British Journal of Mathematics & Computer Science 9(5): 380-393, 2015, Article no.BJMCS.2015.210 ISSN: 2231-0851 SCIENCEDOMAIN international www.sciencedomain.org Inconsistent Countable Set in Second Order ZFC and Nonexistence of the Strongly Inaccessible Cardinals Jaykov Foukzon1∗ 1Center for Mathematical Sciences, Israel Institute of Technology, Haifa, Israel. Article Information DOI: 10.9734/BJMCS/2015/16849 Editor(s): (1) Wei Wu, Applied Mathematics Department, Dalian University of Technology, China. (2) Sheng Zhang, Department of Mathematics, Bohai University, Jinzhou, China. Reviewers: (1) Anonymous, Technical School Centre of Maribor, Slovenia. (2) Anonymous, COMSATS Institute of Information Technology, Pakistan. (3) Anonymous, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana. (4) Cenap zel, AB Department of Math Bolu, Turkey. Complete Peer review History: http://www.sciencedomain.org/review-history.php?iid=1146&id=6&aid=9622 Original Research Article Received: 16 February 2015 Accepted: 24 April 2015 Published: 06 June 2015 Abstract In this article we derived an important example of the inconsistent countable set in second order ZFC (ZFC2) with the full second-order semantics. Main results: (i) ¬Con(ZFC2), (ii) let k be an inaccessible cardinal and Hk is a set of all sets having hereditary size less then k, then ¬Con(ZFC + (V = Hk)). Keywords: Gödel encoding; Completion of ZFC2; Russell ′s paradox ; ω-model; Henkin semantics; full second-order semantics. 1 Introduction Let's 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 the Zermelo set theory, the first constructed axiomatic *Corresponding author:E-mail: jaykovfoukzon@list.ru; Foukzon; BJMCS, 9(5), 380-393, 2015; Article no.BJMCS.2015.210 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 not published paper [1]. However, it is deemed unlikely that even ZFC2 which is a very stronger than ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC2 were inconsistent, that fact would have been uncovered by now. This much is certain - 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. Note that in this paper we view the second order set theory ZFC2 under the Henkin semantics [2],[3] and under the full second-order semantics [4],[5].Thus we interpret the wff's of ZFC2 language with the full second-order semantics as required in [4],[5]. Designation 1.1. We will be denote by ZFCHs2 set theory ZFC2 with the Henkin semantics and we will be denote by ZFCfss2 set theory ZFC2 with the full second-order semantics. Remark 1.2.There is no completeness theorem for second-order logic with the full second-order 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 (standard or nonstandard) model MZFC fss 2 of ZFCfss2 [2]. Let Z be the conjunction of all the axioms of ZFC fss 2 . 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(ZFC fss 2 + ∃MZFC fss 2 ) =⇒ ¬Con(ZFC + (V = Hk)).In this paper we prove that ZFCfss2 is inconsistent. We will start from a simple naive consideration.Let = be the countable collection of all sets X such that ZFCfss2 ` ∃!XΨ (X) ,where Ψ (X) is any 1-place open wff i.e., ∀Y {Y ∈ = ↔ ∃Ψ (*) ∃!X [Ψ (X) ∧ Y = X]} . (1.1) Let X /∈` ZFC fss 2 Y be a predicate such that X /∈` ZFC fss 2 Y ↔ ZFCfss2 ` X /∈ Y.Let < be the countable collection of all sets such that ∀X [ X ∈ < ↔ X /∈` ZFC fss 2 X ] . (1.2) From (1.2) one obtain < ∈ < ↔ < /∈` ZFC fss 2 <. (1.3) But obviously this is a contradiction. However contradiction (1.3) it is not a contradiction inside ZFCfss2 for the reason that predicate X /∈` ZFC fss 2 Y not is a predicate of ZFCfss2 and therefore countable collections = and < not is a sets of ZFCfss2 . Nevertheless by using Gödel encoding the above stated contradiction can be shipped in special consistent completion of ZFCfss2 . Remark 1.3. We note that in order to deduce ¬Con(ZFCHs2 ) from Con(ZFCHs2 ) 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 [6].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 ZFC Hs 2 -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 381 Foukzon; BJMCS, 9(5), 380-393, 2015; Article no.BJMCS.2015.210 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 ZFCHs2 -proof of it. Remark 1.4. However assumption ∃MZFC Hs 2 st is not necessary. 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 for some linear order A [6],[7]. Thus one can to choose Gödel encoding inside M ZHs2 st . Remark 1.5. However there is no any problem as mentioned above in second order set theory ZFC2 with the full second-order semantics becouse corresponding second order arithmetic Z fss 2 is categorical. Remark 1.6. 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 omega 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 omega 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. 2 Derivation of the Inconsistent Countable Set in ZFCHs2 + ∃MZFC Hs 2 Let Th be some fixed, but unspecified, consistent formal theory. For later convenience, we assume that the encoding is done in some fixed formal second order theory S and that Th contains S.The sense in which S is contained in Th is better exemplified than explained: if S is a formal system of a second order arithmetic ZHs2 and Th is, say, ZFC Hs 2 , then Th contains S in the sense that there is a well-known embedding, or interpretation, of S in Th. Since encoding is to take place in S, it will have to have a large supply of constants and closed terms to be used as codes. (e.g. in formal arithmetic, one has 0, 1, ... .) S will also have certain function symbols to be described shortly.To each formula, Φ, of the language of Th is assigned a closed term, [Φ]c, called the code of Φ. We note that if Φ (x) is a formula with free variable x, then [Φ (x)]c is a closed term encoding the formula Φ (x) with x viewed as a syntactic object and not as a parameter. Corresponding to the logical connectives and quantifiers are function symbols, neg (*), imp (*), etc., such that, for all formulae Φ, Ψ : S ` neg ([Φ]c) = [¬Φ]c , S ` imp ([Φ]c , [Ψ]c) = [Φ→ Ψ]c etc. Of particular importance is the substitution operator, represented by the function symbol sub (*, *). For formulae Φ (x), terms t with codes [t]c : 382 Foukzon; BJMCS, 9(5), 380-393, 2015; Article no.BJMCS.2015.210 S ` sub ([Φ (x)]c , [t]c) = [Φ (t)]c . (2.1) It well known [8] that one can also encode derivations and have a binary relation ProvTh (x, y) (read "x proves y " or "x is a proof of y") such that for closed t1, t2 : S ` ProvTh (t1, t2) iff t1 is the code of a derivation in Th of the formula with code t2 . It follows that Th ` Φ iff S ` ProvTh (t, [Φ]c) (2.2) for some closed term t.Thus one can define PrTh (y)↔ ∃xProvTh (x, y) , (2.3) and therefore one obtain a predicate asserting provability. We note that is not always the case that [8]: Th ` Φ iff S ` PrTh ([Φ]c) . (2.4) It well known [8] that the above encoding can be carried out in such a way that the following important conditions D1,D2 and D3 are meet for all sentences [8]: D1.Th ` Φ implies S ` PrTh ([Φ]c) , D2.S ` PrTh ([Φ]c)→ PrTh ([PrTh ([Φ]c)]c) , D3.S ` PrTh ([Φ]c) ∧PrTh ([Φ→ Ψ]c)→ PrTh ([Ψ]c) . (2.5) Conditions D1,D2 and D3 are called the Derivability Conditions. Lemma 2.1. Assume that: (i) Con (Th) and (ii) Th ` PrTh ([Φ]c) , where Φ is a closed formula.Then Th 0 PrTh ([¬Φ]c) . Proof. LetConTh (Φ) be a formula{ ConTh (Φ) , ∀t1∀t2¬ [ProvTh (t1, [Φ]c) ∧ProvTh (t2, neg ([Φ]c))]↔ ¬∃t1¬∃t2 [ProvTh (t1, [Φ]c) ∧ProvTh (t2, neg ([Φ]c))] . (2.6) where t1, t2 is a closed term. We note that Th+Con (Th) ` ConTh (Φ) for any closed Φ. Suppose that Th ` PrTh ([¬Φ]c) ,then (ii) gives Th ` PrTh ([Φ]c) ∧PrTh ([¬Φ]c) . (2.7) From (2.3) and (2.7) we obtain ∃t1∃t2 [ProvTh (t1, [Φ]c) ∧ProvTh (t2, neg ([Φ]c))] . (2.8) But the formula (2.6) contradicts the formula (2.8). Therefore Th 0 PrTh ([¬Φ]c) . Lemma 2.2. Assume that : (i)Con(Th) and (ii) Th ` PrTh ([¬Φ]c) , where Φ is a closed formula.Then Th 0 PrTh ([Φ]c) . Assumption 2.1. Let Thbe an second order theory with the Henkin semantics. We assume now that : (i) the language of Th consists of: numerals 0,1,... countable set of the numerical variables: {v0, v1, ...} countable set of the set variables: = {x, y, z,X, Y, Z,<, ...} countable set of the n-ary function symbols: fn0 , f n 1 , ... countable set of the n-ary relation symbols: Rn0 , R n 1 , ... 383 Foukzon; BJMCS, 9(5), 380-393, 2015; Article no.BJMCS.2015.210 connectives: ¬,→ quantifier:∀. (ii) Th contains ZFC2, (iii) Th has an an ω-model MThω or (iv) Th has an nonstandard model MThNst. . Definition 2.1. An Th-wff Φ (well-formed formula Φ) is closed i.e. Φ is a sentence if it has no free variables; a wff is open if it has free variables.We'll use the slang 'k-place open wff ' to mean a wff with k distinct free variables. Definition 2.2.We said that,Th# is a nice theory or a nice extension of the Th iff (i) Th# contains Th;(ii) Let Φ be any closed formula, then Th ` PrTh ([Φ]c) implies Th# ` Φ. Definition 2.3.We said that,Th# is a maximally nice theory or a maximally nice extension of the Th iff Th# is consistent and for any consistent nice extension Th′ of the Th : Ded ( Th# ) j Ded (Th′) implies Ded ( Th# ) = Ded (Th′) . Remark 2.1. We note that a theory Th# depend on model MThω or M Th Nst., i.e. Th # = Th# [ MThω ] or Th# = Th# [ MThNst ] correspondingly. We will consider the case Th# , Th# [ MThω ] without loss of generality. Proposition 2.1. Assume that (i) Con (Th) and (ii ) Th has an ω-model MThω .Then theory Th can be extended to a maximally consistent nice theory Th# , Th# [ MThω ] . Proof. LetΦ1... Φi... be an enumeration of all wff's of the theory Th (this can be achieved if the set of propositional variables can be enumerated). Define a chain ℘ = {Thi|i ∈ N} ,Th1 = Th of consistent theories inductively as follows: assume that theory Thi is defined. (i) Suppose that a statement (2.9) is satisfied Thi ` PrThi ([Φi] c) and [Thi 0 Φi] ∧ [ MThω |= Φi ] . (2.9) Then we define a theory Thi+1 as follows Thi+1 , Thi ∪ {Φi} .Using Lemma 2.1 we will rewrite the condition (2.9) symbolically as follows{ Thi ` Pr#Thi ([Φi] c) , Pr#Thi ([Φi] c) ⇐⇒ PrThi ([Φi] c) ∧ [ MThω |= Φi ] . (2.10) (ii) Suppose that a statement (2.11) is satisfied Thi ` PrThi ([¬Φi] c) and [Thi 0 ¬Φi] ∧ [ MThω |= ¬Φi ] . (2.11) Then we define theory Thi+1 as follows: Thi+1 , Thi ∪ {¬Φi} . Using Lemma 2.2 we will rewrite the condition (2.11) symbolically as follows{ Thi ` Pr#Thi ([¬Φi] c) , Pr#Thi ([¬Φi] c) ⇐⇒ PrThi ([¬Φi] c) ∧ [ MThω |= ¬Φi ] . (2.12) (iii) Suppose that a statement (2.13) is satisfied Thi ` PrThi ([Φi] c) and Thi ` PrThi ([Φi] c) =⇒ Φi. (2.13) We will rewrite the condition (2.13) symbolically as follows 384 Foukzon; BJMCS, 9(5), 380-393, 2015; Article no.BJMCS.2015.210 { Thi ` Pr∗Thi ([Φi] c) Pr∗Thi ([Φi] c) ⇐⇒ PrThi ([Φi] c) ∧ [PrThi ([Φi] c) =⇒ Φi] (2.14) Then we define a theory Thi+1 as follows: Thi+1 , Thi. (iv) Suppose that a statement (2.15) is satisfied Thi ` PrThi ([¬Φi] c) and Thi ` PrThi ([¬Φi] c) =⇒ ¬Φi. (2.15) We will rewrite the condition (2.15) symbolically as follows{ Thi ` Pr∗Thi ([Φi] c) Pr∗Thi ([¬Φi] c) ⇐⇒ PrThi ([¬Φi] c) ∧ [PrThi ([¬Φi] c) =⇒ ¬Φi] (2.16) Then we define a theory Thi+1 as follows: Thi+1 , Thi.We define now a theory Th# as follows: Th# , ⋃ i∈N Thi. (2.17) First, notice that each Thi is consistent. This is done by induction on i and by Lemmas 2.1-2.2. By assumption, the case is true when i = 1.Now, suppose Thi is consistent. Then its deductive closure Ded (Thi) is also consistent. If a statement (2.14) is satisfied,i.e. Th ` PrTh ([Φi]c) and Th ` Φi, then clearly Thi+1 , Thi∪{Φi} is consistent since it is a subset of closure Ded (Thi) .If a statement (2.15) is satisfied,i.e. Th ` PrTh ([¬Φi]c) and Th ` ¬Φi, then clearly Thi+1 , Thi ∪ {¬Φi} is consistent since it is a subset of closure Ded (Thi) .Otherwise:(i) if a statement (2.9) is satisfied,i.e. Th ` PrTh ([Φi]c) and [Thi 0 Φi]∧ [ MThω |= Φi ] then clearly Thi+1 , Thi ∪{Φi} is consistent by Lemma 2.1 and by one of the standard properties of consistency: ∆∪ {A} is consistent iff ∆ 0 ¬A; (ii) if a statement (2.11) is satisfied,i.e. Thi ` PrThi ([¬Φi] c) and [Thi 0 ¬Φi]∧ [ MThω |= ¬Φi ] then clearly Thi+1 , Thi ∪ {¬Φi} is consistent by Lemma 2.2 and by one of the standard properties of consistency: ∆∪{¬A} is consistent iff ∆ 0 A.Next, notice Ded ( Th# ) is maximally consistent nice extension of the Ded (Th) .Ded ( Th# ) is consistent because, by the standard Lemma 2.3 belov, it is the union of a chain of consistent sets. To see that Ded ( Th# ) is maximal, pick any wff Φ. Then Φ is some Φi in the enumerated list of all wff's. Therefore for any Φ such that Thi ` PrThi ([Φ] c) or Thi ` PrThi ([¬Φ] c), either Φ ∈ Th# or ¬Φ ∈ Th#.Since Ded (Thi+1) j Ded ( Th# ) , we have Φ ∈ Ded ( Th# ) or ¬Φ ∈ Ded ( Th# ) ,which implies that Ded ( Th# ) is maximally consistent nice extension of the Ded (Th) . Lemma 2.3. The union of a chain ℘ = {Γi|i ∈ N} of consistent sets Γi, ordered by j, is consistent. Definition 2.4. We define now predicate PrTh# ([Φi] c) asserting provability in Th# : PrTh# ([Φi] c) ⇐⇒ [ Pr#Thi ([Φi] c) ] ∨ [ Pr∗Thi ([Φi] c) ] , PrTh# ([¬Φi] c) ⇐⇒ [ Pr#Thi ([¬Φi] c) ] ∨ [ Pr∗Thi ([¬Φi] c) ] . (2.18) Definition 2.5. Let Ψ = Ψ (x) be one-place open wff such that the conditions: (∗) Th ` ∃!xΨ [Ψ (xΨ)] or (∗∗) Th ` PrTh ([∃!xΨ [Ψ (xΨ)]]c) and MThω |= ∃!xΨ [Ψ (xΨ)] is satisfied. Then we said that, a set y is a Th#-set iff there is exist one-place open wff Ψ (x) such that y = xΨ. We write y [ Th# ] iff y is a Th#-set. Remark 2.2. Note that [(∗) ∨ (∗∗)] =⇒ Th# ` ∃!xΨ [Ψ (xΨ)] . 385 Foukzon; BJMCS, 9(5), 380-393, 2015; Article no.BJMCS.2015.210 Remark 2.3. Note that y [ Th# ] ⇐⇒ ∃Ψ [(y = xΨ) ∧PrTh# ([∃!xΨ [Ψ (xΨ)]] c)] Definition 2.6. Let = be a collection such that : ∀x [ x ∈ = ↔ x is a Th#-set ] . Proposition 2.2. Collection = is a Th#-set. Proof. Let us consider an one-place open wff Ψ (x) such that conditions (∗) or (∗∗) is satisfied, i.e. Th# ` ∃!xΨ [Ψ (xΨ)] .We note that there exists countable collection Ψ of the one-place open wff's Ψ = {Ψn (x)}n∈N such that: (i) Ψ (x) ∈Ψ and (ii) Th ` ∃!xΨ [[Ψ (xΨ)] ∧ {∀n (n ∈ N) [Ψ (xΨ)↔ Ψn (xΨ)]}] or Th ` ∃!xΨ [PrTh ([Ψ (xΨ)]c) ∧ {∀n (n ∈ N) PrTh ([Ψ (xΨ)↔ Ψn (xΨ)]c)}] and MThω |= ∃!xΨ [[Ψ (xΨ)] ∧ {∀n (n ∈ N) [Ψ (xΨ)↔ Ψn (xΨ)]}] (2.19) or of the equivalent form Th ` ∃!x1 [[Ψ1 (x1)] ∧ {∀n (n ∈ N) [Ψ1 (x1)↔ Ψn,1 (x1)]}] or Th ` ∃!xΨ [PrTh ([Ψ (x1)]c) ∧ {∀n (n ∈ N) PrTh ([Ψ (x1)↔ Ψn (x1)]c)}] and MThω |= ∃!xΨ [[Ψ (x1)] ∧ {∀n (n ∈ N) [Ψ (x1)↔ Ψn (x1)]}] (2.20) where we set Ψ (x) = Ψ1 (x1) ,Ψn (x1) = Ψn,1 (x1) and xΨ = x1. We note that any collection Ψk = {Ψn,k (x)}n∈N , k = 1, 2, ... such above defines an unique set xΨk ,i.e. Ψk1 ⋂ Ψk2 = ∅ iff xΨk1 6= xΨk2 .We note that collections Ψk , k = 1, 2, .. is no part of the ZFC2,i.e. collection Ψk there is no set in sense of ZFC2. However that is no problem, because by using Gödel numbering one can to replace any collection Ψk , k = 1, 2, .. by collection Θk = g (Ψk ) of the corresponding Gödel numbers such that Θk = g (Ψk ) = {g (Ψn,k (xk))}n∈N , k = 1, 2, ... . (2.21) It is easy to prove that any collection Θk = g (Ψk ) , k = 1, 2, .. is a Th #-set.This is done by Gödel encoding [8],[9] of the statament (2.19) by Proposition 2.1 and by axiom schema of separation [10]. Let gn,k = g (Ψn,k (xk)) , k = 1, 2, .. be a Gödel number of the wff Ψn,k (xk) .Therefore g (k) = {gn,k}n∈N , where we set k =Ψk , k = 1, 2, .. and ∀k1∀k2 [ {gn,k1}n∈N ⋂ {gn,k2}n∈N = ∅↔ xk1 6= xk2 ] . (2.22) Let { {gn,k}n∈N } k∈N be a family of the all sets {gn,k}n∈N . By axiom of choice [10] one obtain unique set =′ = {gk}k∈N such that ∀k [ gk ∈ {gn,k}n∈N ] .Finally one obtain a set = from a set =′ by axiom schema of replacement [10]. Thus one can define a Th#-set <c $ = : ∀x [x ∈ <c ↔ (x ∈ =) ∧PrTh# ([x /∈ x] c)] . (2.23) Proposition 2.3. Any collection Θk = g (Ψk ) , k = 1, 2, .. is a Th #-set. Proof. We define gn,k = g (Ψn,k (xk)) = [Ψn,k (xk)] c , vk = [xk] c . Therefore gn,k = g (Ψn,k (xk))↔ Fr (gn,k, vk) (see [9]). Let us define now predicate Π (gn,k, vk) { Π (gn,k, vk)↔ PrTh ([∃!xk [Ψ1,k (x1)]]c)∧ ∧∃!xk (vk = [xk]c) [∀n (n ∈ N) [PrTh ([[Ψ1,k (xk)]]c)↔ PrTh (Fr (gn,k, vk))]] . (2.24) 386 Foukzon; BJMCS, 9(5), 380-393, 2015; Article no.BJMCS.2015.210 We define now a set Θk such that{ Θk = Θ ′ k ∪ {gk} , ∀n (n ∈ N) [gn,k ∈ Θ′k ↔ Π (gn,k, vk)] (2.25) But obviously definitions (2.19) and (2.25) is equivalent by Proposition 2.1. Proposition 2.4. (i) Th# ` ∃<c,(ii) <c is a countable Th#-set. Proof.(i) Statement Th# ` ∃<c follows immediately by using statement ∃= and axiom schema of separation [4]. (ii) follows immediately from countability of a set =. Proposition 2.5. A set <c is inconsistent. Proof.From formla (2.18) one obtain Th# ` <c ∈ <c ↔ PrTh# ([<c /∈ <c] c) . (2.21) From formula (2.21) and Proposition 2.1 one obtain Th# ` <c ∈ <c ↔ <c /∈ <c (2.22) and therefore Th# ` (<c ∈ <c) ∧ (<c /∈ <c) . (2.23) But this is a contradiction. Proposition 2.6.Assume that (i) Con (Th) and (ii ) Th has an nonstandard model MThNst.Then theory Th can be extended to a maximally consistent nice theory Th# , Th# [ MThNst ] . Proof. Let Φ1... Φi... be an enumeration of all wff's of the theory Th (this can be achieved if the set of propositional variables can be enumerated). Define a chain ℘ = {Thi|i ∈ N} ,Th1 = Th of consistent theories inductively as follows: assume that theory Thi is defined. (i) Suppose that a statement (2.24) is satisfied Thi ` PrThi ([Φi] c) and [Thi 0 Φi] ∧ [ MThNst |= Φi ] . (2.24) Then we define a theory Thi+1 as follows Thi+1 , Thi ∪ {Φi} .Using Lemma 2.1 we will rewrite the condition (2.24) symbolically as follows{ Thi ` Pr#Thi ([Φi] c) , Pr#Thi ([Φi] c) ⇐⇒ PrThi ([Φi] c) ∧ [ MThNst |= Φi ] . (2.25) (ii) Suppose that a statement (2.26) is satisfied Thi ` PrThi ([¬Φi] c) and [Thi 0 ¬Φi] ∧ [ MThNst |= ¬Φi ] . (2.26) Then we define theory Thi+1 as follows: Thi+1 , Thi ∪ {¬Φi} . Using Lemma 2.2 we will rewrite the condition (2.26) symbolically as follows{ Thi ` Pr#Thi ([¬Φi] c) , Pr#Thi ([¬Φi] c) ⇐⇒ PrThi ([¬Φi] c) ∧ [ MThω |= ¬Φi ] . (2.27) (iii) Suppose that a statement (2.28) is satisfied Thi ` PrThi ([Φi] c) and Thi ` PrThi ([Φi] c) =⇒ Φi. (2.28) 387 Foukzon; BJMCS, 9(5), 380-393, 2015; Article no.BJMCS.2015.210 We will rewrite the condition (2.28) symbolically as follows{ Thi ` Pr∗Thi ([Φi] c) , Pr∗Thi ([Φi] c) ⇐⇒ PrThi ([Φi] c) ∧ [PrThi ([Φi] c) =⇒ Φi] (2.29) Then we define a theory Thi+1 as follows: Thi+1 , Thi. (iv) Suppose that a statement (2.30) is satisfied Thi ` PrThi ([¬Φi] c) and Thi ` PrThi ([¬Φi] c) =⇒ ¬Φi. (2.30) We will rewrite the condition (2.30) symbolically as follows{ Thi ` Pr∗Thi ([Φi] c) , Pr∗Thi ([¬Φi] c) ⇐⇒ PrThi ([¬Φi] c) ∧ [PrThi ([¬Φi] c) =⇒ ¬Φi] (2.31) Then we define a theory Thi+1 as follows: Thi+1 , Thi.We define now a theory Th# as follows: Th# , ⋃ i∈N Thi. (2.32) First, notice that each Thi is consistent. This is done by induction on i and by Lemmas 2.1-2.2. By assumption, the case is true when i = 1.Now, suppose Thi is consistent. Then its deductive closure Ded (Thi) is also consistent. If a statement (2.28) is satisfied,i.e. Th ` PrTh ([Φi]c) and Th ` Φi, then clearly Thi+1 , Thi∪{Φi} is consistent since it is a subset of closure Ded (Thi) .If a statement (2.30) is satisfied,i.e. Th ` PrTh ([¬Φi]c) and Th ` ¬Φi, then clearly Thi+1 , Thi ∪ {¬Φi} is consistent since it is a subset of closure Ded (Thi) .Otherwise:(i) if a statement (2.24) is satisfied,i.e. Th ` PrTh ([Φi]c) and [Thi 0 Φi]∧ [ MThω |= Φi ] then clearly Thi+1 , Thi ∪{Φi} is consistent by Lemma 2.1 and by one of the standard properties of consistency: ∆∪ {A} is consistent iff ∆ 0 ¬A; (ii) if a statement (2.26) is satisfied,i.e. Thi ` PrThi ([¬Φi] c) and [Thi 0 ¬Φi]∧ [ MThω |= ¬Φi ] then clearly Thi+1 , Thi ∪ {¬Φi} is consistent by Lemma 2.2 and by one of the standard properties of consistency: ∆∪{¬A} is consistent iff ∆ 0 A.Next, notice Ded ( Th# ) is maximally consistent nice extension of the Ded (Th) .Ded ( Th# ) is consistent because, by the standard Lemma 2.3 belov, it is the union of a chain of consistent sets. To see that Ded ( Th# ) is maximal, pick any wff Φ. Then Φ is some Φi in the enumerated list of all wff's. Therefore for any Φ such that Thi ` PrThi ([Φ] c) or Thi ` PrThi ([¬Φ] c), either Φ ∈ Th# or ¬Φ ∈ Th#.Since Ded (Thi+1) j Ded ( Th# ) , we have Φ ∈ Ded ( Th# ) or ¬Φ ∈ Ded ( Th# ) ,which implies that Ded ( Th# ) is maximally consistent nice extension of the Ded (Th) . Definition 2.7. We define now predicate PrTh# ([Φi] c) asserting provability in Th# : PrTh# ([Φi] c) ⇐⇒ [ Pr#Thi ([Φi] c) ] ∨ [ Pr∗Thi ([Φi] c) ] , PrTh# ([¬Φi] c) ⇐⇒ [ Pr#Thi ([¬Φi] c) ] ∨ [ Pr∗Thi ([¬Φi] c) ] . (2.33) Definition 2.8. Let Ψ = Ψ (x) be one-place open wff such that the conditions: (∗) Th ` ∃!xΨ [Ψ (xΨ)] or (∗∗) Th ` PrTh ([∃!xΨ [Ψ (xΨ)]]c) and MThNst |= ∃!xΨ [Ψ (xΨ)] is satisfied. Then we said that, a set y is a Th#-set iff there is exist one-place open wff Ψ (x) such that y = xΨ. We write y [ Th# ] iff y is a Th#-set. Remark 2.4. Note that [(∗) ∨ (∗∗)] =⇒ Th# ` ∃!xΨ [Ψ (xΨ)] . 388 Foukzon; BJMCS, 9(5), 380-393, 2015; Article no.BJMCS.2015.210 Remark 2.5. Note that y [ Th# ] ⇐⇒ ∃Ψ [(y = xΨ) ∧PrTh# ([∃!xΨ [Ψ (xΨ)]] c)] Definition 2.9. Let = be a collection such that : ∀x [ x ∈ = ↔ x is a Th#-set ] . Proposition 2.7. Collection = is a Th#-set. Proof. Let us consider an one-place open wff Ψ (x) such that conditions (∗) or (∗∗) is satisfied, i.e. Th# ` ∃!xΨ [Ψ (xΨ)] .We note that there exists countable collection Ψ of the one-place open wff's Ψ = {Ψn (x)}n∈N such that: (i) Ψ (x) ∈Ψ and (ii) Th ` ∃!xΨ [ [Ψ (xΨ)] ∧ { ∀n ( n ∈MZ Hs 2 st ) [Ψ (xΨ)↔ Ψn (xΨ)] }] or Th ` ∃!xΨ [ PrTh ([Ψ (xΨ)] c) ∧ { ∀n ( n ∈MZ Hs 2 st ) PrTh ([Ψ (xΨ)↔ Ψn (xΨ)]c) }] and MThNst |= ∃!xΨ [ [Ψ (xΨ)] ∧ { ∀n ( n ∈MZ Hs 2 st ) [Ψ (xΨ)↔ Ψn (xΨ)] }] (2.34) or of the equivalent form  Th ` ∃!x1 [ [Ψ1 (x1)] ∧ { ∀n ( n ∈MZ Hs 2 st ) [Ψ1 (x1)↔ Ψn,1 (x1)] }] or Th ` ∃!xΨ [ PrTh ([Ψ (x1)] c) ∧ { ∀n ( n ∈MZ Hs 2 st ) PrTh ([Ψ (x1)↔ Ψn (x1)]c) }] and MThNst |= ∃!xΨ [ [Ψ (x1)] ∧ { ∀n ( n ∈MZ Hs 2 st ) [Ψ (x1)↔ Ψn (x1)] }] (2.35) where we set Ψ (x) = Ψ1 (x1) ,Ψn (x1) = Ψn,1 (x1) and xΨ = x1. We note that any collection Ψk = {Ψn,k (x)}n∈N , k = 1, 2, ... such above defines an unique set xΨk ,i.e. Ψk1 ⋂ Ψk2 = ∅ iff xΨk1 6= xΨk2 .We note that collections Ψk , k = 1, 2, .. is no part of the ZFC Hs 2 ,i.e. collection Ψk there is no set in sense of ZFCHs2 . However that is no problem, because by using Gödel numbering one can to replace any collection Ψk , k = 1, 2, .. by collection Θk = g (Ψk ) of the corresponding Gödel numbers such that Θk = g (Ψk ) = {g (Ψn,k (xk))}n∈N , k = 1, 2, ... . (2.36) It is easy to prove that any collection Θk = g (Ψk ) , k = 1, 2, .. is a Th #-set. This is done by Gödel encoding [8],[9] of the statament (2.19) by Proposition 2.6 and by axiom schema of separation [4]. Let gn,k = g (Ψn,k (xk)) , k = 1, 2, .. be a Gödel number of the wff Ψn,k (xk) .Therefore g (k) = {gn,k}n∈N , where we set k =Ψk , k = 1, 2, .. and ∀k1∀k2 [ {gn,k1}n∈N ⋂ {gn,k2}n∈N = ∅↔ xk1 6= xk2 ] . (2.37) Let { {gn,k}n∈N } k∈N be a family of the all sets {gn,k}n∈N . By axiom of choice [10] one obtain unique set =′ = {gk}k∈N such that ∀k [ gk ∈ {gn,k}n∈N ] . F inallyoneobtainaset= from a set =′ by axiom schema of replacement [10].Thus one can define a Th#-set <c $ = : ∀x [x ∈ <c ↔ (x ∈ =) ∧PrTh# ([x /∈ x] c)] . (2.38) 389 Foukzon; BJMCS, 9(5), 380-393, 2015; Article no.BJMCS.2015.210 Proposition 2.8. Any collection Θk = g (Ψk ) , k = 1, 2, .. is a Th #-set. Proof. We define gn,k = g (Ψn,k (xk)) = [Ψn,k (xk)] c , vk = [xk] c . Therefore gn,k = g (Ψn,k (xk))↔ Fr (gn,k, vk) (see [9]). Let us define now predicate Π (gn,k, vk){ Π (gn,k, vk)↔ PrTh ([∃!xk [Ψ1,k (x1)]]c)∧ ∧∃!xk (vk = [xk]c) [ ∀n ( n ∈MZ Hs 2 st ) [PrTh ([[Ψ1,k (xk)]] c)↔ PrTh (Fr (gn,k, vk))] ] . (2.39) We define now a set Θk such that{ Θk = Θ ′ k ∪ {gk} , ∀n (n ∈ N) [gn,k ∈ Θ′k ↔ Π (gn,k, vk)] (2.40) But obviously definitions (2.39) and (2.40) is equivalent by Proposition 2.6. Proposition 2.9. (i) Th# ` ∃<c,(ii) <c is a countable Th#-set. Proof.(i) Statement Th# ` ∃<c follows immediately by using statement ∃= and axiom schema of separation [10]. (ii) follows immediately from countability of a set =. Proposition 2.10. A set <c is inconsistent. Proof.From formla (2.18) one obtain Th# ` <c ∈ <c ↔ PrTh# ([<c /∈ <c] c) . (2.41) From formula (2.41) and Proposition 2.6 one obtain Th# ` <c ∈ <c ↔ <c /∈ <c (2.42) and therefore Th# ` (<c ∈ <c) ∧ (<c /∈ <c) . (2.43) But this is a contradiction. 3 Derivation of the Inconsistent Countable Set in ZFC2 with the Full Semantics Let Th be an second order theory with the full second order semantics.We assume now that: (i) Th contains ZFCfss2 ,(ii) Th has no any model. Definition 3.1. Using formula (2.3) one can define predicate PrωTh (y) really asserting provability in ZFCfss2 PrωTh (y)↔ ∃x ( x ∈MZ2ω ) ProvTh (x, y) , (3.1) Theorem 3.1.[11].(Löb's Theorem for ZFC2) Let Φ be any closed formula with code y = [Φ]c ∈MZ2ω , then Th ` PrωTh ([Φ]c) implies Th ` Φ (see [12] Theorem 5.1). Proof. Assume that (#) Th ` PrωTh ([Φ]c) . 390 Foukzon; BJMCS, 9(5), 380-393, 2015; Article no.BJMCS.2015.210 Note that (1) Th 0 ¬Φ. Otherwise one obtain Th ` PrωTh ([¬Φ]c) ∧PrωTh ([Φ]c) , but this is a contradiction. (2) Assume now that (2.i) Th ` PrωTh ([Φ]c) and (2.ii) Th 0 Φ. From (1) and (2.ii) follows that (3) Th 0 ¬Φ and Th 0 Φ. Let Th¬Φ be a theory (4)Th¬Φ , Th∪{¬Φ} .From (3) follows that (5) Con (Th¬Φ) . From (4) and (5) follows that (6) Th¬Φ ` PrωTh¬Φ ([¬Φ] c) . From (4) and (#) follows that (7) Th¬Φ ` PrωTh¬Φ ([Φ] c) . From (6) and (7) follows that (8) Th¬Φ ` PrωTh¬Φ ([Φ] c) ∧PrωTh¬Φ ([¬Φ] c) ,but this is a contradiction. Definition 3.2. Let Ψ = Ψ (x) be one-place open wff such that the conditions: (∗) Th ` ∃!xΨ [Ψ (xΨ)] or (∗∗) Th ` PrωTh ([∃!xΨ [Ψ (xΨ)]]c) is satisfied. Then we said that, a set y is a Th-set iff there is exist one-place open wff Ψ (x) such that y = xΨ. We write y [Th] iff y is a Th-set. Remark 3.1. Note that [(∗) ∨ (∗∗)] =⇒ Th ` ∃!xΨ [Ψ (xΨ)] . Remark 3.2. Note that y [Th] ⇐⇒ ∃Ψ [(y = xΨ) ∧PrωTh ([∃!xΨ [Ψ (xΨ)]]c)] Definition 3.3. Let = be a collection such that : ∀x [x ∈ = ↔ x is a Th-set] . Proposition 3.2. Collection = is a Th-set. Definition 3.4. We define now a Th-set <c $ = : ∀x [x ∈ <c ↔ (x ∈ =) ∧PrωTh ([x /∈ x]c)] . (3.2) Proposition 3.3. (i) Th ` ∃<c,(ii) <c is a countable Th-set. Proof.(i) Statement Th ` ∃<c follows immediately by using statement ∃= and axiom schema of separation [4]. (ii) follows immediately from countability of a set =. Proposition 3.4. A set <c is inconsistent. Proof.From formla (3.2) one obtain Th ` <c ∈ <c ↔ PrωTh ([<c /∈ <c]c) . (3.3) From formula (3.3) and definition 3.1 one obtain Th ` <c ∈ <c ↔ <c /∈ <c (3.4) and therefore Th ` (<c ∈ <c) ∧ (<c /∈ <c) . (3.5) 391 Foukzon; BJMCS, 9(5), 380-393, 2015; Article no.BJMCS.2015.210 But this is a contradiction. Therefore finally we obtain: Theorem 3.2. [12].¬Con(ZFC2). That is well known that under ZFC it can be shown that k is inaccessible if and only if (Vk,∈) is a model of ZFC2 [5],[11].Thus finally we obtain. Theorem 3.3. [12].¬Con(ZFC + (V = Hk)). 4 Conclusion In this paper we have proved that the second order ZFC with the full second-order semantic is inconsistent,i.e. ¬Con(ZFC2).Main result is: let k be an inaccessible cardinal and Hk is a set of all sets having hereditary size less then k, then ¬Con(ZFC + (V = Hk)).This result also was obtained in [7],[11],[12] by using essentially another approach. For the first time this result has been declared to AMS in [13],[14]. An important applications in topology and homotopy theory are obtained in [15],[16],[17]. Acknowledgments A reviewers provided important clarifications. Competing Interests The author declares that no competing interests exist. References [1] Nelson E. Warning signs of a possible collapse of contemporary mathematics. 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Handbook of mathematical logic. Edited by J. Barwise. North-Holland Publishing Company; 1977. [9] Mendelson E. Introduction to mathematical logic. June 1, 1997. ISBN-10: 0412808307. ISBN13: 978-0412808302. [10] Takeuti G, Zaring WM. Introduction to axiomatic set theory. Springer-Verlag; 1971. [11] Foukzon J. Generalized Lob's theorem. Strong reflection principles and large cardinal axioms. Consistency Results in Topology. Available: http://arxiv.org/abs/1301.5340 [12] Foukzon J. Inconsistent Countable Set. Available: http://vixra.org/abs/1302.0048 [13] Foukzon J, Men'kova ER. Generalized Löb's Theorem. strong reflection principles and large cardinal axioms. Advances in Pure Mathematics. 2013;3(3). [14] Foukzon J. Strong reflection principles and large cardinal axioms. Fall Southeastern Sectional Meeting University of Louisville, Louisville, KY October 5-6, 2013 (Saturday -Sunday) Meeting #1092. Available:http://www.ams.org/amsmtgs/2208 abstracts/1092-03-13.pdf [15] Foukzon J. Consistency results in topology and homotopy theory. Pure and Applied Mathematics Journal. 2015;4(1-1):1-5. Published online October 29, 2014. DOI: 10.11648/j.pamj.s.2015040101.11. ISSN: 2326-9790 (Print); ISSN: 2326-9812 (Online). [16] Foukzon J. Generalized lob's theorem. Strong reflection. principles and large cardinal axioms. Consistency Results in Topology, IX Iberoamerican Conference on Topology and its Applications 24-27 June, Almeria, Spain. Book of abstracts. 2014;66. [17] Foukzon J. Generalized lob's theorem. Strong reflection principles and large cardinal axioms. Consistency Results in Topology, International Conference on Topology and its Applications, July 3-7, Nafpaktos, Greece. Book of abstracts. 2014;81. ----------------------------------------------– c©2015 Foukzon; 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. Peer-review history: The peer review history for this paper can be accessed here (Please copy paste the total link in your browser address bar) www.sciencedomain.org/review-history.php?iid=1146&id=6&aid=9622