜ   ∉ℜ   and From (99)-(100) we obtain ℜ st st st st st st st st   ∈ℜ   ) ∧ (ℜ   ∉ℜ   ). therefore ZFCst ( ℜ st st st st. b) Assume now that. ((. )). ((.   ∉ℜ    ∧  Bew    BewZFC # ℜ st st ZFCst # ℜ st ∉ ℜ st st   . ) ) ⇒ ℜ.  st.    . (101) ∉ℜ st .   Hs ∉ ℜ   Hs . From (101) and (98) we obtain Then from (101) we obtain  ℜ 2 2  Hs  Hs  Hs  Hs   Hs       Hs which immediately , so  Hs ℜ ∉ ℜ , ℜ ∈ ℜ  Hs ℜ ∈ ℜ ZFC 2. 2. 2. ZFC 2. gives us a contradiction . Hs. ZFC 2. DOI: 10.4236/apm.2019.99034. 704. 2. ( ℜ.  Hs 2. 2. 2. ) (. 2. ).   Hs ∧ ℜ   Hs ∉ ℜ   Hs . ∈ℜ 2 2 2 Advances in Pure Mathematics.

(21) J. Foukzon, E. Men’kova. 3.3. Derivation of the Inconsistent Provably Definable Set in ZFCNst Designation 3.3.1. i) Let PA be a first order theory which contain usual postulates of Peano arithmetic [8] and recursive defining equations for every primitive recursive function as desired. ZFC be a nonstandard model of ZFC and let M stPA be a standard ii) Let M Nst ZFC and denote such nonstanmodel of PA . We assume now that M stPA ⊂ M Nst ZFC  PA . dard model of ZFC by M Nst   ZFC  PA . iii) Let ZFC Nst be the theory ZFC = ZFC + M Nst Nst   ZFC   iv) Let ℑNst be the set of the all sets of M st  PA provably definable in ZFC Nst , and let ℜ Nst = { X ∈ ℑNst : Nst ( X ∉ X )} where Nst A means “sentence A derivable in ZFC Nst ”, or some appropriate modification thereof. We replace now (45) by formula. {. }. ∀Y Y ∈ ℑNst ↔ Nst ∃Ψ ( ⋅) ∃!X  Ψ ( X ) ∧ Y = X   ,. (102). and we replace (46) by formula ∀X  X ∈ ℜ Nst ↔ ( X ∈ ℑNst ) ∧ Nst ( X ∉ X )  .. (103). Assume that ZFCNst  ℜ Nst ∈ ℑNst . Then, we have that: ℜ Nst ∈ ℜ Nst iff Nst ( ℜ Nst ∉ ℜ Nst ) , which immediately gives us ℜ Nst ∈ ℜ Nst iff ℜ Nst ∉ ℜ Nst . But this is a contradiction, i.e., ZFCNst  ( ℜ Nst ∈ ℜ Nst ) ∧ ( ℜ Nst ∉ ℜ Nst ) . We choose now Nst A in the following form Nst A  BewZFCNst ( #A ) ∧  BewZFCNst ( #A ) A .. (104). Here BewZFCNst ( #A ) is a canonical Gödel formula which says to us that there. exists proof in ZFC Nst of the formula A with Gödel number #A ∈ M stPA . Remark 3.3.1. Notice that definition (104) holds as definition of predicate really asserting provability in ZFC Nst . Designation 3.3.2. i) Let g ZFCNst ( u ) be a Gödel number of given an expression u of ZFC Nst . ii) Let FrNst ( y, v ) be the relation: y is the Gödel number of a wff of ZFC Nst that contains free occurrences of the variable with Gödel number v [6] [10].. iii) Let ℘Nst ( y, v,ν 1 ) be a Gödel number of the following wff: g ZFCNst ( X ) = ν , ∃!X  Ψ ( X ) ∧ Y = X  , where g ZFCNst ( Ψ ( X ) ) = y , g ZFCNst (Y ) = ν 1 . iv) Let PrZFCNst ( z ) be a predicate asserting provability in ZFC Nst . Remark 3.3.2. Let ℑNst be the countable collection of all sets X such that. ZFC Nst  ∃!X Ψ ( X ) , where Ψ ( X ) is a 1-place open wff i.e.,. {. }. ∀Y Y ∈ ℑNst ⇔ ZFCNst  ∃Ψ ( X ) ∃!X  Ψ ( X ) ∧ Y = X  .. (105). We rewrite now (105) in the following form. {. (. ).  ( y, v ) ∀Y Y ∈ ℑNst ⇔ g ZFCNst (Y= ) ν 1 ∧ ∃y Fr Nst. (. ). ∧ g ZFCNst ( X ) = ν ∧  PrZFCNst (℘Nst ( y, v,ν 1 ) ). (106). }. ∧  PrZFCNst (℘Nst ( y, v,ν 1 ) ) ⇒ ∃!X  Ψ ( X ) ∧ Y = X    DOI: 10.4236/apm.2019.99034. 705. Advances in Pure Mathematics.

(22) J. Foukzon, E. Men’kova. Designation 3.3.3. Let ℘Nst ( z ) be a Gödel number of the following wff: Z ∉ Z , where g ZFCNst ( Z ) = z . Remark 3.3.3. Let ℜ Nst above by formula (103), i.e.,. ∀Z  Z ∈ ℜ Nst ↔ ( Z ∈ ℑNst ) ∧ Nst ( Z ∉ Z )  .. (107). We rewrite now (107) in the following form. (. ). ∀Z  Z ∈ ℜ Nst ↔ Z ∈ ℑNst ∧ g ZFCNst ( Z )= z ∧ PrZFCNst (℘Nst ( z ) )  ∧  PrZFCNst (℘Nst ( z ) ) ⇒ Z ∉ Z  .. (108). Theorem 3.3.1. ZFC Nst  ℜ Nst ∈ ℜ Nst ∧ ℜ Nst ∉ ℜ Nst .. 3.4. Generalized Tarski’s Undefinability Lemma Remark 3.4.1. Remind that: i) if Th is a theory, let TTh be the set of Godel numbers of theorems of Th [10], ii) the property x ∈ TTh is said to be is expressible in Th by wff True ( x1 ) if the following properties are satisfied [10]: a) if n ∈ TTh then Th  True ( n ) , b) if n ∉ TTh then Th  ¬True ( n ) . Remark 3.4.2. Notice it follows from (a) ∧ (b) that ¬ ( Th  True ( n ) ) ∧ ( Th  ¬True ( n ) )  .. Theorem 3.4.1. (Tarski’s undefinability Lemma) [10]. Let Th be a consistent theory with equality in the language L in which the diagonal function D is representable and let g Th ( u ) be a Gödel number of given an expression u of Th . Then the property x ∈ TTh is not expressible in Th . Proof. By the diagonalization lemma applied to ¬True ( x1 ) there is a sentence F such that: c) Th  F ⇔ ¬True ( q ) , where q is the Godel number of F , i.e. g Th ( F ) = q . Case 1. Suppose that Th  F , then q ∈ TTh . By (a), Th  True ( q ) . But, from Th  F and (c), by biconditional elimination, one obtains Th  ¬True ( q ) . Hence Th is inconsistent, contradicting our hypothesis. Case 2. Suppose that Th  F , then q ∉ TTh . By (b), Th  ¬True ( q ) . Hence, by (c) and biconditional elimination, Th  F . Thus, in either case a contradiction is reached. Definition 3.4.1. If Th is a theory, let TTh be the set of Godel numbers of theorems of Th and let g Th ( u ) be a Gödel number of given an expression u of Th . The property x ∈ TTh is said to be is a strongly expressible in Th by wff True∗ ( x1 ) if the following properties are satisfied:. −1 a) if n ∈ TTh then Th  True∗ ( n ) ∧ ( True∗ ( n ) ⇒ g Th ( n )) , ∗ b) if n ∉ TTh then Th  ¬True ( n ) .. Theorem 3.4.2. (Generalized Tarski’s undefinability Lemma). Let Th be a. consistent theory with equality in the language L in which the diagonal function D is representable and let g Th ( u ) be a Gödel number of given an expression u of Th . Then the property x ∈ TTh is not strongly expressible in Th . Proof. By the diagonalization lemma applied to ¬True∗ ( x1 ) there is a sen-. tence F ∗ such that: c) Th  F ∗ ⇔ ¬True∗ ( q ) , where q is the Godel number DOI: 10.4236/apm.2019.99034. 706. Advances in Pure Mathematics.

(23) J. Foukzon, E. Men’kova. of F ∗ , i.e. g Th ( F ∗ ) = q . Case 1. Suppose that Th  F ∗ , then q ∈ TTh . By (a), Th  True∗ ( q ) . But, from Th  F ∗ and (c), by biconditional elimination, one obtains Th  ¬True∗ ( q ) . Hence Th is inconsistent, contradicting our hypothesis.. Case 2. Suppose that Th  F ∗ , then q ∉ TTh . By (b), Th  ¬True∗ ( q ) . Hence, by (c) and biconditional elimination, Th  F ∗ . Thus, in either case a. contradiction is reached. Remark 3.4.3. Notice that Tarski’s undefinability theorem cannot blocking the biconditionals. ℜ ∈ ℜ ⇔ ℜ ∉ ℜ, ℜ st ∈ ℜ st ⇔ ℜ st ∉ ℜ st ,. (109). ℜ Nst ∈ ℜ Nst ⇔ ℜ Nst ∉ ℜ Nst , see Subsection 2.2.. 3.5. Generalized Tarski’s Undefinability Theorem Remark 3.5.1. I) Let Th1# be the theory Th1#  ZFC 2Hs .  ( Th # ) , we establish a countable sequence In addition under assumption Con 1. of the consistent extensions of the theory Th1# such that: # # i) Th1#    Thi#  Thi+ 1    Th ∞ , where # # ii) Thi+ 1 is a finite consistent extension of Th i , iii) Th ∞# =  i∈ Thi# ,. iv) Th ∞# proves the all sentences of Th1# , which is valid in M, i.e., M  A ⇒ Th ∞#  A , see see Subsection 4.1, Proposition 4.1.1. # #  ZFCst . II) Let Th1,st be Th1,st.  ( Th # ) , we establish a countable seIn addition under assumption Con 1,st quence of the consistent extensions of the theory Th1# such that: i) Th1,# st    Thi#, st  Thi#+1, st    Th ∞# , st , where. ii) Thi#+1, st is a finite consistent extension of Thi#, st ,. iii) Th ∞# , st =  i∈ Thi#, st , # iv) Th ∞# ,st proves the all sentences of Th1,st , which valid in M stZFC , i.e., M stZFC  A ⇒ Th ∞# , st  A , see Subsection 4.1, Proposition 4.1.1. III) Let Th1,# Nst be Th1,# Nst  ZFCNst ..  ( Th # ) , we establish a countable seIn addition under assumption Con 1, Nst quence of the consistent extensions of the theory Th1# such that:. i) Th1,# Nst    Thi#, Nst  Thi#+1, st    Th ∞# , Nst , where ii) Thi#+1, Nst is a finite consistent extension of Thi#, Nst ,. iii) Th ∞# , st =  i∈ Thi#, st # ZFC iv) Th ∞# ,st proves the all sentences of Th1,st , which valid in M Nst [ PA] , i.e.,. ZFC M Nst [ PA]  A ⇒ Th∞# , Nst  A , see Subsection 4.1, Proposition 4.1.1. Remark 3.5.2. I) Let ℑi , i = 1, 2, be the set of the all sets of M provably de# finable in Thi ,. {. }. ∀Y Y ∈ ℑi ↔ i ∃Ψ ( ⋅) ∃!X  Ψ ( X ) ∧ Y = X  . and let ℜ= i DOI: 10.4236/apm.2019.99034. { x ∈ ℑi : i ( x ∉ x )} 707. (110). where i A means sentence A derivable in Advances in Pure Mathematics.

(24) J. Foukzon, E. Men’kova. Thi# . Then we have that ℜi ∈ ℜi iff i ( ℜi ∉ ℜi ) , which immediately gives us. ℜi ∈ ℜi iff ℜi ∉ ℜi . We choose now i A, i = 1, 2, in the following form i A  Bewi ( #A ) ∧  Bewi ( #A ) A .. (111). Here Bewi ( #A ) , i = 1, 2, is a canonical Gödel formulae which says to us that there exists proof in Thi# , i = 1, 2, of the formula A with Gödel number #A . II) Let ℑi , st , i = be the set of the all sets of M stZFC provably definable 1, 2, # in Thi , st ,. }. {. ∀Y Y ∈ ℑi , st ↔ i , st ∃Ψ ( ⋅) ∃!X  Ψ ( X ) ∧ Y = X  .. (112). and let ℜi , st= { x ∈ ℑi , st : i , st ( x ∉ x )} where i , st A means sentence A derivable in Thi#, st . Then we have that ℜi , st ∈ ℜi , st iff i , st ( ℜi , st ∉ ℜi , st ) , which immediately gives us ℜi , st ∈ ℜi , st iff ℜi , st ∉ ℜi , st . We choose now i , st A, i = 1, 2, in the following form i , st A  Bewi , st ( #A ) ∧  Bewi , st ( #A ) ⇒ A .. (113). Here Bewi , st ( #A ) , i = 1, 2, is a canonical Gödel formulae which says to us that there exists proof in Thi#, st , i = 1, 2, of the formula A with Gödel number #A .. ZFC III) Let ℑi , Nst , i = 1, 2, be the set of the all sets of M Nst [ PA] provably de-. finable in Thi#, Nst ,. {. }. ∀Y Y ∈ ℑi , Nst ↔ i , Nst ∃Ψ ( ⋅) ∃!X  Ψ ( X ) ∧ Y = X  .. (114). and let ℜi , Nst= { x ∈ ℑi , Nst : i , Nst ( x ∉ x )} where i , Nst A means sentence A derivable in Thi#, Nst . Then we have that ℜi , Nst ∈ ℜi , Nst iff i , Nst ( ℜi , Nst ∉ ℜi , Nst ) , which immediately gives us ℜi , Nst ∈ ℜi , Nst iff ℜi , Nst ∉ ℜi , Nst . We choose now i , Nst A, i = 1, 2, in the following form i , Nst A  Bewi , Nst ( #A ) ∧  Bewi , Nst ( #A ) ⇒ A .. (115). Here Bewi , Nst ( #A ) , i = 1, 2, is a canonical Gödel formulae which says to us that there exists proof in Thi#, Nst , i = 1, 2, of the formula A with Gödel number #A . Remark 3.5.3. Notice that definitions (111), (113) and (115) hold as definitions of predicates really asserting provability in Thi# , Thi#, st and Thi#, Nst , i = 1, 2, correspondingly. Remark 3.5.4. Of course the all theories Thi# , Thi#, st , Thi#, Nst , i = 1, 2, are inconsistent, see subsection 4.1. Remark 3.5.5. I) Let ℑ∞ be the set of the all sets of M provably definable in Th ∞# ,. {. }. ∀Y Y ∈ ℑ∞ ↔ ∞ ∃Ψ ( ⋅) ∃!X  Ψ ( X ) ∧ Y = X  .. (116). and let ℜ∞= { x ∈ ℑ∞ : ∞ ( x ∉ x )} , where ∞ A means “sentence A derivable in Th ∞# ”. Then, we have that ℜ∞ ∈ ℜ∞ iff ∞ ( ℜ∞ ∉ ℜ∞ ) , which immediately DOI: 10.4236/apm.2019.99034. 708. Advances in Pure Mathematics.

(25) J. Foukzon, E. Men’kova. gives us ℜ∞ ∈ ℜ∞ iff ℜ∞ ∉ ℜ∞ . We choose now ∞ A, i = 1, 2, in the following form ∞ A  ∃i  Bewi ( #A ) ∧  Bewi ( #A ) ⇒ A  .. (117). II) Let ℑ∞ ,st be the set of the all sets of M stZFC provably definable in Th ∞# ,st ,. }. {. ∀Y Y ∈ ℑ∞ , st ↔ ∞ , st ∃Ψ ( ⋅) ∃!X  Ψ ( X ) ∧ Y = X  .. (118). and let ℜ∞ ,st be the set ℜ∞ , st= { x ∈ ℑ∞ , st : ∞ , st ( x ∉ x )} , where ∞ ,st A means “sentence A derivable in Th ∞# ,st ”. Then, we have that ℜ∞ , st ∈ ℜ∞ , st iff ∞ , st ( ℜ∞ , st ∉ ℜ∞ , st ) , which immediately gives us ℜ∞ , st ∈ ℜ∞ , st iff ℜ∞ , st ∉ ℜ∞ , st .. We choose now ∞ , st A, i = 1, 2, in the following form. ∞ , st A  ∃i  Bewi , st ( #A ) ∧  Bewi , st ( #A ) ⇒ A  .. (119). ZFC III) Let ℑ∞ , Nst be the set of the all sets of M Nst [ PA] provably definable in. Th ∞# , Nst ,. {. }. ∀Y Y ∈ ℑ∞ , Nst ↔ ∞ , Nst ∃Ψ ( ⋅) ∃!X  Ψ ( X ) ∧ Y = X  .. (120). and let ℜ∞ , Nst be the set ℜ∞ , Nst= { x ∈ ℑ∞ , Nst : ∞ , Nst ( x ∉ x )} where ∞ , Nst A means “sentence A derivable in Th ∞# , Nst ”. Then, we have that ℜ∞ , Nst ∈ ℜ∞ , Nst iff ∞ , Nst ( ℜ∞ , Nst ∉ ℜ∞ , Nst ) , which immediately gives us. ℜ∞ , Nst ∈ ℜ∞ , Nst ℜ∞ , Nst ∉ ℜ∞ , Nst . We choose now ∞ , Nst A, i = 1, 2, in the following form ∞ , Nst A  ∃i  Bewi , Nst ( #A ) ∧  Bewi , Nst ( #A ) ⇒ A  .. iff. (121). Remark 3.5.6. Notice that definitions (117), (119) and (121) hold as definitions of a predicate really asserting provability in Th ∞# , Th ∞# , st and Th ∞# , Nst correspondingly. Remark 3.5.7. Of course all the theories Th ∞# , Th ∞# , st and Th ∞# , Nst are inconsistent, see subsection 4.1. Remark 3.5.8. Notice that under naive consideration the set ℑ∞ and ℜ∞ can be defined directly using a truth predicate, which of course is not available in the language of ZFC2Hs (but iff ZFC2Hs is consistent) by well-known Tarski’s undefinability theorem [10]. Theorem 3.5.1. Tarski’s undefinability theorem: I) Let ThL be first order theory with formal language L , which includes negation and has a Gödel numbering g (  ) such that for every L -formula A ( x ) there is a formula B such that B ↔ A ( g ( B ) ) holds. Assume that ThL has a standard model M stThL. and Con ( ThL ,st ) where. ThL , st  ThL + ∃M stThL .. (122). Let T ∗ be the set of Gödel numbers of L -sentences true in M stThL . Then there is no L -formula True ( n ) (truth predicate) which defines T ∗ . That is, there is no L -formula True ( n ) such that for every L -formula A, True ( g ( A ) ) ⇔ A. (123). holds. DOI: 10.4236/apm.2019.99034. 709. Advances in Pure Mathematics.

(26) J. Foukzon, E. Men’kova. II) Let Th Hs be second order theory with Henkin semantics and formal. language L , which includes negation and has a Gödel numbering g (  ) such that for every L -formula A ( x ) there is a formula B such that B ↔ A ( g ( B ) ) holds. and Con ( ThLHs , st ) , where. Assume that Th Hs has a standard model M stThL. Hs. Hs. (124). Hs Th,Hs + ∃M stThL st  Th. Let T ∗ be the set of Gödel numbers of the all L -sentences true in M. Then there is no L -formula True ( n ) (truth predicate) which defines T ∗ . That is, there is no L -formula True ( n ) such that for every L -formula A, True ( g ( A ) ) ⇔ A. (125). holds. Remark 3.5.9. Notice that the proof of Tarski’s undefinability theorem in this form is again by simple reductio ad absurdum. Suppose that an L -formula True(n) defines T ∗ . In particular, if A is a sentence of ThL then Th True ( g ( A ) ) holds in  iff A is true in M st L . Hence for all A, the Tarski T-sentence True ( g ( A ) ) ⇔ A is true in M stThL . But the diagonal lemma yields a counterexample to this equivalence, by giving a “Liar” sentence S such that S ⇔ ¬True ( g ( S ) ) holds in M stThL . Thus no L -formula True ( n ) can define T ∗ . Remark 3.5.10. Notice that the formal machinery of this proof is wholly elementary except for the diagonalization that the diagonal lemma requires. The proof of the diagonal lemma is likewise surprisingly simple; for example, it does not invoke recursive functions in any way. The proof does assume that every L -formula has a Gödel number, but the specifics of a coding method are not required. Remark 3.5.11. The undefinability theorem does not prevent truth in one consistent theory from being defined in a stronger theory. For example, the set of (codes for) formulas of first-order Peano arithmetic that are true in  is definable by a formula in second order arithmetic. Similarly, the set of true formulas of the standard model of second order arithmetic (or n-th order arithmetic for any n) can be defined by a formula in first-order ZFC. Remark1.3.5.12. Notice that Tarski’s undefinability theorem cannot blocking the biconditionals. ℜi ∈ ℜi ⇔ ℜi ∉ ℜi , i ∈  ,. (126). ℜ∞ ∈ ℜ∞ ⇔ ℜ∞ ∉ ℜ∞ , etc.,. see Remark 3.5.14 below. Remark 3.5.13. I) We define again the set ℑ∞ but now by using generalized truth predicate True∞# ( g ( A ) , A ) such that. True∞# ( g ( A ) , A ) ⇔ ∃i  Bewi ( #A ) ∧  Bewi ( #A ) ⇒ A  ⇔ True∞ ( g ( A ) ) ∧ True∞ ( g ( A ) ) ⇒ A ⇔ A, True∞ ( g ( A ) ) ⇔ ∃iBewi ( #A ) . DOI: 10.4236/apm.2019.99034. 710. (127). Advances in Pure Mathematics.

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