Advances in Pure Mathematics, 2013, 3, 368-373 http://dx.doi.org/10.4236/apm.2013.33053 Published Online May 2013 (http://www.scirp.org/journal/apm) Copyright © 2013 SciRes. APM Generalized Löb's Theorem. Strong Reflection Principles and Large Cardinal Axioms J. Foukzon1, E. R. Men'kova2 1Israel Institute of Technology, Haifa, Israel 2Lomonosov Moscow State University, Moscow, Russia Email: jaykovfoukzon@list.ru, E_Menkova@mail.ru Received February 9, 2013; revised March 13, 2013; accepted April 14, 2013 Copyright © 2013 J. Foukzon, E. R. Men'kova. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this article, a possible generalization of the Löb's theorem is considered. Main result is: let κ be an inaccessible cardinal, then ( )Con ZFC κ¬ + ∃ . Keywords: Löb's Theorem; Second Godel Theorem; Consistency; Formal System; Uniform Reflection Principles; ω-Model of ZFC; Standard Model of ZFC; Inaccessible Cardinal 1. Introduction Let Th be some fixed, but unspecified, consistent formal theory. Theorem 1 [1]. (Löb's Theorem). If ( )Prov ,Th nTh x x n φ∃ →  where x is the Gödel number of the proof of the formula with Gödel number n, and n is the numeral of the Gödel number of the formula nφ , then nTh φ . Taking into account the second Gödel theorem it is easy to be able to prove ( )Prov ,Th nx x n φ∃ →  , for disprovable (refutable) and undecidable formulas nφ . Thus summarized, Löb's theorem says that for refutable or undecidable formula φ , the intuition "if exists proof of φ then φ " is fails. Definition 1. Let ThMω be an ω -model of the Th. We said that, Th# is a nice theory over Th or a nice extension of the Th iff: 1) Th# contains Th; 2) Let Φ be any closed formula, then [ ]( )Pr &c ThThTh Mω   Φ     Φ  implies #Th Φ . Definition 2. We said that, Th# is a maximally nice theory over Th or a maximally nice extension of the Th iff Th# is consistent and for any consistent nice extension Th′ of the Th: ( ) ( )#Ded DedTh Th′⊆ implies ( ) ( )#Ded DedTh Th′= . Theorem 2. (Generalized Löb's Theorem). Assume that 1) Con(Th) and 2) Th has an ω -model ThMω . Then theory Th can be extended to a maximally consistent nice theory Th#. 2. Preliminaries Let Th be some fixed, but unspecified, consistent formal theory. For later convenience, we assume that the encoding is done in some fixed formal theory S and that Th contains S. We do not specify S-it is usually taken to be a formal system of arithmetic, although a weak set theory is often more convenient. The sense in which S is contained in Th is better exemplified than explained: If S is a formal system of arithmetic and Th is, say, ZFC, then Th contains S in the sense that there is a well-known embedding, or interpretation, of S in Th. Since encoding is to take place in S, it will have to have a large supply of constants and closed terms to be used as codes. (e.g. in formal arithmetic, one has 0, 1, ) S will also have certain function symbols to be described shortly. To each formula, Φ , of the language of Th is assigned a closed term, [ ]cΦ , called the code of Φ . [N. B. If ( )xΦ is a formula with a free variable x, then ( ) cxΦ   is a closed term encoding the formula ( )xΦ with x viewed as a syntactic object and not as a parameter.] Corresponding to the logical connectives and quantifiers are function symbols, ( ) ( )neg ,imp⋅ ⋅ , etc., such that, for all formulae [ ]( ) [ ] [ ] [ ]( ) [ ] , : neg , imp , c c c c c S S Φ Ψ Φ = ¬Φ Φ Ψ = − Φ →Ψ− etc. J. FOUKZON, E. R. MEN'KOVA Copyright © 2013 SciRes. APM 369 Of particular importance is the substitution operator, represented by the function symbol ( )sub ,⋅ ⋅ . For formulae ( )xΦ , terms t with codes [ ]ct : ( ) [ ]( ) ( )sub ,c ccS x t tΦ = Φ  −    . (2.1) Iteration of the substitution operator sub allows one to define function symbols 3 4sub ,sub , ,subn such that ( ) [ ] [ ] [ ]( ) ( ) 1 2 1 2 1 2 sub , , , , , , , , , , c cc c n n n c n S x x x t t t t t t  Φ   = Φ −      (2.2) It well known [2,3] that one can also encode derivations and have a binary relation ( )Prov ,Th x y (read "x proves y" or "x is a proof of y") such that for closed ( )1 2 1 2, : Prov ,Tht t S t t− iff 1t is the code of a derivation in Th of the formula with code 2t . It follows that [ ]( )Prov , cThTh S tΦ↔ Φ  (2.3) for some closed term t. Thus one can define predicate ( )PrTh y : ( ) ( )Pr Prov ,Th Thy x x y↔∃ , (2.4) and therefore one obtain a predicate asserting provability. Remark 2.1. We note that is not always the case that [2,3]: [ ]( )Pr cThTh i SΦ ↔ Φ  . (2.5) It well known [3] that the above encoding can be carried out in such a way that the following important conditions 1, 2D D and 3D are met for all sentences [2,3]: [ ]( ) [ ]( ) [ ]( ) [ ]( ) [ ]( ) [ ]( ) 1. implies Pr , 2. Pr Pr Pr , 3. Pr Pr Pr . c Th cc c Th Th Th c c Th Th c Th D Th S D S D S Φ Φ   Φ → Φ     Φ ∧ Φ →Ψ → Ψ     (2.6) Conditions 1, 2D D and 3D are called the Derivability Conditions. Assumption 2.1. We assume now that: 1) the language of Th consists of: numerals 0, 1, countable set of the numerical variables: { }0 1, ,ν ν  countable set F of the set variables: { }, , , , , , ,F x y z X Y Z= R countable set of the n-ary function symbols: 0 1, , n nf f  countable set of the n-ary relation symbols: 0 1, , n nR R  connectives: ,¬ → quantifier: ∀ . 2) Th contains ZFC 3) Th has an ω -model ThMω . Theorem 2.1. (Löb's Theorem). Let be 1) ( )Con Th and 2) φ be closed. Then [ ]( )Pr iffcThTh Thφ φ φ→  . (2.7) It well known that replacing the induction scheme in Peano arithmetic PA by the ω -rule with the meaning "if the formula ( )A n is provable for all n, then the formula ( )A x is provable": ( ) ( ) ( ) ( ) 0 , 1 , , , , A A A n xA x∀   (2.8) leads to complete and sound system PA∞ where each true arithmetical statement is provable. S. Feferman showed that an equivalent formal system #Th can be obtained by erecting on Th PA= a transfinite progression of formal systems PA∞ according to the following scheme ( )( ) ( ){ } 0 1 Pr , c PA PA PA PA PA x A x xA x PA PA αα α λ α α λ + < = = + ∀ →∀   =   (2.9) where ( )A x is a formula with one free variable and λ is a limit ordinal. Then , O Th PA Oαα∈= being Kleene's system of ordinal notations, is equivalent to #Th PA∞= . It is easy to see that # #Th PA= , i.e. #Th is a maximally nice extension of the PA. 3. Generalized Löb's Theorem Definition 3.1. An wffTh − Φ (well-formed formula Φ ) is closed i.e., Φ is a Th-sentence iff it has no free variables; a wff Ψ is open if it has free variables. We'll use the slang "k-place open wff" to mean a wff with k distinct free variables. Given a model ThM of the Th and a Th-sentence Φ , we assume known the meaning of M Φ -i.e. Φ is true in ThM , (see for example [4-6]). Definition 3.2. Let ThMω be an ω -model of the Th. We shall say that, #Th is a nice theory over Th or a nice extension of the Th iff: 1) #Th contains Th; 2) Let Φ be any closed formula, then [ ]( )P &r TcTh hTh Mω      Φ Φ  implies #Th Φ . Definition 3.3. We shall say that #Th is a maximally nice theory over Th or a maximally nice extension of the Th iff #Th is consistent and for any consistent nice extension Th′ of the Th: ( ) ( )#Ded DedTh Th′⊆ implies ( ) ( )#Ded DedTh Th′= . J. FOUKZON, E. R. MEN'KOVA Copyright © 2013 SciRes. APM 370 Lemma 3.1. Assume that: 1) ( )Con Th ; and 2) [ ]( )Pr cThTh Φ , where Φ is a closed formula. Then [ ]( )Pr cThTh ¬Φ . Proof. Let ( )ConTh Φ be the formula ( ) [ ]( ) [ ]( )( ) [ ]( ) [ ]( )( ) 1 2 1 2 1 2 1 2 Con Prov , Prov ,neg Prov , Prov ,neg Th c c Th Th c c Th Th t t t t t t t t Φ  ∀ ∀ ¬ Φ ∧ Φ    ↔ ¬∃ ¬∃ Φ ∧ Φ    (3.1) where 1 2,t t is a closed term. We note that under canonical observation, one obtains ( ) ( )Con ConThTh Th+ Φ for any closed wff Φ . Suppose that [ ]( )Pr cThTh ¬Φ , then assumption (ii) gives [ ]( ) [ ]( )Pr Prc cTh ThTh Φ ∧ ¬Φ . (3.2) From (3.1) and (3.2) one obtain [ ]( ) [ ]( )( )1 2 1 2Prov , Prov ,negc cTh Tht t t t ∃ ∃ Φ ∧ Φ   . (3.3) But the Formula (3.3) contradicts the Formula (3.1). Therefore: [ ]( )Pr cThTh ¬Φ . Lemma 3.2. Assume that: 1) ( )Con Th ; and 2) [ ]( )Pr cThTh ¬Φ , where Φ is a closed formula. Then [ ]( )Pr cThTh Φ . Theorem 3.1. [7,8]. (Generalized Löb's Theorem). Assume that: ( )Con Th . Then theory Th can be extended to a maximally consistent nice theory #Th over Th. Proof. Let 1 iΦ Φ  be an enumeration of all wff's of the theory Th (this can be achieved if the set of propositional variables can be enumerated). Define a chain { } 1,iTh i Th Th℘= ∈ = of consistent theories inductively as follows: assume that theory iTh is defined. 1) Suppose that a statement (3.4) is satisfied [ ]( ) [ ] Pr and & c Th i Th i i i Th Th Mω Φ Φ Φ     . (3.4) Then we define theory 1iTh + as follows { }1i i iTh Th+ Φ  . 2) Suppose that a statement (3.5) is satisfied [ ]( ) [ ] Pr and & c Th i Th i i i Th Th Mω ¬Φ   ¬Φ ¬Φ    . (3.5) Then we define theory 1iTh + as follows: { }1i i iTh Th+ ¬Φ  . 3) Suppose that a statement (3.6) is satisfied [ ]( )Pr cTh iTh Φ and i iTh Φ . (3.6) Then we define theory 1iTh + as follows: { }1i i iTh Th+ Φ  . 4) Suppose that a statement (3.7) is satisfied [ ]( )Pr cTh iTh ¬Φ and iTh ¬Φ . (3.7) Then we define theory 1iTh + as follows: 1i iTh Th+  . We define now theory #Th as follows: # i i Th Th ∈   . (3.8) First, notice that each iTh is consistent. This is done by induction on i and by Lemmas 3.1-3.2. By assumption, the case is true when 1i = . Now, suppose iTh is consistent . Then its deductive closure ( )Ded iTh is also consistent. If a statement (3.6) is satisfied i.e., [ ]( )Pr cTh iTh Φ and iTh Φ , then clearly { }1i i iTh Th+ Φ  is consistent since it is a subset of closure ( )Ded iTh . If a statement (3.7) is satisfied, i.e., [ ]( )Pr cTh iTh ¬Φ and i iTh ¬Φ , then clearly { }1i i iTh Th+ ¬Φ  is consistent since it is a subset of closure ( )Ded iTh . Otherwise: 1) if a statement (3.4) is satisfied, i.e. [ ]( )ThPr i ci iTh Φ and i iTh Φ , then clearly { }1i i iTh Th+ Φ  is consistent by Lemma 3.1 and by one of the standard properties of consistency: { }A∆ is consistent iff A∆ ¬ ; 2) if a statement (3.5) is satisfied, i.e. [ ]( )Pr cTh iTh ¬Φ and i iTh ¬Φ , then clearly { }1i i iTh Th+ ¬Φ  is consistent by Lemma 3.2 and by one of the standard properties of consistency: { }A∆ ¬ is consistent iff A∆  . Next, notice ( )#Ded Th is a maximally consistent nice extension of the set ( )Ded Th . A set ( )#Ded Th is consistent because, by the standard Lemma 3.3 below, it is the union of a chain of consistent sets. To see that ( )#Ded Th is maximal, pick any wff Φ . Then Φ is some iΦ in the enumerated list of all wff's. Therefore J. FOUKZON, E. R. MEN'KOVA Copyright © 2013 SciRes. APM 371 for any Φ such that [ ]( )Pr cThTh Φ or [ ]( )Pr ,cThTh ¬Φ either #ThΦ∈ or #.Th¬Φ∈ Since ( ) ( )#1Ded Ded ,iTh Th+ ⊆ we have ( )#Ded ThΦ∈ or ( )#Ded Th¬Φ∈ , which implies that ( )#Ded Th is maximally consistent nice extension of the ( )Ded Th . Lemma 3.3. The union of a chain { }i i℘= Γ ∈ of the consistent sets iΓ , ordered by ⊆ , is consistent. Definition 3.4. (a) Assume that a theory Th has an ω -model ThMω and Φ is a Th-sentence. Let ωΦ be a Th-sentence Φ with all quantifiers relativised to ω -model ThMω ; (b) Assume that a theory Th has a standard model ThSM and Φ is an Th-sentence. Let SMΦ be a Th-sentence Φ with all quantifiers relativized to a model ThSM [9]. Remark 3.1. In some special cases we denote a sentence ωΦ by a symbol ThMω Φ   and we denote a sentence SMΦ by symbol ThM Φ   correspondingly. Definition 3.5. (a) Assume that Th has an ω -model ThMω . Let Thω be a theory Th relativized to a model ThMω , that is, any Thω -sentence has a form ωΦ for some Th-sentence Φ [9]; (b) Assume that Th has an standard model ThSM . Let SMTh be a theory Th relativized to a model ThSM , that is, any SMTh -sentence has a form SMΦ for some Thsentence Φ [9]. Remark 3.2. In some special cases we denote a theory Thω by symbol ThTh Mω   and we denote a theory SMTh by symbol ThTh M   correspondingly. Theorem 3.2. (Strong Reflection Principle). (i) Assume that: Th has an ω -model ThMω . Then for any Thω -sentence ωΦ [ ]( )Pr iff .cThTh Thωω ωω ωΦ Φ  (3.9) (ii) Assume that: Th has model ThSMM . Then for any SMTh -sentence SMΦ [ ]( )Pr iff .cSM SM SM SM SMTh Th ThΦ Φ  (3.10) Proof. (i) The one direction is obvious. For the other, assume that [ ]( )Pr ,cThTh Thω ωω ωωΦ Φ  , (3.11) and Thω ω¬Φ . Then [ ]( )Pr cThTh ωω ω¬Φ . (3.12) Note that ( )ωCon Th holds since ThMω∃ . Let ω ConTh be the formula [ ]( ) [ ]( ) [ ]( )( ) [ ]( ) [ ]( ) [ ]( )( ) 1 2 3 3 1 2 1 2 3 3 1 2 Con Prov , Prov ,neg Prov , Prov ,neg . c Th c c Th Th c c c Th Th t t t t t t t t t t t t ω ω ω ω ω ω ω ω ω ω ω ↔ ∀ ∀ ∀ = Φ  ¬ Φ ∧ Φ   ↔ ¬∃ ¬∃ ¬∃ = Φ  × Φ ∧ Φ   (3.13) where 1 2 3, ,t t t is a closed term. Note that for any ω - model ThMω by the canonical observation one obtains the equivalence ( )Con ConThTh ωω ↔ (see [2]). But the Formulae (3.11)-(3.12) contradicts the Formula (3.13). Therefore [ ]( ), Pr and .cThh ThT ωω ω ω ωω Φ ¬Φ ¬Φ   Then theory Th Thω ω ω′ = +¬Φ is consistent and from the above observation one obtains that: ( )Con ConThTh ωω ′′ ↔ , where [ ]( ) [ ]( ) [ ]( )( ) 1 2 3 3 1 2 Con Prov , Prov ,neg . c Th c c Th Th t t t t t t ω ω ω ω ω ω ′ ′ ′ ↔ ¬∃ ¬∃ ¬∃ = Φ  × Φ ∧ Φ   (3.14) On the other hand one obtains [ ]( ) [ ]( )Pr , Prc cTh ThTTh hω ωω ω ω ω′ ′′ ′Φ ¬Φ  . (3.15) But the Formulae (3.15) contradicts the Formula (3.14). This contradiction completed the proof. Proof (ii) similarly as the proof (i) above. Definition 3.6. Let Th be a theory such that the Assumption 1.1 is satisfied. (a) Let ( );Th ThCon Th Mω ωΞ ≡ be a sentence in Th asserting that Th has ω -model ThMω . (b) Let ( );SMTh ThSMCon Th MΞ ≡ be a sentence in Th asserting that Th has standard model ThSMM . Assumption 3.1. We assume that (i) a sentence ThωΞ is expressible in Th, i.e., a sentence ThωΞ is expressible by using the lenguage Th of the Th; (ii) a sentence SMThΞ is expressible in Th, i.e., a sentence SMThΞ is expressible by using the lenguage Th of the Th. Remark 3.3. Note that (i) for any ω -model ThMω of the Th by the canonical observation (see [2]) one obtains the equivalence ( ) ( ) Con ; Con Con Th Th Th Th M Th M Th M ω ω ω     ↔   ↔  , (3.16) (see remark 3.1) and the equivalence J. FOUKZON, E. R. MEN'KOVA Copyright © 2013 SciRes. APM 372 Th Con PrTh Th cTh M Th M M ω ω ω           ↔ ¬       (3.17) (see remark 3.2), where  is a closed formula refutable in Th. (ii) for any standard model ThMω of the Th by the canonical observation (see [2] chapter), one obtains the equivalence ( ) ( )Con ; Con Con Th SM Th Th SM SM Th M Th M Th M      ↔ ↔  (3.18) (see remark 3.1) and the equivalence Con PrTh SMSM cTh Th SMTh M M       ↔ ¬        (3.19) (see remark 3.2), where  is a closed formula refutable in Th. Lemma 3.4. (I) Assume that Th has ω -model ThMω . Let 1Th be a theory 1 ThTh Th ω= + Ξ . Then 1Th is consistent. (II) Assume that Th has standard model ThSM . Let 2Th be a theory 2 SM ThTh Th= +Ξ . Then 2Th is consistent. Proof. (I) Assume that a theory ( )1 ;Th ThTh Th Th Con Th Mω ω= + Ξ ≡ + is inconsistent: ( )1Con Th¬ . This means that there is no any model ThM of Th in which ( ); ThCon Th Mω is true and in particular that is Th has no any ω -model 1, ThM ω of Th in which ( ); ThCon Th Mω is true, i.e., ( )1, 1, 1,;ThTh Th Th ThM M Con Th M Mωω ω ω ω   Ξ ≡    and therefore one obtains for any ω -model 1, ThM ω of Th that ( )1, 1,Con ; ,Th Th ThM Th M Mω ω ω ¬   (3. 20) and in particular ( )1, 1, 1,Con ; ,Th Th ThM Th M Mω ω ω ¬   (3. 21) From (3.21) using (3.16)-(3.17) and one obtains 1, 1, 1, 1, 1,Th Con Pr .Th Th cTh Th Th Th M M M M M ω ω ω ω ω             ¬ ↔          (3. 22) From (3.22) and Theorem 3.2(I) one obtains 1, 1, . cTh ThM Mω ω           (3. 23) Obviously (3.23) contradicts to the assumption that Th has an ω -model ThMω . This contradiction completed the proof. Theorem 3.3. (I) Th has no any ω -model ThMω . (II) Th has no any standard model ThSM . Proof. (I) By Lemma 3.4(I) one obtains that ( )1 1 .Th Con Th But Godel's Second Incompleteness Theorem applied to 1Th asserts that ( )1Con Th is unprovable in 1Th . This contradiction completed the proof. Proof. (II) Similarly as above. Remark 3.4. We emphasize that it is well known that axiom ZFCSM∃ a single statement in ZFC see [10], Ch. II, section 7. We denote this statement through all this paper by symbol ( ); ZFCCon ZFC SM . Theorem 3.4. ZFC has no anyω-model ZFCMω . Proof. Immediately follows from Theorem 3.3 (I) and Remark 3.4. Theorem 3.5. ZFC has no any standard model. ZFCSM . Proof. Immediately follows from Theorem 3.3 (II) and Remark 3.4. Theorem 3.6. ZFC is incompatible with all the usual large cardinal axioms [11] which imply the existence standard model of ZFC. Proof. Theorem 3.6 immediately follows from Theorem 3.5. Theorem 3.7. Let κ be an inaccessible cardinal. Then ( )Con ZFC κ¬ +∃ . Proof. Let Hκ be a set of all sets having hereditary size less then κ. 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