1. Introduction
Main Results
Let us remind that accordingly to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell’s paradox. In 1908, two ways of avoiding the paradox were proposed, Russell’s type theory and Zermelo set theory, the first constructed axiomatic set theory. Zermelo’s axioms went well beyond Frege’s axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo-Fraenkel set theory ZFC. “But how do we know that ZFC is a consistent theory, free of contradictions? The short answer is that we don’t; it is a matter of faith (or of skepticism)”—E. Nelson wrote in his paper [1] . However, it is deemed unlikely that even ZFC2 which is significantly stronger than ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC and ZFC2 were consistent, that fact would have been uncovered by now. This much is certain—ZFC and ZFC2 are immune to the classic paradoxes of naive set theory: Russell’s paradox, the Burali-Forti paradox, and Cantor’s paradox.
Remark 1.1.1. The inconsistency of the second-order set theory ZFC2082 originally have been uncovered in [2] and officially announced in [3] , see also ref. [4] [5] [6] .
Remark 1.1.2. In order to derive a contradiction in second-order set theory ZFC2 with the Henkin semantics [7] , we remind the definition given in P. Cohen handbook [8] (see [8] Ch. III, sec. 1, p. 87). P. Cohen wrote: “A set which can be obtained as the result of a transfinite sequence of predicative definitions Godel called ‘constructible’”. His result then is that the constructible sets are a model for ZF and that in this model GCH and AC hold. The notion of a predicative construction must be made more precise, of course, but there is essentially only one way to proceed. Another way to explain constructibility is to remark that the constructible sets are those sets which just occur in any model in which one admits all ordinals. The definition we now give is the one used in [9] .
Definition 1.1.1. [8] . Let X be a set. The set is defined as the union of X and the set Y of all sets y for which there is a formula in ZF such that if denotes A with all bound variables restricted to X, then for some , in X,
(1)
Observe , if X is infinite (and we assume AC). It should be clear to the reader that the definition of , as we have given it, can be done entirely within ZF and that is a single formula 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 is expressible in ZF.”
Remark 1.1.3. We will say that a set y is definable by the formula relative to a given set X.
Remark 1.1.4. Note that a simple generalisation of the notion of the definability which has been by Definition 1.1.1 immediately gives Russell’s paradox in second order set theory ZFC2 with the Henkin semantics [7] .
Definition 1.1.2. [6] . i) We will say that a set y is definable relative to a given set X iff there is a formula in ZFC then for some , in X there exists a set z such that the condition is satisfied and or symbolically
(2)
It should be clear to the reader that the definition of , as we have given it, can be done entirely within second order set theory ZFC2 with the Henkin semantics [7] denoted by and that is a single formula in .
ii) We will denote the set Y of all sets y definable relative to a given set X by .
Definition 1.1.3. Let be a set of the all sets definable relative to a given set X by the first order 1-place open wff’s and such that
(3)
Remark 1.1.5. (a) Note that since is a set definable by the first order 1-place open wff :
(4)
Theorem 1.1.1. [6] . Set theory is inconsistent.
Proof. From (3) and Remark 1.1.2 one obtains
(5)
From (5) one obtains a contradiction
(6)
Remark 1.1.6. Note that in paper [6] we dealing by using following definability condition: a set y is definable if there is a formula in ZFC such that
(7)
Obviously in this case a set is a countable set.
Definition 1.1.4. Let be the countable set of the all sets definable by the first order 1-place open wff’s and such that
(8)
Remark 1.1.7. (a) Note that since is a set definable by the first order 1-place open wff :
(9)
one obtains a contradiction .
In this paper we dealing by using following definability condition.
Definition 1.1.5. i) Let be a standard model of ZFC. We will say that a set y is definable relative to a given standard model of ZFC if there is a formula in ZFC such that if denotes A with all bound variables restricted to , then for some , in there exists a set z such that the condition is satisfied and or symbolically
(10)
It should be clear to the reader that the definition of , as we have given it, can be done entirely within second order set theory ZFC2 with the Henkin semantics.
ii) In this paper we assume for simplicity but without loss of generality that
(11)
Remark 1.1.8. Note that in this paper we view i) the first order set theory ZFC under the canonical first order semantics ii) the second order set theory ZFC2 under the Henkin semantics [7] and iii) the second order set theory ZFC2 under the full second-order semantics [8] [9] [10] [11] [12] but also with a proof theory based on formal Urlogic [13] .
Remark 1.1.9. Second-order logic essentially differs from the usual first-order predicate calculus in that it has variables and quantifiers not only for individuals but also for subsets of the universe and variables for n-ary relations as well [7] - [13] . The deductive calculus of second order logic is based on rules and axioms which guarantee that the quantifiers range at least over definable subsets [7] . As to the semantics, there are two types of models: i) Suppose is an ordinary first-order structure and is a set of subsets of the domain A of . The main idea is that the set-variables range over , i.e.
.
We call a Henkin model, if satisfies the axioms of and truth in is preserved by the rules of . We call this semantics of second-order logic the Henkin semantics and second-order logic with the Henkin semantics the Henkin second-order logic. There is a special class of Henkin models, namely those where is the set of all subsets of A.
We call these full models. We call this semantics of second-order logic the full semantics and second-order logic with the full semantics the full second-order logic.
Remark 1.1.10. We emphasize that the following facts are the main features of second-order logic:
1) The Completeness Theorem: A sentence is provable in if and only if it holds in all Henkin models [7] - [13] .
2) The Löwenheim-Skolem Theorem: A sentence with an infinite Henkin model has a countable Henkin model.
3) The Compactness Theorem: A set of sentences, every finite subset of which has a Henkin model, has itself a Henkin model.
4) The Incompleteness Theorem: Neither 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 such that the semiring of natural numbers is the only full model of up to isomorphism.
8) There is a finite second-order axiom system RCF2 such that the field of the real numbers is the only full model of RCF2 up to isomorphism.
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.
(12)
where X is a second-order variable, and where abbreviates “X is a functional relation”, see [12] .
Thus we interpret the wff’s of ZFC2 language with the full second-order semantics as required in [12] [13] but also with a proof theory based on formal urlogic [13] .
Designation 1.1.1. We will denote: i) by set theory with the Henkin semantics,
ii) by set theory with the full second-order semantics,
iii) by set theory and
iv) by set theory , where is a standard model of the theory .
Remark 1.1.12. There is no completeness theorem for second-order logic with the full second-order semantics. Nor do the axioms of 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 of [11] .
Let Z be the conjunction of all the axioms of . We assume now that: Z is true, i.e. . 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 is a strongly inaccessible if and only if is a model of . Thus
(13)
In this paper we prove that:
i) ii) and iii) is inconsistent, where is a standard model of the theory .
Axiom [8] . There is a set and a binary relation which makes a model for ZFC.
Remark 1.1.13. i) We emphasize that it is well known that axiom a single statement in ZFC see [8] , Ch. II, Section 7. We denote this statement thought all this paper by symbol . The completeness theorem says that .
ii) Obviously there exists a single statement in such that .
We denote this statement through all this paper by symbol and there exists a single statement in . We denote this statement through all this paper by symbol .
Axiom [8] . There is a set such that if R is then is a model for ZFC under the relation R.
Definition 1.1.6. [8] . The model is called a standard model since the relation used is merely the standard -relation.
Remark 1.1.14. Note that axiom doesn’t imply axiom , see ref. [8] .
Remark 1.1.15. We remind that in Henkin semantics, each sort of second-order variable has a particular domain of its own to range over, which may be a proper subset of all sets or functions of that sort. Leon Henkin (1950) defined these semantics and proved that Gödel’s completeness theorem and compactness theorem, which hold for first-order logic, carry over to second-order logic with Henkin semantics. This is because Henkin semantics are almost identical to many-sorted first-order semantics, where additional sorts of variables are added to simulate the new variables of second-order logic. Second-order logic with Henkin semantics is not more expressive than first-order logic. Henkin semantics are commonly used in the study of second-order arithmetic. Väänänen [13] argued that the choice between Henkin models and full models for second-order logic is analogous to the choice between ZFC and ( is von Neumann universe), as a basis for set theory: “As with second-order logic, we cannot really choose whether we axiomatize mathematics using or ZFC. The result is the same in both cases, as ZFC is the best attempt so far to use as an axiomatization of mathematics”.
Remark 1.1.16. Note that in order to deduce: i) from ,
ii) from , by using Gödel encoding, one needs something more than the consistency of , e.g., that has an omega-model or an standard model i.e., a model in which the integers are the standard integers and the all wff of , ZFC, etc. represented by standard objects. To put it another way, why should we believe a statement just because there’s a -proof of it? It’s clear that if is inconsistent, then we won’t believe -proofs. What’s slightly more subtle is that the mere consistency of isn’t quite enough to get us to believe arithmetical theorems of ; we must also believe that these arithmetical theorems are asserting something about the standard naturals. It is “conceivable” that might be consistent but that the only nonstandard models it has are those in which the integers are nonstandard, in which case we might not “believe” an arithmetical statement such as “ is inconsistent” even if there is a -proof of it.
Remark 1.1.17. Note that assumption is not necessary if nonstandard model is a transitive or has a standard part , see [14] [15] .
Remark 1.1.18. Remind that if M is a transitive model, then is the standard . 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 of the second-order arithmetic the terms comprise the initial segment isomorphic to . This initial segment is called the standard cut of the . The order type of any nonstandard model of is equal to , see ref. [16] , for some linear order A.
Thus one can choose Gödel encoding inside the standard model .
Remark 1.1.19. However there is no any problem as mentioned above in second order set theory with the full second-order semantics because corresponding second order arithmetic is categorical.
Remark 1.1.20. Note if we view second-order arithmetic as a theory in first-order predicate calculus. Thus a model of the language of second-order arithmetic consists of a set M (which forms the range of individual variables) together with a constant 0 (an element of M), a function S from M to M, two binary operations + and × on M, a binary relation < on M, and a collection D of subsets of M, which is the range of the set variables. When D is the full power set of M, the model 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. , 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, 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 , which is the usual set of natural numbers with its usual structure and all its subsets, is called the intended or standard model of second-order arithmetic.
2. Generalized Löb’s Theorem. Remarks on the Tarski’s Undefinability Theorem
2.1. Remarks on the Tarski’s Undefinability Theorem
Remark 2.1.1. In paper [2] under the following assumption
(14)
it has been proved that there exists countable Russell’s set such that the following statement is satisfied:
(15)
From (15) it immediately follows a contradiction
(16)
From (16) and (14) by reductio and absurdum it follows
(17)
Theorem 2.1.1. [17] [18] [19] . (Tarski’s undefinability theorem). Let be first order theory with formal language , which includes negation and has a Gödel numbering such that for every -formula there is a formula B such that holds. Assume that has a standard model and where
(18)
Let be the set of Gödel numbers of -sentences true in . Then there is no -formula (truth predicate) which defines . That is, there is no -formula such that for every -formula A,
(19)
where the abbreviation means that A holds in standard model , i.e. . Therefore implies that
(20)
Thus Tarski’s undefinability theorem reads
(21)
Remark 2.1.2. i) By the other hand the Theorem 2.1.1 says that given some really consistent formal theory that contains formal arithmetic, the concept of truth in that formal theory 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 , 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 is inconsistent this is not surprising if we define a formula by drawing only on a language .
iii) Note that if under assumption we define a formula by drawing only on a language by reductio ad absurdum it follows
(22)
Remark 2.1.3. i) Let be a theory . In this paper under assumption we define a formula by drawing only on a language by using Generalized Löb’s theorem [4] [5] . Thus by reductio ad absurdum it follows
(23)
ii) However note that in this case we obtain by using approach that completely different in comparison with approach based on derivation of the countable Russell’s set with conditions (15).
2.2. Generalized Löb’s Theorem
Definition 2.2.1. Let be first order theory and . A theory is complete if, for every formula A in the theory’s language , that formula A or its negation is provable in , i.e., for any wff A, always or .
Definition 2.2.2. Let be first order theory and . We will say that a theory is completion of the theory if i) , ii) a theory is complete.
Theorem 2.2.1. [4] [5] . Assume that: , where . Then there exists completion of the theory such that the following conditions hold:
i) For every formula A in the language of ZFC that formula or formula is provable in i.e., for any wff A, always or .
ii) , where for any m a theory is finite extension of the theory .
iii) Let be recursive relation such that: y is a Gödel number of a proof of the wff of the theory and x is a Gödel number of this wff. Then the relation is expressible in the theory by canonical Gödel encoding and really asserts provability in .
iv) Let be relation such that: y is a Gödel number of a proof of the wff of the theory and x is a Gödel number of this wff. Then the relation is expressible in the theory by the following formula
(24)
v) The predicate really asserts provability in the set theory .
Remark 2.2.1. Note that the relation is expressible in the theory since a theory is a finite extension of the recursively axiomatizable theory ZFC and therefore the predicate exists since any theory is recursively axiomatizable.
Remark 2.2.2. Note that a theory obviously is not recursively axiomatizable nevertheless Gödel encoding holds by Remark 2.2.1.
Theorem 2.2.2. Assume that: , where . Then truth predicate is expressible by using only first order language by the following formula
(25)
Proof. Assume that:
(26)
It follows from (26) there exists such that and therefore by (24) we obtain
(27)
From (24) immediately by definitions one obtains (25).
Remark 2.2.3. Note that Theorem 2.1.1 in this case reads
(28)
Theorem 2.2.3. .
Proof. Assume that: . From (25) and (28) one obtains a contradiction (see Remark 2.1.3) and therefore by reductio ad absurdum it follows .
Theorem 2.2.4. [4] [5] . Let be a nonstandard model of ZFC and let be a standard model of PA.
We assume now that and denote such nonstandard model of the set theory ZFC by . Let be the theory . Assume that: , where . Then there exists completion of the theory such that the following conditions hold:
i) For every formula A in the language of ZFC that formula or formula is provable in i.e., for any wff A, always or .
ii) , where for any m a theory is finite extension of the theory .
iii) Let be recursive relation such that: y is a Gödel number of a proof of the wff of the theory and x is a Gödel number of this wff. Then the relation is expressible in the theory by canonical Gödel encoding and really asserts provability in .
iv) Let be relation such that: y is a Gödel number of a proof of the wff of the theory and x is a Gödel number of this wff. Then the relation is expressible in the theory by the following formula
(29)
v) The predicate really asserts provability in the set theory .
Remark 2.2.4. Note that the relation is expressible in the theory since a theory is a finite extension of the recursively axiomatizable theory ZFC and therefore the predicate exists since any theory is recursively axiomatizable.
Remark 2.2.5. Note that a theory obviously is not recursively axiomatizable nevertheless Gödel encoding holds by Remark 2.2.1.
Theorem 2.2.5. Assume that: , where , .
Then truth predicate is expressible by using first order language by the following formula
(30)
Proof. Assume that:
(31)
It follows from (29) there exists such that and therefore by (31) we obtain
(32)
From (32) immediately by definitions one obtains (30).
Remark 2.2.6. Note that Theorem 2.1.1 in this case reads
(33)
Theorem 2.2.6. .
Proof. Assume that: . From (30) and (33) one obtains a contradiction and therefore by reductio ad absurdum it follows .
Theorem 2.2.7. Assume that: , where . Then there exists completion of the theory such that the following conditions hold:
i) For every first order wff formula A (wff1 A) in the language of that formula or formula is provable in i.e., for any wff1 A, always or .
ii) , where for any m a theory is finite extension of the theory .
iii) Let be recursive relation such that: y is a Gödel number of a proof of the wff1 of the theory and x is a Gödel number of this wff1. Then the relation is expressible in the theory by canonical Gödel encoding and really asserts provability in .
iv) Let be relation such that: y is a Gödel number of a proof of the wff of the set theory and x is a Gödel number of this wff1. Then the relation is expressible in the set theory by the following formula
(34)
v) The predicate really asserts provability in the set theory .
Remark 2.2.7. Note that the relation is expressible in the theory since a theory is a finite extension of the finite axiomatizable theory and therefore the predicate exists since any theory is recursively axiomatizable.
Remark 2.2.8. Note that a theory obviously is not recursively axiomatizable nevertheless Gödel encoding holds by Remark 2.2.1.
Theorem 2.2.8. Assume that: , where
.
Then truth predicate is expressible by using first order language by the following formula
(35)
where A is wff1.
Proof. Assume that:
(36)
It follows from (34) there exists such that and therefore by (36) we obtain
(37)
From (37) immediately by definitions one obtains (35).
Remark 2.2.9. Note that in considered case Tarski’s undefinability theorem (2.1.1) reads
(38)
where A is wff1.
Theorem 2.2.9. .
Proof. Assume that: . From (35) and (38) one obtains a contradiction and therefore by reductio ad absurdum it follows .
3. Derivation of the Inconsistent Provably Definable Set in Set Theory , and
3.1. Derivation of the Inconsistent Provably Definable Set in Set Theory
Definition 3.1.1. i) Let be a wff of . We will say that is a first order n-place open wff if contains free occurrences of the first order individual variables and quantifiers only over any first order individual variables .
ii) Let be the countable set of the all first order provable definable sets X, i.e. sets such that , where is a first order 1-place open wff that contains only first order variables (we will denote such wff for short by wff1), with all bound variables restricted to standard model , i.e.
(39)
or in a short notation
(40)
Notation 3.1.1. In this subsection we often write for short instead but this should not lead to a confusion.
Assumption 3.1.1. We assume now for simplicity but without loss of generality that
(41)
and therefore by definition of model one obtains .
Let be a predicate such that . Let be the countable set of the all sets such that
(42)
From (42) one obtains
(43)
But obviously (43) immediately gives a contradiction
(44)
Remark 3.1.1. Note that a contradiction (44) in fact is a contradiction inside for the reason that predicate is expressible by first order
language as predicate of (see subsection 1.2, Theorem 1.2.8 (ii)-(iii)) and therefore countable sets and are sets in the sense of the set theory .
Remark 3.1.2. Note that by using Gödel encoding the above stated contradiction can be shipped in special completion of , see subsection 1.2, Theorem 1.2.8.
Remark 3.1.3. i) Note that Tarski’s undefinability theorem cannot block the equivalence (43) since this theorem is no longer holds by Proposition 2.2.1. (Generalized Löbs Theorem).
ii) In additional note that: since Tarski’s undefinability theorem has been proved under the same assumption by reductio ad absurdum it follows again , 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 . Let be the set of the all sets of M provably definable in , and let where means “sentence A derivable in ”, or some appropriate modification thereof. We replace now formula (39) by the following formula
(45)
and we replace formula (42) by the following formula
(46)
Definition 3.1.2. We rewrite now (45) in the following equivalent form
(47)
where the countable set is defined by the following formula
(48)
Definition 3.1.3. Let be the countable set of the all sets such that
(49)
Remark 3.1.5. Note that since is a set definable by the first order 1-place open wff1:
(50)
From (49) and Remark 3.1.4 one obtains
(51)
But (51) immediately gives a contradiction
(52)
Remark 3.1.6. Note that contradiction (52) is a contradiction inside for the reason that the countable set is a set in the sense of the set theory .
In order to obtain a contradiction inside without any reference to Assumption 3.1.1 we introduce the following definitions.
Definition 3.1.4. We define now the countable set by the following formula
(53)
Definition 3.1.5. We choose now in the following form
(54)
Here is a canonical Gödel formula which says to us that there exists proof in of the formula A with Gödel number .
Remark 3.1.7. Note that the Definition 3.1.5 holds as definition of predicate really asserting provability of the first order sentence A in .
Definition 3.1.6. Using Definition 3.1.5, we replace now formula (48) by the following formula
(55)
Definition 3.1.7. Using Definition 3.1.5, we replace now formula (49) by the following formula
(56)
Definition 3.1.8. Using Proposition 2.1.1 and Remark 2.1.10 [6] , we replace now formula (53) by the following formula
(57)
Definition 3.1.9. Using Definitions 3.1.4-3.1.6, we define now the countable set by formula
(58)
Remark 3.1.8. Note that from the second order axiom schema of replacement (12) it follows directly that is a set in the sense of the set theory .
Definition 3.1.10. Using Definition 3.1.8 we replace now formula (56) by the following formula
(59)
Remark 3.1.9. Notice that the expression (60)
(60)
obviously is a well formed formula of and therefore a set is a set in the sense of .
Remark 3.1.10. Note that since is a set definable by 1-place open wff
(61)
Theorem 3.1.1. Set theory is inconsistent.
Proof. From (59) we obtain
(62)
a) Assume now that:
(63)
Then from (62) we obtain and
,
therefore and so
(64)
From (63)-(64) we obtain
and thus .
b) Assume now that
(65)
Then from (65) we obtain . From (65) and (62) we obtain , so , which immediately gives us a contradiction .
Definition 3.1.11. We choose now in the following form
(66)
or in the following equivalent form
(67)
similar to (46). Here is a Gödel formula which really asserts provability in of the formula A with Gödel number .
Remark 3.1.11. Notice that the Definition 3.1.12 with formula (66) holds as definition of predicate really asserting provability in .
Definition 3.1.12. Using Definition 3.1.11 with formula (66), we replace now formula (48) by the following formula
(68)
Definition 3.1.13. Using Definition 3.1.11 with formula (66), we replace now formula (49) by the following formula
(69)
Definition 3.1.14. Using Definition 3.1.11 with formula (66), we replace now formula (53) by the following formula
(70)
Definition 3.1.15. Using Definitions 3.1.12-3.1.16, we define now the countable set by formula
(71)
Remark 3.1.12. Note that from the axiom schema of replacement (12) it follows directly that is a set in the sense of the set theory .
Definition 3.1.16. Using Definition 3.1.15 we replace now formula (69) by the following formula
(72)
Remark 3.1.13. Notice that the expressions (73)
(73)
obviously are a well formed formula of and therefore collection is a set in the sense of .
Remark 3.1.14. Note that since is a set definable by 1-place open wff1
(74)
Theorem 3.1.2. Set theory is inconsistent.
Proof. From (72) we obtain
(75)
a) Assume now that:
(76)
Then from (75) we obtain and therefore , thus we obtain
(77)
From (76)-(77) we obtain and , thus and finally we obtain .
b) Assume now that
(78)
Then from (78) we obtain . From (78) and (75) we obtain , thus and which immediately gives us a contradiction .
3.2. Derivation of the Inconsistent Provably Definable Set in Set Theory ZFCst
Let be the countable set of all sets X such that , where is a 1-place open wff of ZFC i.e.,
(79)
Let be a predicate such that . Let be the countable set of the all sets such that
(80)
From (80) one obtains
(81)
But (81) immediately gives a contradiction
(82)
Remark 3.2.1. Note that a contradiction (82) is a contradiction inside
for the reason that predicate is expressible by using first order
language as predicate of (see subsection 4.1) and therefore countable sets and are sets in the sense of the set theory .
Remark 3.2.2. Note that by using Gödel encoding the above stated contradiction can be shipped in special completion of , see subsection 1.2, Theorem 1.2.2 (i).
Designation 3.2.1. i) Let be a standard model of ZFC and
ii) let be the theory ,
iii) let be the set of the all sets of provably definable in , and let , where means: “sentence A derivable in ”, or some appropriate modification thereof.
We replace now (79) by formula
(83)
and we replace (80) by formula
(84)
Assume that . Then, we have that: iff , which immediately gives us iff . But this is a contradiction, i.e., . We choose now in the following form
(85)
Here is a canonical Gödel formula which says to us that there exists proof in of the formula A with Gödel number .
Remark 3.2.3. Notice that Definition 3.2.6 holds as definition of predicate really asserting provability in .
Definition 3.2.1. We rewrite now (83) in the following equivalent form
(86)
where the countable collection is defined by the following formula
(87)
Definition 3.2.2. Let be the countable collection of the all sets such that
(88)
Remark 3.2.4. Note that since is a collection definable by 1-place open wff
(89)
Definition 3.2.3. By using formula (85) we rewrite now (86) in the following equivalent form
(90)
where the countable collection is defined by the following formula
(91)
Definition 3.2.4. Using formula (85), we replace now formula (88) by the following formula
(92)
Definition 3.2.5. Using Proposition 2.1.1 and Remark 2.2.2 [6] , we replace now formula (89) by the following formula
(93)
Definition 3.2.6. Using Definitions 3.2.3-3.2.5, we define now the countable set by formula
(94)
Remark 3.2.5. Note that from the axiom schema of replacement it follows directly that is a set in the sense of the set theory .
Definition 3.2.7. Using Definition 3.2.6 we replace now formula (92) by the following formula
(95)
Remark 3.2.6. Notice that the expression (96)
(96)
obviously is a well formed formula of and therefore collection is a set in the sense of .
Remark 3.2.7. Note that since is a collection definable by 1-place open wff
(97)
Theorem 3.2.1. Set theory is inconsistent.
Proof. From (95) we obtain
(98)
a) Assume now that:
(99)
Then from (98) we obtain and , therefore and so
(100)
From (99)-(100) we obtain and therefore .
b) Assume now that
(101)
Then from (101) we obtain . From (101) and (98) we obtain , so which immediately gives us a contradiction .
3.3. Derivation of the Inconsistent Provably Definable Set in ZFCNst
Designation 3.3.1. i) Let be a first order theory which contain usual postulates of Peano arithmetic [8] and recursive defining equations for every primitive recursive function as desired.
ii) Let be a nonstandard model of ZFC and let be a standard model of . We assume now that and denote such nonstandard model of ZFC by .
iii) Let be the theory .
iv) Let be the set of the all sets of provably definable in , and let where means “sentence A derivable in ”, or some appropriate modification thereof. We replace now (45) by formula
(102)
and we replace (46) by formula
(103)
Assume that . Then, we have that: iff , which immediately gives us iff . But this is a contradiction, i.e., . We choose now in the following form
(104)
Here is a canonical Gödel formula which says to us that there exists proof in of the formula A with Gödel number .
Remark 3.3.1. Notice that definition (104) holds as definition of predicate really asserting provability in .
Designation 3.3.2. i) Let be a Gödel number of given an expression u of .
ii) Let be the relation: y is the Gödel number of a wff of that contains free occurrences of the variable with Gödel number v [6] [10] .
iii) Let be a Gödel number of the following wff: , where , , .
iv) Let be a predicate asserting provability in .
Remark 3.3.2. Let be the countable collection of all sets X such that , where is a 1-place open wff i.e.,
(105)
We rewrite now (105) in the following form
(106)
Designation 3.3.3. Let be a Gödel number of the following wff: , where .
Remark 3.3.3. Let above by formula (103), i.e.,
(107)
We rewrite now (107) in the following form
(108)
Theorem 3.3.1. .
3.4. Generalized Tarski’s Undefinability Lemma
Remark 3.4.1. Remind that: i) if is a theory, let be the set of Godel numbers of theorems of [10] , ii) the property is said to be is expressible in by wff if the following properties are satisfied [10] :
a) if then , b) if then .
Remark 3.4.2. Notice it follows from (a) (b) that
.
Theorem 3.4.1. (Tarski’s undefinability Lemma) [10] . Let be a consistent theory with equality in the language in which the diagonal function D is representable and let be a Gödel number of given an expression u of . Then the property is not expressible in .
Proof. By the diagonalization lemma applied to there is a sentence such that: c) , where q is the Godel number of , i.e. .
Case 1. Suppose that , then . By (a), . But, from and (c), by biconditional elimination, one obtains . Hence is inconsistent, contradicting our hypothesis.
Case 2. Suppose that , then . By (b), . Hence, by (c) and biconditional elimination, . Thus, in either case a contradiction is reached.
Definition 3.4.1. If is a theory, let be the set of Godel numbers of theorems of and let be a Gödel number of given an expression u of . The property is said to be is a strongly expressible in by wff if the following properties are satisfied:
a) if then ,
b) if then .
Theorem 3.4.2. (Generalized Tarski’s undefinability Lemma). Let be a consistent theory with equality in the language in which the diagonal function D is representable and let be a Gödel number of given an expression u of . Then the property is not strongly expressible in .
Proof. By the diagonalization lemma applied to there is a sentence such that: c) , where q is the Godel number of , i.e. .
Case 1. Suppose that , then . By (a), . But, from and (c), by biconditional elimination, one obtains . Hence is inconsistent, contradicting our hypothesis.
Case 2. Suppose that , then . By (b), . Hence, by (c) and biconditional elimination, . Thus, in either case a contradiction is reached.
Remark 3.4.3. Notice that Tarski’s undefinability theorem cannot blocking the biconditionals
(109)
see Subsection 2.2.
3.5. Generalized Tarski’s Undefinability Theorem
Remark 3.5.1. I) Let be the theory .
In addition under assumption , we establish a countable sequence of the consistent extensions of the theory such that:
i) , where
ii) is a finite consistent extension of ,
iii) ,
iv) proves the all sentences of , which is valid in M, i.e., , see see Subsection 4.1, Proposition 4.1.1.
II) Let be .
In addition under assumption , we establish a countable sequence of the consistent extensions of the theory such that:
i) , where
ii) is a finite consistent extension of ,
iii) ,
iv) proves the all sentences of , which valid in , i.e., , see Subsection 4.1, Proposition 4.1.1.
III) Let be .
In addition under assumption , we establish a countable sequence of the consistent extensions of the theory such that:
i) , where
ii) is a finite consistent extension of ,
iii)
iv) proves the all sentences of , which valid in , i.e., , see Subsection 4.1, Proposition 4.1.1.
Remark 3.5.2. I) Let be the set of the all sets of M provably definable in ,
(110)
and let where means sentence A derivable in . Then we have that iff , which immediately gives us iff . We choose now in the following form
(111)
Here is a canonical Gödel formulae which says to us that there exists proof in of the formula A with Gödel number .
II) Let be the set of the all sets of provably definable in ,
(112)
and let where means sentence A derivable in .
Then we have that iff , which immediately gives us iff . We choose now in the following form
(113)
Here is a canonical Gödel formulae which says to us that there exists proof in of the formula A with Gödel number .
III) Let be the set of the all sets of provably definable in ,
(114)
and let where means sentence A derivable in . Then we have that iff , which immediately gives us iff .
We choose now in the following form
(115)
Here is a canonical Gödel formulae which says to us that there exists proof in of the formula A with Gödel number .
Remark 3.5.3. Notice that definitions (111), (113) and (115) hold as definitions of predicates really asserting provability in and correspondingly.
Remark 3.5.4. Of course the all theories are inconsistent, see subsection 4.1.
Remark 3.5.5. I) Let be the set of the all sets of M provably definable in ,
(116)
and let , where means “sentence A derivable in ”. Then, we have that iff , which immediately gives us iff . We choose now in the following form
(117)
II) Let be the set of the all sets of provably definable in ,
(118)
and let be the set , where means “sentence A derivable in ”. Then, we have that iff , which immediately gives us iff . We choose now in the following form
(119)
III) Let be the set of the all sets of provably definable in ,
(120)
and let be the set where means “sentence A derivable in ”. Then, we have that iff , which immediately gives us iff . We choose now in the following form
(121)
Remark 3.5.6. Notice that definitions (117), (119) and (121) hold as definitions of a predicate really asserting provability in and correspondingly.
Remark 3.5.7. Of course all the theories and are inconsistent, see subsection 4.1.
Remark 3.5.8. Notice that under naive consideration the set and can be defined directly using a truth predicate, which of course is not available in the language of (but iff is consistent) by well-known Tarski’s undefinability theorem [10] .
Theorem 3.5.1. Tarski’s undefinability theorem: I) Let be first order theory with formal language , which includes negation and has a Gödel numbering such that for every -formula there is a formula B such that holds. Assume that has a standard model and where
(122)
Let be the set of Gödel numbers of -sentences true in . Then there is no -formula (truth predicate) which defines . That is, there is no -formula such that for every -formula A,
(123)
holds.
II) Let be second order theory with Henkin semantics and formal language , which includes negation and has a Gödel numbering such that for every -formula there is a formula B such that holds.
Assume that has a standard model and , where
(124)
Let be the set of Gödel numbers of the all -sentences true in M. Then there is no -formula (truth predicate) which defines . That is, there is no -formula such that for every -formula 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 . In particular, if A is a sentence of then holds in iff A is true in . Hence for all A, the Tarski T-sentence is true in . But the diagonal lemma yields a counterexample to this equivalence, by giving a “Liar” sentence S such that holds in . Thus no -formula can define .
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
(126)
see Remark 3.5.14 below.
Remark 3.5.13. I) We define again the set but now by using generalized truth predicate such that
(127)
holds.
II) We define the set using generalized truth predicate such that
(128)
holds. Thus in contrast with naive definition of the sets and there is no any problem which arises from Tarski’s undefinability theorem.
III) We define a set using generalized truth predicate such that
(129)
holds. Thus in contrast with naive definition of the sets and there is no any problem which arises from Tarski’s undefinability theorem.
Remark 3.5.14. In order to prove that set theory is inconsistent without any reference to the set , notice that by the properties of the extension it follows that definition given by formula (127) is correct, i.e., for every -formula such that the following equivalence holds.
Theorem 3.5.2. (Generalized Tarski’s undefinability theorem) (see subsection 4.2, Proposition 4.2.1). Let 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 such that for every -formula there is a formula B such that the equivalence holds. Assume that has a standard Model . Then there is no -formula , such that for every -formula A such that , the following equivalence holds
(130)
Theorem 3.5.3. i) Set theory is inconsistent;
ii) Set theory is inconsistent; iii) Set theory is inconsistent; (see subsection 4.2, Proposition 4.2.2).
Proof. i) Notice that by the properties of the extension of the theory it follows that
(131)
Therefore formula (127) gives generalized “truth predicate” for the set theory . By Theorem 3.5.2 one obtains a contradiction.
ii) Notice that by the properties of the extension of the theory it follows that
(132)
Therefore formula (128) gives generalized “truth predicate” for the set theory . By Theorem 3.5.2 one obtains a contradiction.
iii) Notice that by the properties of the extension of the theory it follows that
(133)
Therefore (129) gives generalized “truth predicate” for the set theory . By Theorem 3.5.2 one obtains a contradiction.
3.6. Avoiding the Contradictions from Set Theory , and Set Theory Using Quinean Approach
In order to avoid difficulties mentioned above we use well known Quinean approach [19] .
3.6.1. Quinean Set Theory NF
Remind that the primitive predicates of Russellian unramified typed set theory (TST), a streamlined version of the theory of types, are equality = and membership . TST has a linear hierarchy of types: type 0 consists of individuals otherwise undescribed. For each (meta-) natural number n, type objects are sets of type n objects; sets of type n have members of type . Objects connected by identity must have the same type. The following two atomic formulas succinctly describe the typing rules: and .
The axioms of TST are:
Extensionality: sets of the same (positive) type with the same members are equal;
Axiom schema of comprehension:
If is a formula, then the set exists i.e., given any formula , the formula
(134)
is an axiom where represents the set and is not free in .
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 . It avoids Russell’s paradox by only allowing stratifiable formulas in the axiom of comprehension. For instance is a stratifiable formula, but 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 3c3 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 appearing in satisfies and any atomic formula appearing in satisfies .
Quinean Set Theory NF
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] .
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:
exists for each stratified formula .
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] . 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 is not an axiom of NF, because cannot be stratified.
3.6.2. SET Theory , and Set Theory with Stratified Axiom Schema of Replacement
The stratified axiom schema of replacement asserts that the image of a set under any function definable by stratified formula of the theory will also fall inside a set.
Stratified Axiom schema of replacement:
Let be any stratified formula in the language of whose free variables are among , so that in particular B is not free in . Then
(135)
i.e., if the relation represents a definable function f, A represents its domain, and is a set for every , then the range of f is a subset of some set B.
Stratified Axiom schema of separation:
Let be any stratified formula in the language of whose free variables are among , so that in particular B is not free in . Then
(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 violated any contradictions (82), (126), etc. mentioned above. The existence of the countable Russell sets and is impossible, because cannot be stratified.
4. Generalized Löbs Theorem
4.1. Generalized Löbs Theorem. Second-Order Theories with Henkin Semantics
Remark 4.1.1. In this section we use second-order arithmetic with Henkin semantics. Notice that any standard model of second-order arithmetic consisting of a set of unusual 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 of subsets of , which is the range of the set variables. Omitting D produces a model of the first order Peano arithmetic.
When is the full power set of , the model 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, see Section 3.
Let 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 and that contains . We assume throughout this paper that formal second order theory has an ω-model . The sense in which is contained in is better exemplified than explained: if is a formal system of a second order arithmetic and is, say, , then contains in the sense that there is a well-known embedding, or interpretation, of S in . Since encoding is to take place in , 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 ). will also have certain function symbols to be described shortly. To each formula, , of the language of is assigned a closed term, , called the code of [19] . We note that if is a formula with free variable x, then is a closed term encoding the formula with x viewed as a syntactic object and not as a parameter. Corresponding to the logical connectives and quantifiers are the function symbols, , , etc., such that for all first order formulae , etc. Of particular importance is the substitution operator, represented by the function symbol . For formulae , terms t with codes :
(137)
It well known that one can also encode derivations and have a binary relation (read “x proves y” or “x is a proof of y”) such that for closed iff is the code of a derivation in of the formula with code . It follows that
(138)
for some closed term t. Thus we can define
(139)
and therefore we obtain a predicate asserting provability.
Remark 4.1.2. I) We note that it is not always the case that:
(140)
unless is fairly sound, e.g. this is the case when and replaced by and correspondingly (see Designation 4.1.1 below).
II) Notice that it is always the case that:
(141)
i.e. that is the case when predicate :
(142)
really asserting provability.
It well known that the above encoding can be carried out in such a way that the following important conditions and are meeting for all sentences:
(143)
Conditions and are called the Derivability Conditions.
Remark 4.1.3. From (141)-(142) it follows that
(144)
Conditions and are called a Strong Derivability Conditions.
Definition 4.1.1. Let be well formed formula (wff) of . Then wff is called -sentence iff it has no free variables.
Designation 4.1.1 i) Assume that a theory has an ω-model and
is a -sentence, then: (we will write instead )
is a -sentence with all quantifiers relativized to ω-model [11] and is a theory relativized to model , i.e., any -sentence has the form for some -sentence .
ii) Assume that a theory has a standard model and is a -sentence, then:
(we will write instead ) is a -sentence with
all quantifiers relativized to a standard model , and is a theory relativized to model , i.e., any -sentence has a form for some -sentence .
iii) Assume that a theory has a non-standard model and is a -sentence, then:
(we will write instead ) is a -sentence with
all quantifiers relativized to non-standard model , and is a theory relativized to model , i.e., any -sentence has a form for some -sentence .
iv) Assume that a theory has a model and is a -sentence, then: is a -sentence with all quantifiers relativized to model , and is a theory relativized to model , i.e. any -sentence has a form for some -sentence .
Designation 4.1.2. i) Assume that a theory with a language has an ω-model and there exists -sentence such that: a) expressible by language and
b) asserts that has a model ; we denote such -sentence by .
ii) Assume that a theory with a language has a non-standard model and there exists -sentence such that: a) expressible by language and
b) asserts that has a non-standard model ; we denote such -sentence by .
iii) Assume that a theory with a language has an model and there exists -sentence such that: a) expressible by language and
b) asserts that has a model ; we denote such -sentence by .
Remark 4.1.4. We emphasize that: i) it is well known that there exists a ZFC-sentence [8] ,
ii) obviously there exists a -sentence and there exists a -sentence .
Designation 4.1.3. Assume that . Let be the formula:
(145)
and where is a closed term.
Lemma 4.1.1. I) Assume that: i) a theory is recursively axiomatizable.
ii) ,
iii) and
iv) , where is a closed formula.
Then .
II) Assume that: i) a theory is recursively axiomatizable.
ii)
iii) and
iv) , where is a closed formula.
Then .
Proof. I) Let be the formula:
(146)
where is a closed term. From (i)-(ii) it follows that theory is consistent. We note that for any closed . Suppose that , then (iii) gives
(147)
From (139) and (147) we obtain
(148)
But the formula (146) contradicts the formula (148). Therefore .
Remark 4.1.5. In additional note that under the following conditions:
i) a theory is recursively axiomatizable,
ii) , and
iii) predicate really asserts provability, one obtains
(149)
and therefore by reductio ad absurdum again one obtains .
II) Let be the formula:
(150)
This case is trivial because formula by the Strong Derivability Condition , see formulae (144), really asserts provability of the -sentence . But this is a contradiction.
Lemma 4.1.2. I) Assume that: i) a theory is recursively axiomatizable.
ii) ,
iii) and
iv) , where is a closed formula. Then ,
II) Assume that: i) a theory is recursively axiomatizable.
ii)
iii) and
iv) ,
where is a closed formula. Then .
Proof. Similarly as Lemma 4.1.1 above.
Example 4.1.1. i) Let be Peano arithmetic and .
Assume that: i)
ii) where is a model of .
Then obviously since and therefore by Lemma 4.1.1 .
ii) Let and let be a theory and . Then obviously
(151)
and therefore
(152)
and
(153)
However by Löbs theorem
(154)
iii) Let and . Then obviously since and therefore by Lemma 4.1.1 we obtain.
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