There is No Standard Model of ZFC and ZFC 2

Transcription

1 Journal of Advances in Mathematics and Computer Science 6: 1-0, 018; Article no.jamcs ISSN: Past name: British Journal of Mathematics & Computer Science, Past ISSN: There is No Standard Model of ZFC and ZFC Jaykov Foukzon 1 and Elena Men kova 1 Israel Institute of Technology, Haifa, Israel. All-Russian Research Institute for Optical and Physical Measurements, Moscow, Russia. Authors contributions This work was carried out in collaboration between both authors. Author JF designed the study, carried out the model analysis and wrote the first draft of the manuscript. Author EM wrote Section 3 of the manuscript and managed the literature searches. Both authors read and approved the final manuscript. Article Information DOI: /JAMCS/018/38773 Editors: 1 Dariusz Jacek Jakbczak, Assistant Professor, Chair of Computer Science and Management in this Department, Technical University of Koszalin, Poland. Reviewers: 1 iaolan Liu, Sichuan University of Science Engineering, China. Derya Doan Durgun, Manisa Celal Bayar University, Turkey. Complete Peer review History: Received: 4 th November 017 Accepted: 8 th January 018 Original Research Article Published: 30 th January 018 Abstract In this paper we view the first order set theory ZF C under the canonical first order semantics and the second order set theory ZF C under the Henkin semantics. Main results are: i Let Mst ZF C be a standard model of ZF C, then ConZF C + Mst ZF C. ii Let M ZF C st be a standard model of ZF C with Henkin semantics, then ConZF C + M ZF C st. iii Let k be inaccessible cardinal then ConZF C + κ. In order to obtain the statements i and ii examples of the inconsistent countable set in a set theory ZF C + Mst ZF C and in a set theory ZF C + M ZF C st were derived. It is widely believed that ZF C + Mst ZF C and ZF C + M ZF C st are inconsistent, i.e. ZF C and ZF C have a standard models. Unfortunately this belief is wrong. Keywords: Gödel encoding; Russell s paradox; standard model; Henkin semantics; inaccessible cardinal. 010 Mathematics Subject Classification: 53C5; 83C05; 57N16. *Corresponding author: jaykovfoukzon@list.ru

2 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs Introduction 1.1 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 the Russell s paradox. In 1908, two ways of avoiding the paradox were proposed, the Russell s type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelo s axioms went well beyond Frege s axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo Fraenkel set theory ZF C. But how do we know that ZF C 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 ZF C which is significantly stronger than ZF C harbors an unsuspected contradiction; it is widely believed that if ZF C and ZF C were inconsistent, that fact would have been uncovered by now. This much is certain ZF C and ZF C is immune to the classic paradoxes of naive set theory: the Russell s paradox, the Burali-Forti paradox, and Cantor s paradox. Remark Note that in this paper we view i the first order set theory ZF C under the canonical first order semantics, ii the second order set theory ZF C under the Henkin semantics [], [3], [4], [5], [6]. Remark 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 [], [6]. The deductive calculus DED of second order logic is based on rules and axioms which guarantee that the quantifiers range at least over definable subsets [6]. As to the semantics, there are two tipes of models: i Suppose U is an ordinary first-order structure and S is a set of subsets of the domain A of U. The main idea is that the set-variables range over S, i.e. U, S = Φ S S S [ U, S = Φ S]. We call U, S the Henkin model, if U, S satisfies the axioms of DED and truth in U, S is preserved by the rules of DED. 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 U, S where S is the set of all subsets of A. We call these full models. We call this semantics of second-order logic the full semantics and secondorder logic with the full semantics the full second-order logic. Remark We emphasize that the following facts are the main features of second-order logic: 1.The Completeness Theorem: A sentence is provable in DED if and only if it holds in all Henkin models [], [6]..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 DED 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.

3 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs 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 Z such that the semiring N of natural numbers is the only full model up to isomorphism of Z. 8. There is a finite second-order axiom system RCF such that the field R of real numbers is the only up to isomorphism full model of RCF. Remark For let second-order ZF C be, as usual, the theory that results obtained from ZF C when the axiom schema of replacement is replaced by its second-order universal closure, i.e. [F unc = u r [r s s u s, r ]], where is a second-order variable, and where F unc abbreviates is a functional relation, see [7]. Designation We will denote i by ZF C Hs set theory ZF C with the Henkin semantics, ii by ZF C Hs set theory ZF C Hs iii by ZF C st set theory + M ZF CHs st, and ZF C + Mst ZF C, where Mst T h is a standard model of the theory T h. Axiom M ZF C. [8]. There is a set M ZF C and a binary relation ϵ M ZF C M ZF C which makes M ZF C a model for ZF C. Remark i We emphasize that it is well known that axiom M ZF C a single statement in ZF C see [8], Ch.II, section 7. We denote this statement throught all this paper by symbol Con ZF C;M ZF C. The completness theorem says that M ZF C Con ZF C. ii Obviously there exists a single statement in ZF C Hs ZF CHs such that M Con ZF C Hs. We denote this statement throught all this paper by symbol Con ZF C Hs ZF CHs ;M and there exists a single statement M ZHs in Z Hs. We denote this statement throught all this paper by symbol Con Z Hs ;M ZHs. Axiom Mst ZF C. [[8]]. There is a set Mst ZF C such that if R is x, y x y x M ZF C st then M ZF C st is a model for ZF C under the relation R. Definition [8].The model Mst ZF C used is merely the standard - relation. } y Mst ZF C, and M ZHs st is called a standard model since the relation Remark [8]. Note that axiom M ZF C doesn t imply axiom M ZF C st. Remark Note that in order to deduce: i ConZF C Hs from ConZF C Hs, and ii ConZF C from ConZF C, by using Gödel encoding, one needs something more than the ZF CHs consistency of ZF C Hs, e.g., that ZF C Hs has an omega-model Mω or an standard model ZF CHs Mst i.e., a model in which the integers are the standard integers. To put it another way, why should we believe a statement just because there s a ZF C Hs -proof of it? It s clear that if ZF C Hs is inconsistent, then we won t believe ZF C Hs -proofs. What s slightly more subtle is 3

4 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs that the mere consistency of ZF C isn t quite enough to get us to believe arithmetical theorems of ZF C Hs ; we must also believe that these arithmetical theorems are asserting something about the standard naturals. It is conceivable that ZF C Hs might be consistent but that the only ZF CHs nonstandard models MNst it has are those in which the integers are nonstandard, in which case we might not believe an arithmetical statement such as ZF C Hs is inconsistent even if there is a ZF C Hs -proof of it. Derivation of the Inconsistent Definable Set in Set Theory ZF C Hs and in Set Theory ZF C st.1 Derivation of the inconsistent definable set in set theory ZF C Hs We assume now that Con Z Hs Designation.1.1. Let Γ Hs ;M ZHs st. be the collection of the all 1-place open wff of the set theory ZF C Hs. Definition.1.1. Let Ψ 1, Ψ be 1-place open wff s of the set theory ZF C Hs. i We define now the equivalence relation Γ Hs ii A subset Λ Hs Γ Hs P si 1 Ψ [Ψ 1 Ψ ].1.1 of Γ Hs never for Ψ 1 in Λ Hs such that Ψ 1 Ψ holds for all Ψ 1 and Ψ in Λ Hs, and and Ψ outside Λ Hs, is called an equivalence class of Γ Hs. iii The collection of all possible equivalence classes of Γ Hs Γ Hs / [Ψ ] Hs Ψ Γ Hs iv For any Ψ Γ Hs let [Ψ ] Hs by by, denoted Γ Hs / }..1. } Φ Γ Hs Ψ Φ denotes the equivalence class to which Ψ belongs. All elements of Γ Hs are also elements of the same equivalence class. equivalent to each other Definition.1.. [9]. Let T h be any theory in the recursive language L Th L PA, where L PA is a language of Peano arithmetic. We say that a number-theoretic relation R x 1,..., x n of n arguments is expressible in T h if and only if there is a wff R x 1,..., x n of T h with the free variables x 1,..., x n such that, for any natural numbers k 1,..., k n, the following hold: i If R k 1,..., k n is true, then T h R k1,..., k n ; ii If R k 1,..., k n is false, then T h R k 1,..., k n. Designation.1.. i Let g ZF C Hs theory ZF C Hs ZF C Hs + M ZF CHs st. u be a Gödel number of given an expression u of the set ii Let Fr Hs y, v be the relation : y is the Gödel number of a wff of the set theory ZF C Hs that contains free occurrences of the variable with Gödel number v [9]. iii Note that the relation Fr Hs y, v is expressible in ZF C Hs by a wff Fr Hs y, v 4

5 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs iv Note that for any y, v N by definition of the relation Fr Hs y, v follows that [ ] Fr Hs y, v!ψ g Hs ZF C Ψ = y g Hs ZF C =,.1.3 where Ψ is a unique wff of ZF C Hs which contains free occurrences of the variable with Gödel number v. We denote a unique wff Ψ defined by using equivalence 1..3 by symbol Ψ y,, i.e. Fr Hs y, v!ψ y, [ ] g Hs ZF C Ψ y, = y g Hs ZF C =,.1.4 v Let Hs y, v, 1 be a Gödel number of the following wff:! [Ψ Y = ], where g ZF C Hs Ψ = y, g ZF C Hs =, g ZF C Hs Y = 1. Definition.1.3. Let Γ Hs be the countable collection of the all 1-place open wff s of the set theoryzf C Hs that contains free occurrences of the variable. Definition.1.4. Let g ZF C Hs =. Let Γ Hs be a set of the all Gödel numbers of the 1-place open wff s of the set theoryzf C Hs that contains free occurrences of the variable with Gödel number v, i.e. Γ Hs = y N y, Fr Hs y, v }.1.5 or in the following equivalent form: y y N [y Γ y N ] Hs Fr y, v..1.6 Remark.1.1. Note that from the axiom of separation it follows directly that Γ Hs sense of the set theory ZF C Hs. Definition.1.5. i We define now the equivalence relation Γ Hs Γ Hs.1.7 in the sense of the set theory ZF C Hs by y 1 y [Ψ y1, Ψ y, ]..1.8 is a set in the Note that from the axiom of separation it follows directly that the equivalence relation is a relation in the sense of the set theory ZF C Hs. ii A subset Λ Hs and y outside Λ Hs of Γ Hs, is an equivalence class of Γ Hs such that y 1 y holds for all y 1 and y 1 in Λ Hs.,and never for y 1 in Λ Hs iii For any y Γ Hs let [y] Hs z Γ Hs y z } denote the equivalence class to which y belongs. All elements of Γ Hs equivalent to each other are also elements of the same equivalence class. iv The collection of all possible equivalence classes of Γ Hs Γ Hs / } [y] Hs y Γ Hs..1.9 by, denoted Γ Hs / Remark.1.. Note that from the axiom of separation it follows directly that Γ Hs / is a set in the sense of the set theory ZF C Hs. Definition.1.6. Let I Hs be the countable collection of the all sets definable by 1-place open wff of the set theory ZF C Hs, i.e. 5

6 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs Y Y I Hs Ψ [ [Ψ ] Hs Γ Hs / [! [Ψ Y = ]] Definition.1.7. We rewrite now.1.10 in the following equivalent form Y Y I Hs Ψ [ [Ψ ] Hs Γ Hs / Y =,.1.11 where the countable collection Γ Hs / is defined by Ψ [Ψ ] Γ Hs / [ [Ψ ] Γ Hs /!Ψ Definition.1.8. Let R Hs be the countable collection of the all sets such that [ I Hs R Hs / ] Remark.1.3. Note that R Hs I Hs since R Hs is a collection definable by 1-place open wff is definable by formula Ψ Z, I Hs I Hs [ Z / ]. From.1.13 one obtains R Hs R Hs R Hs / R Hs But.1.14 gives a contradiction R Hs R Hs R Hs / R Hs However contradiction.1.15 it is not a contradiction inside ZF C Hs for the reason that the countable collection I Hs is not a set in the sense of the set theory ZF C Hs. In order to obtain a contradiction inside ZF C Hs we introduce the following definitions. Definition.1.9.We define now the countable set Γ Hs / by y [y] Hs Γ Hs / } [y] Hs Γ Hs Hs / Fr y, v [!Ψ y, ] Remark.1.4. Note that from the axiom of separation it follows directly that Γ / is a set in the sense of the set theory ZF C Hs. Definition We define now the countable set I Hs by formula [ [y] Y Y I Hs y Γ Hs / g Hs ZF C = Y = Note that from the axiom schema of replacement it follows directly that I Hs is a set in the sense of the set theory ZF C Hs. Definition We define now the countable set R Hs by formula [ I Hs R Hs / ] Note that from the axiom schema of separation it follows directly that R Hs is a set in the sense of the set theory ZF C Hs. Remark.1.5. Note that R Hs I Hs since R Hs is definable by the following formula Ψ Z I Hs [ Z / ] Theorem.1.1. Set theory ZF C Hs is inconsistent. Proof. From.1.18 and Remark.1.5 we obtain R Hs R Hs R Hs / R Hs from which 6

7 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs immediately one obtains a contradiction R Hs R Hs R Hs / R Hs Derivation of the inconsistent definable set in set theory ZF C st Designation..1. i Let g ZF Cst u be a Gödel number of given an expression u of the set theory ZF C st ZF C + M ZF C st. ii Let Fr sty, v be the relation : y is the Gödel number of a wff of the set theoryzf C st that contains free occurrences of the variable with Gödel number v [9]. iii Note that the relation Fr sty, v is expressible in ZF C st by a wff Fr sty, v. iv Note that for any y, v N by definition of the relation Fr sty, v follows that Fr sty, v!ψ [g ZF Cst Ψ = y g ZF Cst = ],..1 where Ψ is a unique wff of ZF C st which contains free occurrences of the variable with Gödel number v. We denote a unique wff Ψ defined by using equivalence..1 by symbol Ψ y,, i.e. Fr st y, v!ψ y, [g ZF Cst Ψ y, = y g ZF Cst = ],.. v Let st y, v, 1 be a Gödel number of the following wff:! [Ψ Y = ], where g ZF Cst Ψ = y, g ZF Cst =, g ZF Cst Y = 1.6 in section, see Remark. and Designation.3, see also [8]-[9]. Definition..1. Let Γ st be the countable collection of the all 1-place open wff s of the set theory ZF C st that contains free occurrences of the variable. Definition... Let g ZF Cst =. Let Γ st be a set of the all Gödel numbers of the 1-place open wff s of the set theory ZF C st that contains free occurrences of the variable with Gödel number v, i.e. Γ st = y N y, Fr sty, v},..3 or in the following equivalent form: [ y y N y Γ st y N Fr ] sty, v. Remark..1. Note that from the axiom of separation it follows directly that Γ st sense of the set theory ZF C st. is a set in the Definition..3. i We define now the equivalence relation Γ st Γ st by Ψ 1 Ψ [Ψ 1 Ψ ]..4 ii A subcollection Λ st of Γ st such that Ψ 1 Ψ holds for all Ψ 1 and Ψ in Λ st, and never for Ψ 1 in Λ st and Ψ outside Λ st, is an equivalence class of Γ st. iii For any Ψ Γ st let [Ψ ] st Φ Γ st Ψ Φ } denote the equivalence class to which Ψ belongs. All elements of Γ st equivalent to each other are also elements of the same equivalence class. iv The collection of all possible equivalence classes of Γ st by, denoted Γ st / Γ st / [Ψ ] st Ψ Γ st }...5 7

8 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs Definition..4. i We define now the equivalence relation Γ st the set theory ZF C st by Γ st in the sense of y 1 y [Ψ y1, Ψ y, ]..6 Note that from the axiom of separation it follows directly that the equivalence relation is a relation in the sense of the set theory ZF C st. ii A subset Λ st of Γ st such that y 1 y holds for all y 1 and y 1 in Λ st,and never for y 1 in Λ st and y outside Λ st, is an equivalence class of Γ st. iii For any y Γ st let [y] st z Γ st y z } denote the equivalence class to which y belongs. All elements of Γ st equivalent to each other are also elements of the same equivalence class. iv The collection of all possible equivalence classes of Γ st by, denoted Γ st / Γ st / } [y] st y Γ st...7 Remark... Note that from the axiom of separation it follows directly that Γ st / is a set in the sense of the set theory ZF C st. Definition..5. Let I st be the countable collection of all sets definable by 1-place open wff of the set theory ZF C st, i.e. Y Y I st Ψ [ [Ψ ] st Γ st / [! [Ψ Y = ]]...8 Definition..6. We rewrite now..8 in the following equivalent form Y Y I st Ψ [ [Ψ ] st Γ st / Y =,..9 where the countable collection Γ st / is defined by Ψ [Ψ ] st Γ st / [ [Ψ ] st Γ st /!Ψ Definition..7. Let R st be the countable collection of the all sets such that I st [ R st / ] Remark..3. Note that R st I st since R st is a collection definable by 1-place open wff is definable by formula Ψ Z, I st I st [ Z / ]. From..11 and Remark..3 one obtains directly R st R st R st / R st...1 But..1 immediately gives a contradiction R st R st R st / R st...13 However contradiction..13 it is not a true contradiction inside ZF C st for the reason that the countable collection I st is not a set in the sense of the set theory ZF C st. In order to obtain a true contradiction inside ZF C st we introduce the following definitions. Definition..8. We define now the countable set Γ st / by formula y [y] st Γ st / } [y] st Γ st / Frsty, v [!Ψ y, ]...14 Remark..4. Note that from the axiom of separation it follows directly that Γ st / is a set in the sense of the set theory ZF C st. 8

9 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs Definition..9. We define now the countable set I st by formula Y Y I st y [ [y] st Γ st / gzf Cst = Y =...15 Note that from the axiom schema of replacement it follows directly that I st is a set in the sense of the set theory ZF C st. Definition..10. We define now the countable set R st by formula I st [ R st / ]...16 Note that from the axiom schema of separation it follows directly that R st is a set in the sense of the set theory ZF C st. Remark..5. Note that R st I st since R st is definable by the following formula Ψ Z I st [ Z / ]...17 Theorem..1. [10]. Set theory ZF C st is inconsistent. Proof. From..17 and Remark..5 we obtain R st R st R st / R st from which immediately one obtains a contradiction R st R st R st / R st...18 Remark..6.Theorem..1 originally was proved in papers [10], [11], [1] by using another essentially complicated approach..3 Derivation of the inconsistent definable set in ZF C Nst Definition.3.1. Let P A be a first order theory which contain usual postulates of Peano arithmetic [9] and recursive defining equations for every primitive recursive function as desired. So for any n + 1-place function f defined by primitive recursion over any n-place base function g and n + - place iteration function h there would be the defining equations: i f 0, y 1,..., y n = g y 1,..., y n, ii f x + 1, y 1,..., y n = h x, f x, y 1,..., y n, y 1,..., y n. Designation.3.1. i Let MNst ZF C model of P A. We assume [ now ] that Mst P A P A. theory ZF C by M ZF C Nst ii Let ZF C Nst be the theory be a nonstandard model of ZF C and let M P A st M ZF C Nst ZF C Nst = ZF C + M ZF C Nst be a standard and denote such nonstandard model of the set [ P A ]. Designation.3.. i Let g ZF CNst [ u ] be a Gödel number of given an expression u of the set theory ZF C Nst ZF C + MNst ZF C P A. ii Let Fr Nsty, v be the relation : y is the Gödel number of a wff of the set theory ZF C Nst that contains free occurrences of the variable with Gödel number v [9]. iii Note that the relation Fr Nst y, v is expressible in ZF C Nst by a wff Fr Nst y, v. iv Note that for any y, v N by definition of the relation Fr Nst y, v follows that Fr Nst y, v!ψ [g ZF CNst Ψ = y g ZF CNst = ],.3.1 where Ψ is a unique wff of ZF C st which contains free occurrences of the variable with Gödel number v. We denote a unique wff Ψ defined by using equivalence.3.1 by symbol Ψ y,, 9

10 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs i.e. Fr Nst y, v!ψ y, [g ZF CNst Ψ y, = y g ZF CNst = ]..3. v Let Nst y, v, 1 be a Gödel number of the following wff:! [Ψ Y = ], where g ZF CNst Ψ = y, g ZF CNst =, g ZF CNst Y = 1. Definition.3.. Let Γ Nst be the countable collection of the all 1-place open wff s of the set theory ZF C Nst that contains free occurrences of the variable. Definition.3.3. Let g ZF CNst =.Let Γ Nst be a set of the all Gödel numbers of the 1-place open wff s of the set theory ZF C Nst that contains free occurrences of the variable with Gödel number v, i.e. Γ Nst = y N y, Fr Nst y, v},.3.3 or in the following equivalent form [ y y N y Γ Nst y N Fr ] Nst y, v. Remark.3.1. Note that from the axiom of separation it follows directly that Γ st sense of the set theory ZF C Nst. Definition.3.4. i We define now the equivalence relation Γ Nst Γ Nst Ψ 1 Ψ [Ψ 1 Ψ ].3.4 is a set in the ii A subcollection Λ st of Γ st such that Ψ 1 Ψ holds for all Ψ 1 and Ψ in Λ st, and never for Ψ 1 in Λ Nst and Ψ outside Λ Nst, is an equivalence class of Γ Nst. iii For any Ψ Γ Nst let [Ψ ] Nst Φ Γ Nst } Ψ Φ denote the equivalence class to which Ψ belongs. All elements of Γ st equivalent to each other are also elements of the same equivalence class. iv The collection of all possible equivalence classes of Γ Nst Γ Nst / [Ψ ] Nst Ψ Γ Nst by, denoted Γ Nst }..3.5 Definition.3.5. i We define now the equivalence relation Γ Nst of the set theory ZF C Nst by y 1 y [Ψ y1, Ψ y, ].3.6 / Γ Nst by in the sense Note that from the axiom of separation it follows directly that the equivalence relation is a relation in the sense of the set theory ZF C Nst. ii A subset Λ Nst Λ Nst of Γ Nst and y outside Λ Nst iii For any y Γ Nst belongs. All elements of Γ Nst class. such that y 1 y holds for all y 1 and y 1 in Λ Nst, is an equivalence class of Γ Nst let [y] Nst z Γ Nst. iv The collection of all possible equivalence classes of Γ Nst, and never for y 1 in y z } denote the equivalence class to which y equivalent to each other are also elements of the same equivalence by, denoted Γ Nst / 10

11 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs Γ Nst / } [y] Nst y Γ Nst..3.7 Remark.3.. Note that from the axiom of separation it follows directly that Γ Nst / is a set in the sense of the set theory ZF C Nst. Definition.3.6. Let I Nst be the countable collection of the all sets definable by 1-place open wff of the set theory ZF C Nst, i.e. Y Y I Nst Ψ [ [Ψ ] Nst Γ Nst / [! [Ψ Y = ]]..3.8 Definition.3.7. We rewrite now.3.8 in the following equivalent form Y Y I Nst Ψ [ [Ψ ] Nst Γ Nst / Y =,.3.9 where the countable collection Γ Nst / is defined by formula Ψ [Ψ ] Nst Γ Nst / [ [Ψ ] Nst Γ Nst /!Ψ Definition.3.8. Let R Nst be the countable collection of the all sets such that I Nst [ R Nst / ] Remark.3.3. Note that R Nst I Nst since R Nst is a collection definable by 1-place open wff is definable by formula Ψ Z, I Nst I Nst [ Z / ]. From.3.11 one obtains R Nst R Nst R Nst / R Nst..3.1 But.3.1 gives a contradiction R Nst R Nst R Nst / R Nst However a contradiction.3.13 it is not a true contradiction inside ZF C Nst for the reason that the countable collection I Nst is not a set in the sense of the set theory ZF C Nst. In order to obtain a true contradiction inside ZF C Nst we introduce the following definitions. Definition.3.9.We define now the countable set Γ Nst / by formula y [y] Nst Γ Nst / } [y] Nst Γ Nst / FrNst y, v [!Ψ y, ] Remark.3.4. Note that from the axiom of separation it follows directly that Γ Nst / is a set in the sense of the set theory ZF C st. Definition We define now the countable set I Nst by formula Y Y I Nst y [ [y] Nst Γ Nst / gzf CNst = Y = Note that from the axiom schema of replacement it follows directly that I st is a set in the sense of the set theory ZF C Nst. Definition We define now the countable set R Nst by formula I Nst [ R Nst / ] Note that from the axiom schema of separation it follows directly that R Nst is a set in the sense of the set theory ZF C Nst. 11

12 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs Remark.3.5. Note that R Nst I Nst since R Nst is definable by the following formula Ψ Z I Nst [ Z / ] Theorem.3.1. Set theory ZF C Nst is inconsistent. Proof. From.3.16 and Remark.3.5 we obtain R Nst R Nst R Nst / R Nst from which one obtains a contradiction R Nst R Nst R Nst / R Nst Avoiding the Contradictions from Set Theory ZF C Hs and Set Theory ZF C st Using Quinean Approach In order to avoid difficulties mentioned above we use well known Quinean approach [13]. 3.1 Quinean set theory N F 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 Φx n is a formula, then the set x n Φx n } n+1 exists i.e., given any formula Φx n, the formula A n+1 x n [x n A n+1 Φx n ] is an axiom where A n+1 represents the set x n Φx n } n+1 and is not free in Φx n. Quinean set theory [13] 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 x 1. It avoids Russell s paradox by only allowing stratifiable formulae 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 In New Foundations NF and related set theories, a formula Φ in the language of first-order logic with equality and membership is said to be stratified if and only if there is a function f x which sends each variable appearing in Φ [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 Φ satisfies f x + 1 = f y and any atomic formula x = y appearing in Φ satisfies f x = f y. Quinean set theory. 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. 1

13 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs 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. Axiom schema of comprehension: x Φ s } exists for each stratified formula Φ 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, [14] so that NF can be finitely axiomatized without any reference to the notion of type. 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. 3. Set theory ZF C Hs, ZF C st and set theory ZF C Nst 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 ZF C st will also fall inside a set. Stratified Axiom schema of replacement. Let Φ s x, y, w 1, w,..., w n be any stratified formula in the language of ZF C st whose free variables are among x, y, A, w 1, w,..., w n, so that in particular B is not free in Φ s. Then A w 1 w... w n [ x x A =!yφ s x, y, w 1, w,..., w n = = B x x A = y y B Φ s x, y, w 1, w,..., w n ], 3..1 i.e., if the relation Φ s x, y,... represents a definable function f, A represents its domain, and fx is a set for every x A, then the range of f is a subset of some set B. Stratified Axiom schema of separation. Let Φ s x, w 1, w,..., w n be any stratified formula in the language of ZF C st whose free variables are among x, A, w 1, w,..., w n, so that in particular B is not free in Φ s. Then w 1 w... w n A B x [x B x A Φ s x, w 1, w,..., w n ], 3.. Remark 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... Notice that the stratified axiom schema of replacement separation obviously violeted any contradictions.1.0,..18 and.3.18 mentioned above. The existence of the countable Russell sets R Hs, R st and R Nst impossible, because x / x cannot be stratified. 4 Second-order Set Theory ZF C with the Full Secondorder Semantics 4.1 Second order set theory ZF C with urlogic Remind that urlogic has the following characteristics [6]. 13

14 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs Sentences of urlogic are finite strings of symbols. That a string of symbols is a sentence of urlogic, is a non-mathematical judgement.. Some sentences are accepted as axioms. That a sentence is an axiom, is a non-mathematical judgement. 3. Derivations are made from axioms. The derivations obey certain rules of proof. That a derivation obeys the rules of proof, is a non-mathematical judgement. 4. Derived sentences can be asserted as facts. Remark Let ZF C Ul be second order set theory ZF C with urlogic. Note that in ZF C Ul by using the rules of DED we dealing without any reference to semantics, i.e. satisfiability in some standard model, validity etc. Definition Let Γ Ul be the countable collection of the all 1-place open wff s of the set theory ZF C Ul that contains free occurrences of the variable. Let Ψ 1, Ψ be 1-place open wff s of the set theory ZF C Ul. We define now the equivalence relation Γ Ul Γ Ul by Ψ 1 Ψ [Ψ 1 Ψ ] For any Ψ Γ Ul let [Ψ ] Ul Φ Γ Ul Ψ Φ } denote the equivalence class equivalent to each other are also elements of the same by, denoted Γ Ul / to which Ψ belongs. All elements of Γ Ul equivalence class. The collection of all possible equivalence classes of Γ Ul Γ Ul / [Ψ ] Ul Ψ Γ Ul } Let Fr Ul y, v be the relation : y is the Gödel number of a wff of the set theoryzf C Ul that contains free occurrences of the variable with Gödel number v [9]. Note that the relation Fr Ul y, v is expressible in ZF C Ul Ul by a wff Fr y, v. Note that for any y, v N by definition of the relation Fr Ul y, v follows that [ ] Fr Ul y, v!ψ g ZF C Ul Ψ = y g ZF C Ul =, where Ψ is a unique wff of ZF C Ul which contains free occurrences of the variable with Gödel number v. We denote a unique wff Ψ defined by using equivalence by symbol Ψ Ul y,, i.e. [ Fr Ul y, v!ψ Ul y, g ZF C Ul Ψ Ul y, ] = y g ZF C Ul = Definition Let g ZF C Ul =. Let Γ Ul be a set of the all Gödel numbers of the 1-place open wff s of the set theory ZF C Ul that contains free occurrences of the variable with Gödel number v, i.e. Γ Ul = y N y, Fr Ul y, v }, or in the following equivalent form: [ y y N y Γ Ul y N ] Ul Fr y, v Remark Note that from the axiom of separation it follows directly that Γ Ul sense of the set theory ZF C Ul. Definition i We define now the equivalence relation is a set in the 14

15 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs Γ Ul Γ Ul in the sense of the set theory ZF C Ul by y 1 y [ Ψ Ul y 1, Ψ Ul y, ] For any y 1 Γ Ul v let [y 1] Ul } y Γ Ul y 1 y denote the equivalence class to which y1 belongs. The collection of all possible equivalence classes of Γ Ul by, denoted Γ Ul / Γ Ul / } [y] Ul y Γ Ul Remark Note that from the axiom of separation it follows directly that Γ Hs / is a set in the sense of the set theory ZF C Ul. Definition Let I Ul be the countable collection of all sets definable by 1-place open wff of the set theory ZF C Ul, i.e. Y Y I Ul Ψ [ [Ψ ] Ul Γ Ul / [! [Ψ Y = ]] Definition We rewrite now in the following equivalent form Y Y I Ul Ψ [ [Ψ ] Ul Γ Ul / Y =, where the countable collection Γ Ul / is defined by Ψ [Ψ ] Ul Γ Ul / [ [Ψ ] Ul Γ Ul /!Ψ Definition Let R Ul be the countable collection of all sets such that [ I Ul R Ul / ] Remark Note that R Ul I Ul since R Ul is a collection definable by 1-place open wff Ψ Z, I Ul I Ul [ Z / ] From one obtains R Ul R Ul R Ul / R Ul But gives a contradiction R Ul R Ul R Ul / R Ul However contradiction.1.16 it is not a contradiction inside ZF C Ul for the reason that the countable collection I Ul is not a set in the sense of the set theory ZF C Ul. In order to obtain a contradiction inside ZF C Ul we introduce the following definitions. Definition We define now the countable set Γ Ul y [y] Ul Γ Ul / by / [y] Ul Γ Ul Ul / Fr y, v [!Ψ Ul y, Remark Note that from the axiom of separation it follows directly that Γ Ul / is a set in the sense of the set theory ZF C Ul Definition We define now the countable set I Ul by formula [ [y]ul Y Y I Ul y Γ Ul / g ZF C Ul =. Y = Note that from the axiom schema of replacement it follows directly that I Hs is a set in the sense of the set theory ZF C Ul. Definition We define now the countable set R Ul by formula 15

16 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs [ I Ul R Ul / ] Note that from the axiom schema of separation it follows directly that R Ul is a set in the sense of the set theory ZF C Ul. Remark Note that R Ul I Ul since R Ul is definable by the following formula Ψ Z I Ul [ Z / ] Theorem Set theory ZF C Ul is inconsistent. Proof. From and Remark we obtain R Ul R Ul R Ul / R Ul from which immediately one obtains a contradiction R Ul R Ul R Ul / R Ul Second-order set theory ZF C with the full second-order semantics Remind that the canonical approach of second order logic with full second-order semantics to the foundations of mathematics is that mathematical propositions have the form U = Φ 4..1 where U is a mathematical structure, such as integers, reals etc., and is a mathematical statement written in second order logic. If A is one of the structures, such as N, +,, < or R, +,, <, for which there is a second order sentence Ξ U such that W W = Ξ U W = U, 4.. then 4.. can be expressed as a second order semantic logical truth = Ξ U = Φ Remark Let ZF C fss be second order set theory ZF C with the full second-order semantics. 1 There is no completeness theorem for second-order logic. Nor do the axioms of second-order ZF C fss imply a reflection principle which ensures that if a sentence of second-order set theory is true, then it is true in some standard model. Remark 4... Thus there may be sentences of the language of second-order set theory ZF C fss : i that are true but unsatisfiable, or ii sentences that are valid, but false. Remark For example let Z be the conjunction of all the axioms of second-order ZF C fss. Z is surely true. But the existence of a model for Z requires the existence of strongly inaccessible cardinals. The axioms of ZF C fss don t entail the existence of strongly inaccessible cardinals, and hence the satisfiability of Z is independent of ZF C fss. Thus, Z is true but its unsatisfiability is consistent with ZF C fss. Definition Well formed formula Ψ of ZF C fss is a well formed formula of the first order wff 1 if Ψ contain only first-order variables and first-order quantifiers. Let Γ fss be the countable collection of the all 1-place open wff 1 s of the set theory ZF C fss that contains free occurrences of the first-order variable. Let Ψ 1, Ψ be 1-place open wff 1 s of the set theory ZF C fss. We define now the equivalence 16

17 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs relation Γ fss Γ fss by Ψ 1 Ψ [Ψ 1 Ψ ] 4..4 For any Ψ Γ fss let [Ψ ] fss Φ Γ fss Ψ Φ } denotes the equivalence class to which Ψ belongs. All elements of Γ fss equivalent to each other are also elements of the same equivalence class. The collection of all possible equivalence classes of Γ fss by, denoted Γ fss / Γ fss / [Ψ ] fss Ψ Γ fss } Let Fr fss y, v be the relation : y is the Gödel number of a wff of the set theory ZF C fss that contains free occurrences of the first-order variable with Gödel number v [9]. Note that the relation Fr fss y, v is expressible in ZF C fss by a wff 1 Fr fss y, v. Note that for any y, v N by definition of the relation Fr fss y, v follows that [ ] Fr fss y, v!ψ g fss ZF C Ψ = y g fss ZF C =, 4..6 where Ψ is a unique wff 1 of ZF C fss which contains free occurrences of the variable with Gödel number v. We denote a unique wff Ψ defined by using equivalence 4..6 by symbol Ψ fss y,, i.e. Fr fss y, v!ψ fss y, [ g fss ZF C Ψ y, ] = y g fss ZF C = Remark In order to avoid difficulties mentioned above,see Remark 4..1-Remark 4..3 we dealing with the countable collection Γ fss of the all 1-place open wff1 s of the set theory ZF Cfss. Definition 4... Let g fss ZF C wff 1 s of the set theory ZF C fss Gödel number v, i.e. Γ fss = y N y, Fr fss y, v or in the following equivalent form [ y y N y Γ fss y N =.Let Γ fss be a set of all Gödel numbers of 1-place open that contains free occurrences of the first-order variable with }, 4..8 ] fss Fr y, v Remark Note that from the axiom of separation it follows directly that Γ fss the sense of the set theory ZF C fss. Definition i We define now the equivalence relation Γ fss Γ fss in the sense of the set theory ZF C fss by y 1 y [ Ψ fss y 1, Ψ fss y, ] The collection of all possible equivalence classes of Γ fss by, denoted Γ fss / } Γ fss v / [y] fss y Γ fss is a set in Remark Note that from the axiom of separation it follows directly that Γ fss / is a set in the sense of the set theory ZF C fss. 17

18 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs Definition Let I fss be the countable collection of the all sets definable by 1-place open first order wff of the set theory ZF C fss, i.e. [ Y Y I fss Ψ [Ψ ] fss Γ fss / [! [Ψ Y = ]] Definition We rewrite now in the following equivalent form [ Y Y I fss Ψ [Ψ ] fss Γ fss / Y =, where the countable collection Γ fss / is defined by [ Ψ [Ψ ] fss Γ fss / [Ψ ] fss Γ fss /!Ψ Definition Let R fss be the countable collection of all sets such that [ ] I fss R fss / Remark Note that R fss I fss since R fss is a collection definable by 1-place open wff 1 Ψ Z, I fss From one obtains I fss [ Z / ] R fss R fss R fss / R fss But gives a contradiction R fss R fss R fss / R fss However contradiction..19 it is not a contradiction inside ZF C fss for the reason that the countable collection I fss is not a set in the sense of the set theory ZF C fss. In order to obtain a contradiction inside ZF C fss we introduce the following definitions. Definition We define now the countable set Γ fss / by y [y] Ul Γ fss / [y] fss Γ fss fss / Fr y, v [!Ψ fss y, Remark Note that from the axiom of separation it follows directly that Γ Ul / is a set in the sense of the set theory ZF C fss. Definition We define now the countable set I fss by formula Y [ Y I fss y [y] fss Γ fss / g fss ZF C = Y = Note that from the axiom schema of replacement it follows directly that I fss is a set in the sense of the set theory ZF C fss. 18

19 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs Definition We define now the countable set R fss by formula [ ] I fss R fss /. 4.. Note that from the axiom schema of separation it follows directly that R fss is a set in the sense of the set theory ZF C fss. Remark Note that R fss I Ul since R Ul is definable by the following formula Ψ Z I fss [ Z / ] Theorem Set theory ZF C fss is inconsistent. Proof. From 4.. and Remark we obtain R fss R fss R fss / R Ul from which immediately one obtains a contradiction R fss R fss 5 Conclusions R fss / R fss a In this paper we have proved that set theory ZF C + M ZF C st is inconsistent. b This result originally was obtained in [10], [15] and [11] by using essentially another complicated approach.. Acknowledgement The reviewers provided important clarifications. Competing Interests Authors have declared that no competing interests exist. References [1] Nelson E Warning signs of a possible collapse of contemporary mathematics. In Infinity: New Research Frontiers, by Michael Heller Editor, W. Hugh Woodin, Hardcover. 013;311: ISBN: Available: nelson/papers/warn.pdf [] Henkin L. Completeness in the theory of types. Journal of Symbolic Logic. 1950;15: DOI: /66967.JSTOR66967 [3] Rossberg M. First-order logic, second-order logic, and completeness. In: V. Hendricks et al., eds. First-order logic revisited, Berlin: Logos-Verlag. P. 004; [4] Shapiro S. Foundations without Foundationalism: A Case for Second-order Logic. Oxford University Press; ISBN

20 Foukzon and Men kova; JAMCS, 6: 1-0, 018; Article no.jamcs [5] Rayo A, Uzquiano G. Toward a theory of second-order consequence. Notre Dame Journal of Formal Logic. 1999;403: [6] Vaananen J. Second-order logic and foundations of mathematics. The Bulletin of Symbolic Logic. 001;74: [7] Uzquiano G. Quantification without a domain. New Waves in Philosophy of Mathematics. Springer. 009;37. ISBN , [8] Cohen P. Set Theory and the continuum hypothesis. Reprint of the W. A. Benjamin, Inc., New York; ISBN-13: [9] Mendelson E. Introduction to mathematical logic.; ISBN-10: [10] Foukzon J, Men kova ER. Generalized Löb s theorem. Strong Reflection Principles and Large Cardinal Axioms. Advances in Pure Mathematics. 013;33: Available: [11] Foukzon J. Inconsistent countable set in second order zfc and nonexistence of the strongly inaccessible cardinals. British Journal of Mathematics & Computer Science. 015;95. ISSN: Available: [1] Foukzon J.. Generalized lob s theorem. Strong Reflection Principles and Large Cardinal Axioms. Consistency Results in Topology. V. 017;1. Available: [13] Quine WV. New foundations for mathematical logic. The American Mathematical Monthly, Mathematical Association of America. 1937;44: DOI: / [14] Hailperin T. A set of axioms for logic. Journal of Symbolic Logic. 1944;9:1-19. [15] Foukzon J. Inconsistent countable set in second order ZFC and unexistence of the strongly inaccessible cardinals. Logic Colloquium16, Leeds, UK, July 31-August 6, 016. Abstract of contributed talks. The Bulletin of Symbolic Logic. 017;3:40. c 018 Foukzon and Men kova; This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Peer-review history: The peer review history for this paper can be accessed here Please copy paste the total link in your browser address bar 0