Inconsistent countable set in second order ZFC and not existence of the strongly inaccessible cardinals

In this article we derived an importent example of the inconsistent countable set in second order ZFC ZFC2 with the full second-order semantic. Main results is:(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.


1.Introduction.
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 the Zermelo set theory, the first constructed axiomatic set theory.Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo-Fraenkel set theory ZFC."But how do we know that ZFC is a consistent theory, free of contradictions?The short answer is that we don't; it is a matter of faith (or of skepticism)"-E.Nelson wrote in his not published paper [1].However, it is deemed unlikely that even ZFC 2 which is a very stronger than ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC 2 were inconsistent, that fact would have been uncovered by now.This much is certain -ZFC 2 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 ZFC 2 under the Henkin semantics [2], [3] and under the full second-order semantics [4], [5].Thus we interpret the wff's of ZFC 2 language with the full second-order semantics as required in [4], [5].
Designation 1.1.We will be denote by ZFC 2 Hs set theory ZFC 2 with the Henkin semantics and we will be denote by ZFC 2 fss set theory ZFC 2 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 ZFC 2 fss imply a reflection principle which ensures that if a sentence Z of second-order set theory is true, then it is true in some (standard or nonstandard) model M ZFC 2 fss of ZFC 2 fss [2].Let Z be the conjunction of all the axioms of ZFC 2 fss .We assume now that: Z is true,i.e.Hs is equal to ℕ  A  ℤ for some linear order A [6], [7].Thus one can to choose Gödel encoding inside M st Z 2
Remark 1.5.However there is no any problem as mentioned above in second order set theory ZFC 2 with the full second-order semantics becouse corresponding second order arithmetic Z 2 fss is categorical.Remark 1.6.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 powerset 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 M Z 2 fss , which is the usual set of natural numbers with its usual structure and all its subsets, is called the intended or standard model of second-order arithmetic.
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 Z 2 Hs and Th is, say, ZFC 2 Hs , 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, , of the language of Th is assigned a closed term,  c , called the code of .We note that if x is a formula with free variable x, then x c is a closed term encoding the formula x with x viewed as a syntactic object and not as a parameter.Corresponding to the logical connectives and quantifiers are function symbols, neg, imp, etc., such that, for all formulae ,  : Of particular importance is the substitution operator, represented by the function symbol sub, .For formulae x, terms t with codes t c : It well known [8] that one can also encode derivations and have a binary relation Prov Th x, y (read "x proves y " or "x is a proof of y") such that for closed t 1 , t 2 : S  Prov Th t 1 , t 2  iff t 1 is the code of a derivation in Th of the formula with code t 2 .It follows that for some closed term t.Thus one can define Pr Th y ↔ ∃xProv Th x, y, 2.3 and therefore one obtain a predicate asserting provability.We note that is not always the case that [8]: 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]: 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 : Remark 2.1.We note that a theory Th # depend on model M  Th or M Nst.Th , i.e.Th #  Th # M  Th  or Th #  Th # M Nst Th  correspondingly.We will consider the case Th #  Th # M  Th  without loss of generality.Proposition 2.1.Assume that (i) ConTh and (ii ) Th has an -model M  Th .Then theory Th can be extended to a maximally consistent nice theory Th #  Th # M  Th .Proof.Let  1 . . . i . . .be an enumeration of all wff's of the theory Th (this can be achieved if the set of propositional variables can be enumerated).Define a chain ℘  Th i |i ∈ ℕ, Th 1  Th of consistent theories inductively as follows: assume that theory Th i is defined.(i) Suppose that a statement (2.9) is satisfied Then we define a theory Th i1 as follows Th i1  Th i   i .Using Lemma 2.1 we will rewrite the condition (2.9) symbolically as follows (ii) Suppose that a statement (2.11) is satisfied Then we define theory Th i1 as follows: Th i1  Th i   i .Using Lemma 2.2 we will rewrite the condition (2.11) symbolically as follows (iii) Suppose that a statement (2.13) is satisfied We will rewrite the condition (2.13) symbolically as follows Then we define a theory Th i1 as follows: Th i1  Th i .
(iv) Suppose that a statement (2.15) is satisfied We will rewrite the condition (2.15) symbolically as follows Then we define a theory Th i1 as follows: Th i1  Th i .We define now a theory Th # as follows: Definition 2.5.Let   x be one-place open wff such that the conditions: Let us consider an one-place open wff x such that conditions ( * ) or ( * * ) is satisfied, i.e.Th #  ∃!x  x  .We note that there exists countable collection ℱ  of the one-place open wff's ℱ    n x n∈ℕ such that: (i) x ∈ ℱ  and (ii) or of the equivalent form where we set x   1 x 1 ,  n x 1    n,1 x 1  and x   x 1 .We note that any collection ℱ  k   n,k x n∈ℕ , k  1, 2, . . .such above defines an unique set x  k , i.e.
. is no part of the ZFC 2 , i.e. collection ℱ  k there is no set in sense of ZFC 2 .However that is no problem, because by using Gödel numbering one can to replace any collection It is easy to prove that any collection . is a Th # -set.This is done by Gödel encoding [8], [10] of the statament (2.19) by Proposition 2.1 and by axiom schema of separation [9].Let

2. 22
Let g n,k  n∈ℕ  k∈ℕ be a family of the all sets g n,k  n∈ℕ .By axiom of choice [9] one obtain unique set ℑ ′  g k  k∈ℕ such that ∀kg k ∈ g n,k  n∈ℕ .Finally one obtain a set ℑ from a set ℑ ′ by axiom schema of replacement [9].Thus one can define a Th # -set We define now a set Θ k such that But obviously definitions (2.19) and (2.25) is equivalent by Proposition 2.1.
Proof.(i) Statement Th #  ∃ c follows immediately by using statement ∃ℑ and axiom schema of separation [4].(ii) follows immediately from countability of a set ℑ.
Proposition 2.5.A set  c is inconsistent.Proof.From formla (2.18) one obtain From formula (2.21) and Proposition 2.1 one obtain and therefore But this is a contradiction.Proposition 2.6.Assume that (i) ConTh and (ii ) Th has an -model M Nst Th .Then theory Th can be extended to a maximally consistent nice theory Th #  Th # M Nst Th .Proof.Let  1 . . . i . . .be an enumeration of all wff's of the theory Th (this can be achieved if the set of propositional variables can be enumerated).Define a chain ℘  Th i |i ∈ ℕ, Th 1  Th of consistent theories inductively as follows: assume that theory Th i is defined.(i) Suppose that a statement (2.24) is satisfied Then we define a theory Th i1 as follows Th i1  Th i   i .Using Lemma 2.1 we will rewrite the condition (2.24) symbolically as follows (ii) Suppose that a statement (2.26) is satisfied Then we define theory Th i1 as follows: Th i1  Th i   i .Using Lemma 2.2 we will rewrite the condition (2.26) symbolically as follows (iii) Suppose that a statement (2.28) is satisfied We will rewrite the condition (2.28) symbolically as follows Then we define a theory Th i1 as follows: Th i1  Th i .
(iv) Suppose that a statement (2.30) is satisfied We will rewrite the condition (2.30) symbolically as follows Then we define a theory Th i1 as follows: Th i1  Th i .We define now a theory Th # as follows: 2 and by one of the standard properties of consistency: Δ  A is consistent iff Δ  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 .Then  is some  i in the enumerated list of all wff's.Therefore for any  such that Th i  Pr Th i  c  or Th i  Pr Th i  c , either  ∈ Th # or  ∈ Th # .Since DedTh i1   DedTh # , we have  ∈ DedTh #  or  ∈ DedTh # , which implies that DedTh #  is maximally consistent nice extension of the DedTh.Definition 2.7.We define now predicate Pr Th #  i  c  asserting provability in Th # : where we set x   1 x 1 ,  n x 1    n,1 x 1  and x   x 1 .We note that any collection ℱ  k   n,k x n∈ℕ , k  1, 2, . . .such above defines an unique set x  k , i.e.
We note that collections ℱ  k , k  1, 2, . . is no part of the ZFC 2 Hs , i.e. collection ℱ  k there is no set in sense of ZFC 2 Hs .However that is no problem, because by using Gödel numbering one can to replace any collection ℱ  k , k  1, 2, . .by collection Θ k  gℱ  k  of the corresponding Gödel numbers such that It is easy to prove that any collection Θ k  gℱ  k , k  1, 2, . . is a Th # -set.This is done by Gödel encoding [8], [10]

2. 37
Let g n,k  n∈ℕ  k∈ℕ be a family of the all sets g n,k  n∈ℕ .By axiom of choice [9] one obtain unique set ℑ ′  g k  k∈ℕ such that ∀kg k ∈ g n,k  n∈ℕ .Finally one obtain a set ℑ from a set ℑ ′ by axiom schema of replacement [9].Thus one can define a Th # -set that the mere consistency of ZFC 2 isn't quite enough to get us to believe arithmetical theorems of ZFC 2Hs ; we must also believe that these arithmetical theorems are asserting something about the standard naturals.It is "conceivable" that ZFC 2Hs might be consistent but that the only nonstandard models 2 fss .It is known that the existence of a model for Z requires the existence of strongly inaccessible cardinals, i.e. under ZFC it can be shown that κ is a strongly inaccessible if and only if H κ , ∈ is a model of ZFC 2 fss .Thus ConZFC 2 fss  ∃M ZFC 2 fss   ConZFC  V  H k .In this paper we prove that ZFC 2 fss is inconsistent.We will start from a simple naive consideration.Let ℑ be the countable collection of all sets X such that ZFC 2 fss  ∃!XX, where X is any 1-place open wff i.e., ∀YY ∈ ℑ ↔ ∃∃!XX ∧ Y  X.1.1 Let X ∉  ZFC 2 fss Y be a predicate such that X ∉  ZFC 2 fss Y ↔ ZFC 2 fss  X ∉ Y. Let  be the countable collection of all sets such that ∀X X ∈  ↔ X ∉  ZFC 2 e.g., that ZFC 2 Hs has an omega-model M ZFC 2 Hs or an standard model M st ZFC 2 Hs 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 ZFC 2 Hs -proof of it?It's clear that if ZFC 2 Hs is inconsistent, then we won't believe ZFC 2 Hs -proofs.What's slightly more subtle is Pr Th  c , where  is a closed formula.Then Th  Pr Th  c .Proof.Let Con Th  be a formula Con Th   ∀t 1 ∀t 2 Prov Th t 1 ,  c  ∧ Prov Th t 2 , neg c ↔ ∃t 1 ∃t 2 Prov Th t 1 ,  c  ∧ Prov Th t 2 , neg c .From (2.3) and (2.7) we obtain ∃t 1 ∃t 2 Prov Th t 1 ,  c  ∧ Prov Th t 2 , neg c . 2.8 But the formula (2.6) contradicts the formula (2.8).Therefore Th  Pr Th  c . Lemma 2.2.Assume that: (i) ConTh and (ii) Th  Pr Th  c , where  is a closed formula.Then Th  Pr Th  c . Assumption 2.1.Let Th be 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: v 0 , v 1 , . . . countable set ℱ of the set variables: ℱ  x, y, z, X, Y, Z, , . . . countable set of the n-ary function symbols: f 0 Th contains ZFC 2 , (iii) Th has an an -model M  Th or (iv) Th has an nonstandard model M Nst.
D1, D2 and D3 are called the Derivability Conditions.Lemma 2.1.Assume that: (i) ConTh and (ii) Th  2.6 where t 1 , t 2 is a closed term.We note that Th ConTh  Con Th  for any closed .Suppose that Th  Pr Th  c , then (ii) gives Th  Pr Th  c  ∧ Pr Th  c . 2.7 Th .Definition 2.1.An Th-wff  (well-formed formula ) is closed -i.e. is a sentence -if it has no free variables; a wff is open if it has free variables.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  be any closed formula, then Th  Pr Th  c  implies Th #  .
notice that each Th i is consistent.This is done by induction on i and by Lemmas 2.1-2.2.By assumption, the case is true when i  1.Now, suppose Th i is consistent.Then its deductive closure DedTh i  is also consistent.If a statement (2.14) is satisfied,i.e.Th  Pr Th  i  c  and Th   i , then clearlyTh i1  Th i   i  is consistent since it is a subset of closure DedTh i .If a statement (2.15) is satisfied,i.e.Th  Pr Th  i  c  and Th   i , then clearly Th i1  Th i   i  is consistent since it is a subset of closure DedTh Lemma 2.2 and by one of the standard properties of consistency: Δ  A is consistent iff Δ  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 .Then  is some  i in the enumerated list of all wff's.Therefore for any  such thatTh i  Pr Th i  c  or Th i  Pr Th i  c , either  ∈ Th # or  ∈ Th # .Since DedTh i1   DedTh # , we have  ∈ DedTh #  or  ∈ DedTh # ,which implies that DedTh #  is maximally consistent nice extension of the DedTh.Lemma 2.3.The union of a chain ℘  Γ i |i ∈ ℕ of consistent sets Γ i , ordered by , is consistent.Definition 2.4.We define now predicate Pr Th #  i  c  asserting provability in Th i .Otherwise:(i) if a statement (2.9) is satisfied,i.e.Th  Pr Th  i  c  andTh i   i  ∧ M  Th   i  then clearly Th i1  Th i   i  is consistent by Lemma 2.1 and by one of the standard properties of consistency:Δ  A is consistent iff Δ  A; (ii) if a statement (2.11) is satisfied,i.e.Th i  Pr Th i  i  c  and Th i   i  ∧ M  Th   i  then clearly Th i1  Th i   i  is consistent by # : notice that each Th i is consistent.This is done by induction on i and by Lemmas 2.1-2.2.By assumption, the case is true when i  1.Now, suppose Th i is consistent.Then its deductive closureDedTh i  is also consistent.If a statement (2.28) is satisfied,i.e.Th  Pr Th  i  c  and Th   i , then clearly Th i1  Th i   i  is consistent since it is a subset of closure DedTh i .If a statement (2.30) is satisfied,i.e.Th  Pr Th  i  c  and Th   i , then clearly Th i1  Th i   i  is consistent since it is a subset of closure DedTh i .Otherwise:(i) if a statement (2.24) is satisfied,i.e.Th  Pr Th  i  c  and Th i   i  ∧ M  Th   i  then clearly Th i1  Th i   i is consistent by Lemma 2.1 and by one of the standard properties of consistency: Δ  A is consistent iff Δ  A; (ii) if a statement (2.26) is satisfied,i.e.Th i  Pr Th i 2.8.Let   x be one-place open wff such that the conditions: *  Th  ∃!x  x   or  * *  Th  Pr Th ∃!x  x   c  and M Nst Th  ∃!x  x   is satisfied.Then we said that, a set y is a Th # -set iff there is exist one-place open wff x such that y  x  .We write yTh #  iff y is a Th # -set.Remark 2.4.Note that  *  ∨  * *   Th #  ∃!x  x  .Let us consider an one-place open wff x such that conditions ( * ) or ( * * ) is satisfied, i.e.Th #  ∃!x  x  .We note that there exists countable collection ℱ  of the one-place open wff's ℱ    n x n∈ℕ such that: (i) x ∈ ℱ  and (ii)Th  ∃!x  x   ∧ ∀n n ∈ M st Hs x   ↔  n x   or Th  ∃!x  Pr Th x   c  ∧ ∀n n ∈ M st Hs Pr Th x   ↔  n x   c  and M Nst Th  ∃!x  x   ∧ ∀n n ∈ M st Th  ∃!x 1  1 x 1  ∧ ∀n n ∈ M st Hs  1 x 1  ↔  n,1 x 1  or Th  ∃!x  Pr Th x 1  c  ∧ ∀n n ∈ M st Remark 2.5.Note that yTh #   ∃y  x   ∧ Pr Th # ∃!x  x   c  Definition 2.9.Let ℑ be a collection such that : ∀x x ∈ ℑ ↔ x is a Th # -set .Proposition 2.7.Collection ℑ is a Th # -set.Proof.HsPr Th x 1  ↔  n x 1  c  and M Nst Th  ∃!x  x 1  ∧ ∀n n ∈ M st Z 2 Hs x 1  ↔  n x 1  2.35 [4]the statament(2.19)byProposition2.6 and by axiom schema of separation[4].Let g n,k  g n,k x k , k  1, 2, . .be a Gödel number of the wff  n,k x k .Therefore gℱ k   g n,k  n∈ℕ , where we set ℱ k  ℱ  k , k  1, 2, . .and ∀k 1 ∀k 2 g n,k 1  n∈ℕ gn,k 2