Models of arithmetic and limitation theorems

2019 ◽  
pp. 119-145
Keyword(s):  
Models Of Arithmetic

Models of arithmetic and upper bounds for arithmetic sets

1994 ◽  
Vol 59 (3) ◽  
pp. 977-983 ◽  
Author(s):  
Alistair H. Lachlan ◽  
Robert I. Soare
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Models Of Arithmetic

AbstractWe settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.

On LP-models of arithmetic

2008 ◽  
Vol 73 (1) ◽  
pp. 212-226 ◽  
Author(s):  
J. B. Paris ◽  
A. Sirokofskich
Keyword(s):  
Congruence Relations ◽  
Classical Models ◽  
Models Of Arithmetic

AbstractWe answer some problems set by Priest in [11] and [12], in particular refuting Priest's Conjecture that all LP-models of Th(ℕ) essentially arise via congruence relations on classical models of Th(ℕ). We also show that the analogue of Priest's Conjecture for IΔ0 + Exp implies the existence of truth definitions for intervals [0, a] ⊂eM ⊨ IΔ0 + Exp in any cut [0, a] ⊂eK ⊆eM closed under successor and multiplication.

Types omitted in uncountable models of arithmetic

1975 ◽  
Vol 40 (3) ◽  
pp. 317-320 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Completeness Theorem ◽  
Elementary Extension ◽  
Ramsey's Theorem ◽  
Definable Set ◽  
Ramsey’S Theorem ◽  
Hanf Number ◽  
Definable Sets ◽  
Omitting Types ◽  
Infinite Set ◽  
Models Of Arithmetic

In [4] it is shown that if the structure omits a type Σ, and Σ is complete with respect to Th(), then there is a proper elementary extension of which omits Σ. This result is extended in the present paper. It is shown that Th() has models omitting Σ in all infinite powers.A type is a countable set of formulas with just the variable ν occurring free. A structure is said to omit the type Σ if no element of satisfies all of the formulas of Σ. A type Σ, in the same language as a theory T, is said to be complete with respect to T if (1) T ∪ Σ is consistent, and (2) for every formula φ(ν) of the language of T (with just ν free), either φ or ¬φ is in Σ.The proof of the result of this paper resembles Morley's proof [5] that the Hanf number for omitting types is . It is shown that there is a model of Th() which omits Σ and contains an infinite set of indiscernibles. Where Morley used the Erdös-Rado generalization of Ramsey's theorem, a definable version of the ordinary Ramsey's theorem is used here.The “omitting types” version of the ω-completeness theorem ([1], [3], [6]) is used, as it was in Morley's proof and in [4]. In [4], satisfaction of the hypotheses of the ω-completeness theorem followed from the fact that, in , any infinite, definable set can be split into two infinite, definable sets.

Some theorems on R-maximal sets and major subsets of recursively enumerable sets

1971 ◽  
Vol 36 (2) ◽  
pp. 193-215 ◽  
Author(s):  
Manuel Lerman
Keyword(s):  
Turing Degrees ◽  
Elementary Equivalence ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Nonstandard Models ◽  
Relational Systems ◽  
The Relationship ◽  
Models Of Arithmetic

In [5], we studied the relational systems /Ā obtained from the recursive functions of one variable by identifying two such functions if they are equal for all but finitely many х ∈ Ā, where Ā is an r-cohesive set. The relational systems /Ā with addition and multiplication defined pointwise on them, were once thought to be potential candidates for nonstandard models of arithmetic. This, however, turned out not to be the case, as was shown by Feferman, Scott, and Tennenbaum [1]. We showed, letting A and B be r-maximal sets, and letting denote the complement of X, that /Ā and are elementarily equivalent (/Ā ≡ ) if there are r-maximal supersets C and D of A and B respectively such that C and D have the same many-one degree (C =mD). In fact, if A and B are maximal sets, /Ā ≡ if, and only if, A =mB. We wish to study the relationship between the elementary equivalence of /Ā and , and the Turing degrees of A and B.

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