GENERALIZED LOB ' S THEOREM STRONG REFLECTION PRINCIPLES AND LARGE CARDINAL AXIOMS CONSISTENCY RESULTS IN TOPOLOGY

Article History Received: 24 May 2017 Revised: 13 June 2017 Accepted: 18 July 2017 Published: 9 August 2017


Introduction
Let Th be some fixed, but unspecified, consistent formal theory.

Definition 1. Let
Th M  be an  -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 be any closed formula, then . 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 Theorem 2. (Generalized Löb's Theorem).Assume that 1) Con(Th) and 2) Th has an  -model Th  .Then theory Th can be extended to a maximally consis-tent nice theory Th # .

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,  , of the language of Th is assigned a closed term, Of particular importance is the substitution operator, represented by the function symbol .For formulae Iteration of the substitution operator sub allows one to define function symbols such that sub , It well known [2,3] that one can also encode derivations and have a binary relation Th x y (read "x proves y" or "x is a proof of y") such that for closed and therefore one obtain a predicate asserting provability.Remark 2.1.We note that is not always the case that [2,3]: It well known [3] that the above encoding can be carried out in such a way that the following important conditions and are met for all sentences [2,3]: implies Pr Assumption 2.1.We assume now that: 1) the language of Th consists of: numerals 0, 1,  countable set of the numerical variables: It well known that replacing the induction scheme in Peano arithmetic PA by the  -rule with the meaning "if the formula   A n is provable for all n, then the formula leads to complete and sound system  where each true arithmetical statement is provable.S. Feferman showed that an equivalent formal system Th can be obtained by erecting on a transfinite progression of formal systems  according to the following scheme

 
A x is a formula with one free variable and  where is a limit ordinal.Then being Kleene's , It is easy to see that Th , i.e. is a maximally nice extension of the PA.
Th M  be an Definition 3.2.Let  -model of the Th.We said that, is a nice theory over Th or a nice extension of the Th iff: contains Th;  be any closed formula, then 2) Let We said that is a maximally nice theory over Th or a maximally nice extension of the Th iff is consistent and for any consistent nice exten-sion of the Th: Lemma 3.1.Assume that: 1) Con ; and 2) From (3.1) and (3.2) one obtain But the Formula (3.3) contradicts the Formula (3.1).
# Th Proof.Let 1 i be an enumeration of all wff's of the theory Th (this can be achieved if the set of propositional variables can be enumerated).Define a chain   1 i of consistent theories inductively as follows: assume that theory is defined.
Then we define theory  as follows 2) Suppose that a statement (3.5) is satisfied Then we define theory as follows: .
3) Suppose that a statement (3.6) is satisfied Then we define theory as follows: . 4) Suppose that a statement (3.7) is satisfied Then we define theory as follows: .
We define now theory as follows: , , t t t     is consistent and from the above observation one obtain that: , where On the other hand one obtain But the Formula (3.15), contradicts the Formula (3.14).This contradiction completed the proof.is incompatible with all the usual large cardinal axioms [10,11] which imply the existence of a strong standard model of ZFC.

 
Con ZFC     .Proof.Let H  be a set of all sets having hereditary size less then κ.It easy to see that H  forms a strong standard model of ZFC.Therefore Theorem 3.7 immediately follows from Theorem 3.6.

Conclusion
In this paper we proved so-called strong reflection principles corresponding to formal theories Th which has ω-models Th M  and in particular to formal theories Th, which has a standard models .The assumption that there exists a standard model of Th is stronger than the assumption that there exists a model of Th.This paper examined some specified classes of the standard models of ZFC so-called strong standard models of ZFC.Such models correspond to large cardinals axioms.In particular we proved that theory ( some inaccessible cardinal κ) is 1  .Thus Theorem 3.6 asserts there exist numerical counterexample which would imply that a specific polynomial equation has at least one integer root.
x is the Gödel number of the proof of the formula with Gödel number n, and is the numeral of the Gödel number of the formula n  , then n Th   .Taking into account the second Gödel theorem it is easy to be able to prove n for disprovable (refutable) and undecidable formulas n

Definition 3 . 1 .
An Th  (well-formed formula   ) is closed i.e.,  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 Th M of the Th and a Th-sentence  , we assume known the meaning M

Definition 3 . 4 .Definition 3 . 6 .Con Th , 2 )
is nice extension of the set consistent because, by the standard Lemma 3.3 below, it Copyright © 2013 SciRes.APM is the union of a chain of consistent sets.To see that   # Ded Th is maximal, pick any wff  .Then  is some in the enumerated list of all wff's.Therefore i (a) Assume that a theory Th has an   -model Th M  and  is a Th-sentence.Let   be a Th-sentence with all quantifiers relativized to   -model Th M  [9]; (b) Assume that a theory Th has a standard model Th SM  Th SM And Φ is a Th-sentence.Let SM be a Th-sentence Φ with all quantifiers relativized to the model [9].Definition 3.5.(a) Assume that Th has an  -model Th M  .Let Th  be a theory Th relativized to a model Th M  -i.e., any Th  -sentence has a form  Assume that Th has an standard model .Let SM be a theory Th relativized to a model -i.e., any SM -sentence has a form for some Th-sentence Φ [9].SM  (a) For a given  -model Th M  of the Th and for any Th  -sentence   Th has an  -model one direction is obvious.For the other, as- term.Note that in any  - Th M  by the canonical observation one obtain the model  11)-(3.12)contradicts the Formula (3.13).Therefore

Definition 3 . 7 ..Definition 3 . 8 ..Definition 3 . 10 .
(a) Assume that: (i) Th has an  - model Then we said that M  and (ii)   Th M  -model of the Th and denote such  is a strong  -model of the Th as .Assume that: (i) Th has an standard model and (ii) SM .Then we said that is a strong standard model of the Th and denote such standard model of the Th as   (a) Assume that Th has a strong .Then we said that Th is a strongly consistent.(b) Assume that Th has a strong standard model   Then we said that Th is a strongly SM-consistent Definition 3.9.(a) Assume that Th has a strong  - model ,  Th M  is a Th-sentence.Let ,     and   be a Th-sentence  with all quantifiers relativized to a strong , Assume that Th has a strong standard model   and Φ is a Th-sentence.Let , SM   be a Thsentence Φ with all quantifiers relativized to Assume that Th has a strong .sentence .Let Th be a theory such that Assumption 1.1 is satisfied.Let be a sentence in Th asserting that Th has a strong , in Th* asserting that Th* has a strong  -model ,    .We assume throughout that Th is a strongly consistent, i.e. a sentence any ω-model M  of the Th.
patible with existence of any inaccessible cardinal κ.Note that the statement: .8) First, notice that each i is consistent.This is done by induction on i and by Lemmas 3.1-3.2.By assumption, the case is true when  .Now, suppose is consistent.Then its deductive closure i Th  