Results 1 
4 of
4
The automorphism group of an arithmetically saturated model of Peano arithmetic
 J. London Math. Soc
, 1995
"... One of the main goals in the study of the automorphism group Aut {Jt) of a countable, recursively saturated model Jt of Peano Arithmetic is to determine to what extent (the isomorphism type of) Jt is recoverable from (the isomorphism type of) Aut(^). A countable, recursively saturated model Jt of PA ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
One of the main goals in the study of the automorphism group Aut {Jt) of a countable, recursively saturated model Jt of Peano Arithmetic is to determine to what extent (the isomorphism type of) Jt is recoverable from (the isomorphism type of) Aut(^). A countable, recursively saturated model Jt of PA is characterized up to isomorphism by two invariants: its firstorder theory Th(^) and its standard system SSy {Jt). At present, there seems to be no indication of how to recover any information about Th {Jt) from Aut {Jt) with the exception of whether or not Th {Jt) is True Arithmetic. We define the notion of arithmetically saturated in Definition 1.7; however, a model Jt of PA is arithmetically saturated if and only if it is recursively saturated and the standard cut is a strong cut. The following is our main theorem. THEOREM. Suppose that Jtx and Jt2 are countable, arithmetically saturated models of PA such that Aut(^) s Aut {Jt2). Then SSy {Jtx) = SSy {Jt2). In the Theorem, it suffices to assume that Jt2 is just recursively saturated. For, as shown by Lascar [8], if Jtx and Jt2 are countable, recursively saturated models of PA
Arithmetic and the Incompleteness Theorems
, 2000
"... this paper please consult me first, via my home page. ..."
THE COMPLEXITY OF CLASSIFICATION PROBLEMS FOR MODELS OF ARITHMETIC
, 908
"... Abstract. We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of ..."
Abstract
 Add to MetaCart
Abstract. We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete. 1.
WEAKLY DEFINABLE TYPES
"... ABSTRACT. We study some generalizations of the notion of a definable type, first in an abstract setting in terms of ultrafilters on certain Boolean algebras, and then as applied to model theory. The notion of a weakly definable ultrafilter or type was developed by one of the authors [K] in a study o ..."
Abstract
 Add to MetaCart
ABSTRACT. We study some generalizations of the notion of a definable type, first in an abstract setting in terms of ultrafilters on certain Boolean algebras, and then as applied to model theory. The notion of a weakly definable ultrafilter or type was developed by one of the authors [K] in a study of models of arithmetic. It generalizes the notion of a definable type; and just as this latter notion has interesting properties in a much more general context, especially in stability theory, it seemed worthwhile to investigate weakly definable types in a general modeltheoretic setting. A goal of this paper is to present the results of our investigations on these lines. It is natural to ask why such notions turn up both in arithmetic and in elementary stability theory. Ressayre, for example, in a review [R] of Gaifman's paper [G], says...although the notion of definable type was introduced by Gaifman in the study of PA, which is the most unstable theory, this notion turned out to be a fundamental one for stable theories. And minimal as well as uniform types also correspond more or less to properties important in the stable case. I expect