There is No Standard Model of ZFC and ZFC_2 with Henkin semantics.Generalized Lob's Theorem.Strong Reflection Principles and Large Cardinal Axioms.Consistency Results in Topology

In this article we proved so-called strong reflection principles corresponding to formal theories Th which has omega-models. An posible generalization of the Lob's theorem is considered.Main results is: (1) let $k$ be an inaccessible cardinal, then $\neg Con(ZFC+\exists k)$,(2) there is a Lindel\"of $T_3$ indestructible space of pseudocharacter $\leqslant \aleph_1$ and size $\aleph_2$ in $L$.


. Main results 2mm
Let us remind that accordingly to naive set theory, any definable collection is a set.Let R be the set of all sets that are not members of themselves.If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves.On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition.This contradiction is Russell's paradox.In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and Zermelo set theory, the first constructed axiomatic set theory.Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo-Fraenkel set theory ZFC."But how do we know that ZFC is a consistent theory, free of contradictions?The short answer is that we don't; it is a matter of faith (or of skepticism)"-E.Nelson wrote in his paper [1]cite: Nelson11.However, it is deemed unlikely that even ZFC 2 which is significantly stronger than ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC and ZFC 2 were consistent, that fact would have been uncovered by now.This much is certain -ZFC and ZFC 2 is immune to the classic paradoxes of naive set theory: Russell's paradox, the Burali-Forti paradox, and Cantor's paradox.
Remark 1.1.2.In order to derive a contradiction in second order set theory ZFC 2 with the Henkin semantics [7]cite: Henkin50, we remind the definition given in P. Cohen handbook [8]cite: Cohen66, (see [8]]cite: Cohen66 Ch.III, sec. 1, p. 87).P. Cohen wroted: "A set which can be obtained as the result of a transfinite sequence of predicative definitions Godel called "constructible".His result then is that the 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 jnust occur in any model in which one admits all ordinals.The definition we now give is the one used in [9]cite: Godel68.
Definition 1.1.1.[8]]cite: Cohen66.Let X be a set.The set X is defined as the union of X and the set Y of all sets у for which there is a formula A z, t 1 , . . ., t k in ZF such that if A X denotes A with all bound variables restricted to X, then for some t i , i 1, . . ., k. in X, у z X | A X z, t 1 , . . ., t k . 1 Observe X P x X, X X if X is infinite (and we assume AC).It should be clear to the reader that the definition of X , as we have given it, can be done entirely within ZF and that Y X is a single formula A X, Y in ZF.In general, one's intuition is that all normal definitions can be expressed in ZF, except possibly those which involve discussing the truth or falsity of an infinite sequence of statements.Since this is a very important point we shall give a rigorous proof in a later section that the construction of X is expressible in ZF. " Remark 1.1.3.We will say that a set y is definable by the formula A z, t 1 , . . ., t k relative to a given set X.
Remark 1.1.4.Note that a simple generalsation of the notion of the definability which has been by Definition 1.1.1immediately gives Russell's paradox in second order set theory ZFC 2 with the Henkin semantics [7]cite: Henkin50.
[6]cite: Foukzon19.(i)We will say that a set y is definable relative to a given set X iff there is a formula A z, t 1 , . . ., t k in ZFC then for some t i X, i 1, . . ., k. in X there exists a set z such that the condition A z, t 1 , . . ., t k is satisfied and y z or symbolically z A z, t 1 , . . ., t k y z . 2 It should be clear to the reader that the definition of X , as we have given it, can be done entirely within second order set theory ZFC 2 with the Henkin semantics [7] Hs .In this paper we dealing by using following definability condition.Definition 1.1.5.(i) Let M st M st ZFC 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, t 1 , . . ., t k in ZFC such that if A Mst denotes A with all bound variables restricted to M st , then for some t i M st , i 1, . . ., k. in M st there exists a set z such that the condition A Mst z, t 1 , . . ., t k is satisfied and y z or symbolically z A Mst z, t 1 , . . ., t k y z .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 ZFC 2 with the Henkin semantics.
(ii) In this paper we assume for simplicity but without loss of generality that A Mst z, t 1 , . . ., t k A Mst 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 ZFC 2 under the Henkin semantics [7]cite: Henkin50 and (iii) the second order set theory ZFC 2 under the full second-order semantics [8]cite: Cohen66,[9]cite: Godel68 , [10]cite: Rossberg ,[11]cite: Shapiro00 , [12]cite: RayoUzquiano99 but also with a proof theory based on formal Urlogic [13]cite: Vaananen01 .
Remark 1.1.9.Second-order logic essantially differs from the usual first-order predicate calculus in that it has variables and quantifiers not only for individuals but also for subsets of the universe and variables for n-ary relations as well [7]-[13]cite: Henkin50cite: Cohen66cite: Godel68cite: Rossbergcite: Shapiro00cite: RayoUzquiano99 deductive calculus DED 2 of second order logic is based on rules and axioms which guarantee that the quantifiers range at least over definable subsets [7]cite: Henkin50.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.

The Compactness Theorem:
A set of sentences, every finite subset of which has a Henkin model, has itself a Henkin model.

The Incompleteness Theorem:
Neither DED 2 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 RCF 2 such that the field of the real numbers is the only full model of RCF 2 up to isomorphism.
Remark 1.1.11.For let second-order ZFC be, as usual, the theory that results obtained from ZFC when the axiom schema of replacement is replaced by its second-order universal closure, i.e.
X Func X u r r s s u s, r X , 12 where X is a second-order variable, and where Func X abbreviates " X is a functional relation", see [12]cite: RayoUzquiano99.
Thus we interpret the wff's of ZFC 2 language with the full second-order semantics as required in [12]cite: RayoUzquiano99, [13]cite: Vaananen01 but also with a proof theory based on formal urlogic [13] 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]cite: Vaananen01 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.Hs ; we must also believe that these arithmetical theorems are asserting something about the standard naturals.It is "conceivable" that ZFC 2 Hs might be consistent but that the only nonstandard models M Nst ZFC 2 Hs it has are those in which the integers are nonstandard, in which case we might not "believe" an arithmetical statement such as "ZFC 2 Hs is inconsistent" even if there is a ZFC 2 Hs -proof of it.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 standard reals need be included in a particular transitive model.Note that in any nonstandard model M Nst Remark 1.1.19.However there is no any problem as mentioned above in second order set theory ZFC 2 with the full second-order semantics because corresponding second order arithmetic Z 2 fss 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 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.Remark 2.1.2.(i) By the other hand the Theorem 2.1.1 says that given some really consistent formal theory Th ,st that contins formal arithmetic, the concept of truth in that formal theory Th ,st is not definable using the expressive means that that arithmetic affords.This implies a major limitation on the scope of "self-representation." It is possible to define a formula True n , but only by drawing on a metalanguage whose expressive power goes beyond that of .To define a truth predicate for the metalanguage would require a still higher metametalanguage, and so on.
(ii) However if formal theory Th ,st is inconsistent this is not surpriising if we define a formula True n True n; Th ,st by drawing only on a language .(iii) Note that if under assumption Con Th ,st we define a formula True n; Th ,st by drawing only on a language by reductio ad absurdum it follows In this subsection we often write for short X , X Hs , X Hs instead Mst X ,

X,Mst
Hs , X,Mst Hs but this should not lead to a confusion.Assumption 3.1.1.We assume now for simplicity but without loss of generality that . 59 Remark 3.1.9.Notice that the expression (60) where the countable collection X Hs / X is defined by the following formula where the countable collection X Hs / X is defined by the following formula  Proof.By the diagonalization lemma applied to True x 1 there is a sentence such that: (c) Th True q , where q is the Godel number of , i.e. g Th q.Case 1. Suppose that Th , then q T Th .By (a), Th True q .But, from Th and (c), by biconditional elimination, one obtains Th True q .Hence Th is inconsistent, ontradicting our hypothesis.
Case 2. Suppose that Th , then q T Th .By (b), Th True q .Hence, by (c) and biconditional elimination, Th .Thus, in either case a contradiction is reached.Definition 3.4.1.If Th is a theory, let T Th 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 T Th is said to be is a strongly expressible in Th by wff True x 1 if the following properties are satisfied: (a) if n T Th then Th True n True n g Th 1 n , (b) if n T Th then Th True n .Theorem 3.4.2.(Generalized Tarski's undefinability Lemma).Let Th be a consistent theory with equality in the language in which the diagonal function D is representable and let g Th u be a Gödel number of given an expression u of Th.Then the property x T Th is not strongly expressible in Th.
Proof.By the diagonalization lemma applied to True x 1 there is a sentence such that: (c) Th True q , where q is the Godel number of , i.e. g Th q.Case 1. Suppose that Th , then q T Th .By (a), Th True q .But, from Th and (c), by biconditional elimination, one obtains Th True q .Hence Th is inconsistent, contradicting our hypothesis.
Case 2. Suppose that Th , then q T Th .By (b), Th True q .Hence, by (c) and biconditional elimination, Th .Thus, in either case a contradiction is reached.In addition under assumption Con Th 1 # , we establish a countable sequence of the consistent extensions of the theory Th 1 # such that:

124
Let T be the set of Gödel numbers of the all -sentences true in M. Then there is no -formula True n (truth predicate) which defines T .That is, there is no -formula True n such that for every -formula A, True g A A 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 -formula True(n) defines T .In particular, if A is a sentence of Th then True g A holds in iff A is true in M st Th .Hence for all A, the Tarski T-sentence True g A A is true in M st Th .But the diagonal lemma yields a counterexample to this equivalence, by giving a "Liar" sentence S such that S True g S holds in M st Th .Thus no -formula True n can define T .
Remark 3.5.10.Notice that the formal machinery of this proof is wholly elementary except for the diagonalization that the diagonal lemma requires.The proof of the diagonal lemma is likewise surprisingly simple; for example, it does not invoke recursive functions in any way.The proof does assume that every -formula has a Gödel number, but the specifics of a coding method are not required.
Remark 3.5.11.The undefinability theorem does not prevent truth in one consistent theory from being defined in a stronger theory.For example, the set of (codes for) formulas of first-order Peano arithmetic that are true in is definable by a formula in second order arithmetic.Similarly, the set of true formulas of the standard model of second order arithmetic (or n-th order arithmetic for any n) can be defined by a formula in first-order ZFC.
Remark1.3.5.12.Notice that Tarski's undefinability theorem cannot blocking the biconditionals , etc., 126 see Remark 3.5.14below.Remark 3.5.13.(I)We define again the set but now by using generalized truth predicate True # g A , A such that 129 holds.Thus in contrast with naive definition of the sets ,Nst and ,Nst there is no any problem which arises from Tarski's undefinability theorem.
Remark 3.5.14.In order to prove that set theory ZFC 2 Hs M ZFC 2 Hs is inconsistent without any reference to the set , notice that by the properties of the extension Th # it follows that definition given by formula ( 127) is correct, i.e., for every ZFC 2 Hs -formula such that M ZFC 2 Hs the following equivalence True g , holds.Theorem 3.5.2.(Generalized Tarski's undefinability theorem) (see subsection 4.2, Proposition 4.2.1).Let Th be a first order theory or the second order theory with Henkin semantics and with formal language , which includes negation and has a Gödel encoding g such that for every -formula A x there is a formula B such that the equivalence B A g B holds.Assume that Th has a standard Model M st Th .Then there is no -formula True n , n , such that for every -formula A such that M A, the following equivalence holds Remind that the primitive predicates of Russellian unramified typed set theory (TST), a streamlined version of the theory of types, are equality and membership .TST has a linear hierarchy of types: type 0 consists of individuals otherwise undescribed.For each (meta-) natural number n, type n 1 objects are sets of type n objects; sets of type n have members of type n 1. Objects connected by identity must have the same type.The following two atomic formulas succinctly describe the typing rules: x n y n and x n y n 1 .
The axioms of TST are: Extensionality: sets of the same (positive) type with the same members are equal; Axiom schema of comprehension: If x n is a formula, then the set x n x n n 1 exists i.e., given any formula x n , the formula x n 134 is an axiom where A n 1 represents the set x n x n n 1 and is not free in x n .Quinean set theory.(New Foundations) seeks to eliminate the need for such superscripts.New Foundations has a universal set, so it is a non-well founded set theory.That is to say, it is a logical theory that allows infinite descending chains of membership such as x n x n 1 x 3 x 2 x 1 .It avoids Russell's paradox by only allowing stratifiable formulas in the axiom of comprehension.For instance x y is a stratifiable formula, but x x is not (for details of how this works see below).
Definition 3.6.1.In New Foundations (NF) and related set theories, a formula in the language of first-order logic with equality and membership is said to be stratified iff there is a function σ which sends each variable appearing in [considered as an item of syntax] to a natural number (this works equally well if all integers are used) in such a way that any atomic formula x y appearing in satisfies σ x 1 σ y and any atomic formula x y appearing in satisfies σ x σ y .

Axioms and stratification are:
The well-formed formulas of New Foundations (NF) are the same as the well-formed formulas of TST, but with the type annotations erased.The axioms of NF are [19]cite: Quine37: Extensionality: Two objects with the same elements are the same object.
A comprehension schema: All instances of TST Comprehension but with type indices dropped (and without introducing new identifications between variables).
By convention, NF's Comprehension schema is stated using the concept of stratified formula and making no direct reference to types.Comprehension then becomes.
Stratified Axiom schema of comprehension: x s exists for each stratified formula s .Even the indirect reference to types implicit in the notion of stratification can be eliminated.Theodore Hailperin showed in 1944 that Comprehension is equivalent to a finite conjunction of its instances, so that NF can be finitely axiomatized without any reference to the notion of type [20]cite: Hailperin44.Comprehension may seem to run afoul of problems similar to those in naive set theory, but this is not the case.For example, the existence of the impossible Russell class x x x is not an axiom of NF, because x x cannot be stratified.

5mm . 3.6.2 Set theory ZFC 2
Hs , ZFC st and set theory ZFC Nst with stratified axiom schema of replacement 2mm The stratified axiom schema of replacement asserts that the image of a set under any function definable by stratified formula of the theory ZFC st will also fall inside a set.
Stratified Axiom schema of replacement: Let s x, y, w 1 , w 2 , , w n be any stratified formula in the language of ZFC st whose free variables are among x, y, A, w 1 , w 2 , , w n , so that in particular B is not free in s .Then i.e., if the relation s x, y, . . .represents a definable function f, A represents its domain, and f x is a set for every A, then the range of f is a subset of some set B.

Stratified Axiom schema of separation:
Let s x, w 1 , w 2 , , w n be any stratified formula in the language of ZFC st whose free variables are among x, A, w 1 , w 2 , , w n , so that in particular B is not free in s .Then w 1 w 2 . . .w n A B x x B x A s x, w 1 , w 2 , , w n , 136 Remark 3.6.1.Notice that the stratified axiom schema of separation follows from the stratified axiom schema of replacement together with the axiom of empty set.
Remark 3.6.2.Notice that the stratified axiom schema of replacement (separation) obviously violeted any contradictions ( 82), (126), etc. mentioned above.The existence of the countable Russell sets 2 Hs , st and Nst is impossible, because x x cannot be stratified.Hs consisting of a set of uusual natural numbers (which forms the range of individual variables) together with a constant 0 (an element of ), a function S from to , two binary operations and on , a binary relation on , and a collection D 2 of subsets of , which is the range of the set variables.Omitting D produces a model of the first order Peano arithmetic.
When D 2 is the full powerset of , the model M st 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 Z 2 fss have only one full model.This follows from the fact that the axioms of Peano arithmetic with the ssecond-order induction axiom have inly one model under second-order semantics, see section 3.
Let Th be some fixed, but unspecified, consistent formal theory.For later convenience, we assume that the encoding is done in some fixed formal second order theory S and that Th contains S. We assume throughout this paper that formal second order theory S has an -model M S .The sense in which S is contained in Th is better exemplified than explained: if S is a formal system of a second order arithmetic 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 M S , it will have to have a large supply of constants and closed terms to be used as codes.
(e.g. in formal arithmetic, one has 0, 1, . . ..) S will also have certain function symbols to be described shortly.To each formula, , of the language of Th is assigned a closed term, c , called the code of [19]cite: Quine37.We note that if x is a formula with free variable x, then x c is a closed term encoding the formula x with x viewed as a syntactic object and not as a parameter.Corresponding to the logical connectives and quantifiers are the function symbols, neg , imp , etc., such that all first order formulae , : S neg c c , S imp c , c c etc. Of particular importance is the substitution operator, represented by the function symbol sub , .For formulae x , terms t with codes t c : S sub x c , t c t c .137 It well known 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
Proof.(I) Let 1 . . .i . . .be an enumeration of the all first order closed wff's of the theory Th (this can be achieved if the set of propositional variables, etc. can be enumerated).
Define a chain Th i,st # |i , Th 1,st # Th of consistent theories inductively as follows: assume that theory Th i,st # is defined.Notice that below we write for short Th i,st Th i 1 # is a finite extension of the recursively axiomatizable theory Th.
We will rewrite the conditions ( 155)-(157) using predicate Pr Th i 1 # # symbolically as follows: Con Th i i ; M Th ,

158
(ii) Suppose that the following statement ( 159 M Th i .Then we define theory Th i 1 # as follows: 166 We define now a theory Th # as follows: 167 (1) First, notice that each Th i # is consistent.This is done by induction on i and by Lemmas 4.1.1-4.1.2.By assumption, the case is true when i 1.Now, suppose Th i # is consistent.Then its deductive closure Ded Th i # is also consistent.(6) Next, notice Ded Th # is maximally consistent nice extension of the Ded Th .Ded Th # is consistent because, by the standard Lemma 4.1.3below, 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.
where we have set x 1 x 1 , n x 1 n,1 x 1 and x x 1 .We note that any set k n,k x n , k 1, 2, . . .such as mentioned above, defines an unique set x k , i.e.
We define now a set k such that where we have set x 1 x 1 , n x 1 n,1 x 1 and x x 1 .We note that any set k n,k x n , k 1, 2, . . .such as mentioned above, defines an unique set x k , i.e.
We define now a set k such that We define now a set k such that (2) Assume now that (2.i) Th Pr Th c and (2.ii) Th .From (1) and (2.ii) it follows that (3) Th and Th .Let Th be a theory (4)Th Th From (3) it follows that (5) Con Th .From (4) and ( 5) it follows that (6) Th Pr Th c .From ( 4) and (#) it follows that (7) Th Pr Th c .From ( 6) and ( 7 We remind that a major part of modern mathematical analysis and related areas based not only on set theory ZFC but on strictly stronger set theory: ZFC M st ZFC .In order to avoid difficultnes which arises from Con ZFC M st ZFC in this subsection we introduce the set theory ZFC w with a weakened axiom of infinity.Without loss of generality we consider second-order arithmetic 2 with an restricted induction schema. Second-order arithmetic 2 includes, but is significantly stronger than, its first-order counterpart Peano arithmetic.Unlike Peano arithmetic, second-order arithmetic allows quantification over sets of natural numbers as well as numbers themselves.Because real numbers can be represented as (infinite) sets of natural numbers in well-known ways, and because second order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in second-order arithmetic.For this reason, second-order arithmetic is sometimes called "analysis".
Induction schema of second-order arithmetic 2 .
If φ n is a formula of second-order arithmetic 2 with a free number variable n and possible other free number or set variables (written m and X), the induction axiom for φ is the axiom: m X φ 0 n φ n φ n 1 nφ n .220 The (full) second-order induction scheme consists of all instances of this axiom, over all second-order formulas.One particularly important instance of the induction scheme is when φ is the formula "n X " expressing the fact that n is a member of X (X being a free set variable): in this case, the induction axiom for φ is X 0 X n n X n 1 X n n X .

I I
x I x x I .246 Such a set as usually called an inductive set.
Definition 5.2.4.We will say that x is a non standard set and abbreviate x Nst iff x contain at least one non standard element, i.e., x Nst  x Nst .247 Remark 5.2.4.It follows from Axiom schema of specification and Axiom schema of replacement (245) we can not extract from a non standard set the standard and non standard elements separately, i.e. for any non standard set x Nst there is no exist a set y and z such that x Nst y z, 248 where y contain only standard sets and z contain only standard sets!As it follows from Theorem 5.3.1 any inductive set is a non standard set.Thus Axiom of infinity can be written in the following form 8 .Axiom of infinity Let S x abbreviate x x , where w is some set.Then: Such a set as usually called a non standard inductive set.

Strong axiom of infinity
Let S x abbreviate x x , where w is some set.Then: Definition 5.3.1.We will say that x Nst is inductive if there is an formula x of ZFC that says: 'x Nst is -inductive'; i.e. i.e.W Nst is the set of all elements of I Nst which happen also to be elements of every other inductive set.This clearly satisfies the hypothesis of (5.3.2),since if x W Nst , then x is in every inductive set, and if x is in every inductive set, it is in particular in I Nst , so it must also be in W Nst .
(2) For uniqueness, first note that any set which satisfies ( 252) is itself inductive, since is in all inductive sets, and if an element x is in all inductive sets, then by the inductive property so is its successor.Thus if there were another set (3) For nonstardarntness we assume that ω is a standard set, i.e. there is no nonstandard element in ω.Then ω where is isomorphic to , but this is a contradiction, since M, .Theorem 5.3.1.There exists unique nonstardard set ω such that (252) holds, i.e.
x x ω I Nst I Nst x I Nst .254 Definition 5.3.2.We will say that a set S is -finite if every surjective -function from S onto itself is one-to-one.
Proof.Assuming that any nonstandard natural number is not -finite one obviously obtains a contradiction.
Remark 5.3.1.Assuming that ω is ф standard set then this method mentioned above produce system which satisfy the axioms of second-order arithmetic Z 2 fss , since the axiom of power set allows us to quantify over the power set of ω , as in second-order logic.Thus it completely determine isomorphic systems, and since they are isomorphic under the identity map, they must in fact be equal.

6mm . 6. Conclusion 3mm
In this paper we have proved that the second order ZFC with the full second-order semantic is inconsistent, i.e.Con ZFC 2 fss .Main result is: let k be an inaccessible cardinal and H k is a set of all sets having hereditary size less then k, then Con ZFC M st ZFC M st ZFC H k .This result also was obtained in [3], [4], [5]cite: Lemhoff16cite: Foulzon17cite: Foukzon15 essentially another approach.For the first time this result has been declared to AMS in [23], [24]cite: Foulzon13a cite: Foukzon13b . [16]cite: Bovykin00 , for some linear order A Thus one can choose Gödel encoding inside the standard model M st Z 2 Hs .
Remind that: (i) if Th is a theory, let T Th be the set of Godel numbers of theorems of Th,[10], (ii) the property x T Th is said to be is expressible in Th by wff True x 1 if the following properties are satisfied [10]cite: Rossberg: (a) if n T Th then Th True n , (b) if n T Th 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]cite: Rossberg.Let Th be a consistent theory with equality in the language 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 T Th is not expressible in Th.
In this section we use second-order arithmetic Z 2 Hs with Henkin semantics.Notice that any standard model M st Z 2 Hs of second-order arithmetic Z 2

#i#i
(2) If statements (155)-(157) are satisfied, i.e.Th i 1 is consistent since it is a subset of closure Ded Th i 1 # .(3) If statements (159)-(161) are satisfied, i.e.Th i 1 is consistent since it is a subset of closure Ded Th i 1 # .(4) If the statement (163) is satisfied, i.e.Th i say that, a set y is a Th-set iff there is exist first order one-place open wff x such that y x .We write y Th iff y is a Th-set.Remark 4.3.9.Note that Let be a collection such that : x x x is a Th-set .Proposition 4.3.1.A set is a Th-set.Definition 4.3.6.We define now a Th-set c : Th c , (ii) c is a countable Th-set.Proof.(i) Statement Th c follows immediately by using statement and axiom schema of separation [4], (ii) follows immediately from countability of a set .Proposition 4.3.3.A set c is inconsistent.Proof.From formula (216) one obtains Th a contradiction.Thus finally we obtain: Theorem 4.3.2.[5]cite: Foukzon15.Con ZFC 2 fss .It well known that under ZFC it can be shown that κ is inaccessible iff V κ , is a model of ZFC 2 [12].Thus finally we obtain. .How can we safe the set theory ZFC M st ZFC 3mm 5mm .5.1.The set theory.ZFC w with a weakened axiom of infinity 2mm the standard and nonstandard natural numbers from the infinite nonstandard set I Nst 2mm existence, we will use the Axiom of Infinity combined with the Axiom schema of specification.Let I Nst be an inductive (non standard) set guaranteed by the Axiom of Infinity.Then we use the Axiom Schema of Specification to define our set W Nst x I Nst : J Nst Φ J Nst x J Nst , 253

.
Let ω denote this unique set.
cite: Henkin50 denoted by ZFC 2 Hs and that Y X is a single formula A X, Y in ZFC 2 Hs .(ii)We will denote the set Y of all sets у definable relative to a given set X by Y Hs be a set of the all sets definable relative to a given set X by the first order 1-place open wff's and such that x x Note that in paper [6]cite: Foukzon19 we dealing by using following definability condition: a set у is definable if there is a formula A z in ZFC such that There is no completeness theorem for second-order logic with the full second-order semantics.Nor do the axioms of ZFC 2 Hs .We denote this statement through all this paper by symbol Con ZFC 2 Hs ; M ZFC 2 Hs and there exists a single statement M Z 2 Hs in Z 2 The model M st ZFC 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 st ZFC , see ref. [8]cite: Cohen66.
cite: Vaananen01.Designation 1.1.1.We will denote: (i) by ZFC 2 Hs set theory ZFC 2 with the Henkin semantics, (ii) by ZFC 2 fss set theory ZFC 2 with the full second-order semantics, (iii) by ZFC 2 Hs set theory 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 model M ZFC 2 fss of ZFC 2 fss [11]cite: Shapiro00.Let Z be the conjunction of all the axioms of ZFC 2 fss .We assume now that: Z is true, i.e.Con ZFC 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 Hs .We denote this statement throught all this paper by symbol Con Z 2 Hs ; M Z 2 Hs .Axiom M st ZFC .[8]cite: Cohen66.There is a set M st ZFC such that if R is x, y |x y x M st ZFC y M st ZFC then M st ZFC is a model for ZFC under the relation R. Definition 1.1.6.[8]cite: Cohen66.
Hs from Con ZFC 2 Hs , (ii) ~Con ZFC from Con ZFC , by using Gödel encoding, one needs something more than the consistency of ZFC 2 Hs , e.g., that ZFC 2 Hs has an omega-model M Hs i.e., a model in which the integers are the standard integers and the all wff of ZFC 2 Hs , ZFC, etc. represented by standard objects.To put it another way, why should we believe a statement just because there's a ZFC 2 " Remark 1.1.16.Note that in order to deduce: (i) ~Con 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 that the mere consistency of ZFC 2 isn't quite enough to get us to believe arithmetical theorems of ZFC 2

.2.Generalized Löb's theorem 2mm Definition 2.2.1. Let Th #
be first order theory and Con Th # .A theory Th # is complete if, for every formula A in the theory's language , that formula A or its negation A is provable in Th # , i.e., for any wff A, always Th # A or Th # A.

Definition 2.2.2.Let Th be
first order theory and Con Th .We will say that a theory Th # is completion of the theory Th if (i) Th Th # , (ii) a theory Th # is complete.

ZFC st # m Th m , where for any m a theory Th m 1 is finite extension of the theory Th m . (iii) Let Pr m st 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 Pr m st y, x is expressible in the theory Th m by canonical Gödel encoding and really asserts provability in Th m .
m , where for any m a theory Th m 1 is finite extension of the theory Th m .(iii) Let Pr m Nst 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 Pr m Nst y, x is expressible in the theory Th m by canonical Gödel encoding and really asserts provability in Th m .(iv) Let Pr Nst # 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 relation Pr Nst # y, x is expressible in the theory ZFC Nst # by the following formula # y, x really asserts provability in the set theory ZFC Nst # .Remark 2.2.4.Note that the relation Pr m Nst y, x is expressible in the theory Th m since a theory Th m is an finite extension of the recursively axiomatizable theory ZFC and therefore the predicate Pr m Nst y, x exists since any theory Th m is recursively axiomatizable.

Remark 2.2.5.Note that
Assume that:Con ZFC 2 Hs , where ZFC 2 Hs# of the theory ZFC 2Hs such that the following condtions holds: (i) For every first order wff formula A (wff 1 A) in the language of ZFC 2 Hs is provable in ZFC 2 Hs# i.e., for any wff 1 A, always ZFC 2 Th m , where for any m a theory Th m 1 is finite extension of the theory Th m .(iii)LetPrmst y, x be recursive relation such that: y is a Gödel number of a proof of the wff 1 of the theory Th m and x is a Gödel number of this wff 1 .Then the relation Pr m st y, x is expressible in the theory Th m by canonical Gödel encoding and really asserts provability in Th m .(iv)LetPrst # 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 wff 1 .Then the relation Pr st # y, x is expressible in the set theory ZFC 2 Hs# by the following formula Note that the relation Pr m st y, x is expressible in the theory Th m since a theory Th m is an finite extension of the finite axiomatizable theory ZFC 2Hs and therefore the predicate Pr a theory ZFC Nst # obviously is not recursively axiomatizable nevertheless Gödel encoding holds by Remark 2.2.1.Theorem 2.2.5.Assume that: Con ZFC Nst , where ZFC Nst True x True g A A M Nst ZFC .33Theorem 2.2.6.Con ZFC Nst .Proof.Assume that: Con ZFC Nst .From (30) and (33) one obtains a condradiction Con ZFC Nst Con ZFC Nst and therefore by reductio ad absurdum it follows Con ZFC Nst .Theorem 2.2.7.(v)The predicate Pr st # y, x really asserts provability in the set theory ZFC 2 Hs# .Remark 2.2.7.m Nst 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 2 Hs , where ZFC 2 Hs.Then truth predicate True n is expressible by using first order language by the following formula

Derivation of the inconsistent provably definable set in set theory ZFC
2 Hs , ZFC st and ZFC Nst 3mm 5mm .3.1.

Derivation of the inconsistent provably definable set in set theory
Hs.We will say that is a first order n-place open wff if contains free occurrences of the first order individual variables X 1 , . .., X n and quantifiers only over any first order individual variables Y 1 , . .., Y m .(ii)Let 2 Hs be the countable set of the all first order provable definable sets X, i.e. sets such that ZFC 2 Hs !X X , where X Mst X is a first order 1-place open wff that contains only first order variables (we will denote such wff for short by wff 1 ), with all bound variables restricted to standard model M st M st ZFC 2 ii) In additional note that: since Tarski's undefinability theorem has been proved under the same assumption M st ZFC 2 Hs by reductio ad absurdum it follows again Con ZFC Nst , 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.Let M be Henkin model of ZFC 2 Hs .Let 2 Note that the Definition 3.1.5holdsas definition of predicate really asserting provability of the first order sentence A in ZFC 2Hs .
Remark 3.1.2.Note that by using Gödel encoding the above stated contradiction can be shipped in special completion ZFC 2 Hs# of ZFC 2 Hs , see subsection 1.2, Theorem 1.2.8.Remark 3.1.3.(i) Note that Tarski's undefinability theorem cannot bloked the equivalence (43) since this theorem is no longer holds by Proposition 2.2.1.(Generalized Löbs Theorem).(Here Bew ZFC 2 Hs #A is a canonical Gödel formula which says to us that there exists proof in ZFC 2 Hs of the formula A with Gödel number #A.Remark 3.1.7.

Derivation of the inconsistent provably definable set in set theory
Note that a contradiction (82) is a contradiction inside ZFC st for the reason that predicate X ZFC st Y is expressible by using first order language as predicate of ZFC st (see subsection 4.1) and therefore countable sets st and st are sets in the sense of the set theory ZFC st .Remark 3.2.2.Note that by using Gödel encoding the above stated contradiction can be shipped in special completion ZFC st # of ZFC st , see subsection 1.2, Theorem 1.2.2 (i).Designation 3.2.1 (i) Let M st ZFC be a standard model of ZFC and (ii) let ZFC st be the theory ZFC st ZFC M st ZFC , (iii) let st be the set of the all sets of M st ZFC provably definable in ZFC st , and let st X Bew ZFCst #A is a canonycal Gödel formula which says to us that there exists proof in ZFC st of the formula A with Gödel number #A M st PA .Remark 3.2.3.Notice that Definition 3.2.6 holds as definition of predicate really asserting provability in ZFC st .
61 Theorem 3.1.1.Set theory ZFC 2 Hs .Definition 3.1.11.We choose now A in the following form A Bew ZFC 2 Hs #A , 66 or in the following equivalent form A Bew ZFC 2 Hs #A Bew ZFC 2 Hs #A A 67 similar to (46).Here Bew ZFC 2 Hs #A is a Gödel formula which really asserts provability in ZFC 2 Hs of the formula A with Gödel number #A. st : st X X , where st A means: 'sentence A derivable in ZFC st ', By using formula (85) we rewrite now (86) in the following equivalent form X X 87 Definition 3.2.2.Let st be the countable collection of the all sets such that

Derivation of the inconsistent provably definable set in ZFC Nst 2mm
Note that from the axiom schema of replacement it follows directly that st is a set in the sense of the set theory ZFC st .wellformedformula of ZFC st and therefore collection st is a set in the sense of ZFC 2Hs .Let ZFC Nst be the theory ZFC Nst ZFC M Nst ZFC PA .(iv)LetNstbe the set of the all sets of M st ZFC PA provably definable in ZFC Nst , and let 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 Bew ZFC Nst #A is a canonycal Gödel formula which says to us that there exists proof in ZFC Nst of the formula A with Gödel number #A M st PA .Remark 3.3.1.Notice that definition (104) holds as definition of predicate really asserting provability in ZFC Nst .ii)Let Fr Nst 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]cite: Foukzon19, [10]cite: Rossberg.(iii)Let Nst y, v, 1 be a Gödel number of the following wff: !X Let Pr ZFC Nst z be a predicate asserting provability in ZFC Nst .Remark 3.3.2.Let Nst be the countable collection of all sets X such that ZFC Nst !X X , where X is a 1-place open wff i.e., NstXNst : Designation 3.3.2.(i)Let g ZFC Nst u be a Gödel number of given an expression u of ZFC Nst .( II) Let i,st , i 1, 2, . . .be the set of the all sets of M st ZFC provably definable in Th i,st # , Nst #A , i 1, 2, . . . is a canonical Gödel formulae which says to us that there exists proof in Th i,Nst # , i 1, 2, . . . of the formula A with Gödel number #A.Of course the all theories Th i (I) Let Th be first order theory with formal language , which includes negation and has a Gödel numbering g such that for every -formula A x there is a formula B such that B A g B holds.Assume that Th has a standard model M st Th and Con Th ,st where Let Th Hs be second order theory with Henkin semantics and formal language , which includes negation and has a Gödel numbering g such that for every -formula A x there is a formula B such that B A g B holds.Assume that Th Hs has a standard model M st the all sentences of Th 1 # , which is valid in M, i.e.,MA i #A , i 1, 2, . . . is a canonycal Gödel formulae which says to us that there exists proof in Th i # , i 1, 2, . . . of the formula A with Gödel number #A.(# , Th i,st # , Th i,Nst # , i 1, 2, . ..are inconsistent, see subsection 4.1.Remark 3.5.5.(I) Let be the set of the all sets of M provably definable in Th # , 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 is fairly sound, e.g. this is the case when S and Th replaced by S S M Th and Th Th M Th correspondingly (see Designation 4.1.1below).(II) Notice that it is always the case that: .that is the case when predicate Pr Th y , y M Th : Let us consider an one-place open wff x such that condition (169) is satisfied, i.e.
1 x k 2 .We note that a sets k , k 1, 2, . .are the part of the ZFC 2 Hs or ZFC, i.e. a set k is a set in the sense of ZFC 2 Hs or ZFC.Note that by using Gödel numbering one can replace any set k , k 1, 2, . .by a set Fr g n,k , v k .Let us define now predicate g We rewrite now the condition (183) in the following equivalent form using only the language of the theory Th i # : We will say that, a set y is a Th i # -set if there exist one-place open wff x such that y x .We will write for short y Th i # iff y is a Th i # -set.Let us consider an one-place open wff x such that conditions (183) are satisfied, i.e.
from the set i by the axiom schema of replacement.
n,k n k be a family of the all sets g n,k n .By axiom of choice one obtains a unique set i g k k such that k g k g n,k n .Finally for any i one obtains a set i

Remark 4.1.17. We
rewrite now the condition (197) in the following equivalent form using only the language of the theory Th # : We will say that, a set y is a Th # -set if there exists one-place open wff x such that y x .We write y Th # iff y is a Th # -set.
Th # -set.This is done by Gödel encoding and by axiom schema of separation.Let g n,k g n,k x k , k 1, 2, . .be a Gödel number of the wff n,k x k .
k n,k x n , k 1, 2, ...such as mentioned above defines a unique set x k , i.e.k 1 k 2 iff x k 1 x k 2 .We note that sets k , k 1,2, . .are the part of the ZFC 2 Hs , i.e. a set k is a set in the sense of ZFC 2 Hs .Note that by using Gödel numbering one can replace any set k , k 1, 2, . .by the set k g k of the corresponding Gödel numbers such that k , k 1, 2, . . is a RCF 2 for real closed fields.For full second-order logic there is a notion of "semantical" derivation:We can derive from if every model of is a model of .Of course scanning through all models of is a highly mathematical act.Thus with respect to ZFC 2 fss , this is a semantically defined system and thus it is not standard to speak about it being contradictory if anything, one might attempt to prove that it has no models, which to be what is being done in section 3 and section 4 for ZFC 2 Note that in order to avoid difficulties with "semantical" derivation mentioned above one considers first order theory Th ZFC 2 fss which contains only first order wff of ZFC 2 Hs .Thus in order to prove that ZFC 2 fss has no models or it being contradictory, one might use the same approach, which is done in section 3 and section 4 for ZFC 2Hs .Definition 4.3.2.Let be a wff of ZFC 2 Hs .We will say that is a first order n place open wff if contains free occurrences of the first order individual variables X 1 , . . ., X n and quantifiers only over any first order individual variables Y 1 , . . ., Y m .Definition 4.3.3.Let Th be a first order theory which contains only first order wff of ZFC 2 Hs .Using formula (141) one can define predicate Pr Th # y really asserting provability of the first order sentences in Th ZFC 2 fss : RayoUzquiano99 .(Löb's Theorem for ZFC 2 fss .) Let be any first order closed formula with code y 213Theorem 4.3.1.[12]cite: