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A Note on Real Subsets of A Recursively Saturated Model

Athanassios Tzouvaras

Mathematical Logic Quarterly 37 (13-16):207-216 (1991)

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Automorphisms of Countable Short Recursively Saturated Models of PA.Erez Shochat - 2008 - Notre Dame Journal of Formal Logic 49 (4):345-360.details A model of Peano Arithmetic is short recursively saturated if it realizes all its bounded finitely realized recursive types. Short recursively saturated models of $\PA$ are exactly the elementary initial segments of recursively saturated models of $\PA$. In this paper, we survey and prove results on short recursively saturated models of $\PA$ and their automorphisms. In particular, we investigate a certain subgroup of the automorphism group of such models. This subgroup, denoted $G\|_{M(a)}$, contains all the automorphisms of a countable short (...) Download Export citation Bookmark 2 citations
On expandability of models of peano arithmetic to models of the alternative set theory.Athanassios Tzouvaras - 1992 - Journal of Symbolic Logic 57 (2):452-460.details We give a sufficient condition for a countable model M of PA to be expandable to an ω-model of AST with absolute Ω-orderings. The condition is in terms of saturation schemes or, equivalently, in terms of the ability of the model to code sequences which have some kind of definition in (M, ω). We also show that a weaker scheme of saturation leads to the existence of wellorderings of the model with nice properties. Finally, we answer affirmatively the question of (...) Download Export citation Bookmark
Freiling's axioms of symmetry in a general setting and some applications.Athanassios Tzouvaras - 2001 - Archive for Mathematical Logic 40 (2):131-145.details We formulate C. Freiling's axioms of symmetry for general second-order structures with respect to a certain ideal of small sets contained in them and find several equivalent formulations of the principles. Then we focus on particular models, namely saturated and recursively saturated ones, and show that they are symmetric with respect to appropriate classes of small sets when their second-order part consists of definable sets. Some asymmetric models are also exhibited as well as partial asymmetric ones constructed by forcing. Download Export citation Bookmark 1 citation
Automorphisms of recursively saturated models of arithmetic.Richard Kaye, Roman Kossak & Henryk Kotlarski - 1991 - Annals of Pure and Applied Logic 55 (1):67-99.details We give an examination of the automorphism group Aut of a countable recursively saturated model M of PA. The main result is a characterisation of strong elementary initial segments of M as the initial segments consisting of fixed points of automorphisms of M. As a corollary we prove that, for any consistent completion T of PA, there are recursively saturated countable models M1, M2 of T, such that Aut[ncong]Aut, as topological groups with a natural topology. Other results include a classification (...) Download Export citation Bookmark 32 citations

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