Recursively saturated nonstandard models of arithmetic; addendum

1982 ◽  
Vol 47 (3) ◽  
pp. 493-494
Author(s):  
C. Smoryński
Keyword(s):  
Nonstandard Models ◽  
Recursively Saturated ◽  
Models Of Arithmetic

James H. Schmerl. Peano models with many generic classes. Pacific Journal of Mathematics, vol. 43 (1973), pp. 523–536. - James H. Schmerl. Correction to: “Peano models with many generic classes”. Pacific Journal of Mathematics, vol. 92 (1981), no. 1, pp. 195–198. - James H. Schmerl. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80 (Proceedings, Seminars, and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80). edited by M. Lerman, J. H. Schmerl, and R. I. Soare, Lecture Notes in Mathematics, vol. 859. Springer, Berlin, pp. 268–282. - James H. Schmerl. Recursively saturatedmodels generated by indiscernibles. Notre Dane Journal of Formal Logic, vol. 26 (1985), no. 1, pp. 99–105. - James H. Schmerl. Large resplendent models generated by indiscernibles. The Journal of Symbolic Logic, vol. 54 (1989), no. 4, pp. 1382–1388. - James H. Schmerl. Automorphism groups of models of Peano arithmetic. The Journal of Symbolic Logic, vol. 67 (2002), no. 4, pp. 1249–1264. - James H. Schmerl. Diversity in substructures. Nonstandard models of arithmetic and set theory. edited by A. Enayat and R. Kossak, Contemporary Mathematics, vol. 361, American Mathematical Societey (2004), pp. 45–161. - James H. Schmerl. Generic automorphisms and graph coloring. Discrete Mathematics, vol. 291 (2005), no. 1–3, pp. 235–242. - James H. Schmerl. Nondiversity in substructures. The Journal of Symbolic Logic, vol. 73 (2008), no. 1, pp. 193–211.

2009 ◽  
Vol 15 (2) ◽  
pp. 222-227
Author(s):  
Roman Kossak
Keyword(s):  
Set Theory ◽  
Graph Coloring ◽  
Automorphism Groups ◽  
Peano Arithmetic ◽  
Discrete Mathematics ◽  
Symbolic Logic ◽  
Nonstandard Models ◽  
Recursively Saturated ◽  
University Of Connecticut ◽  
Models Of Arithmetic

Recursively saturated nonstandard models of arithmetic

1981 ◽  
Vol 46 (2) ◽  
pp. 259-286 ◽  
Author(s):  
C. Smoryński
Keyword(s):  
Saturation Properties ◽  
Domain Of Applicability ◽  
Nonstandard Models ◽  
Partial Type ◽  
Truth Definition ◽  
Recursively Saturated ◽  
Models Of Arithmetic

Through the ability of arithmetic to partially define truth and the ability of infinite integers to simulate limit processes, nonstandard models of arithmetic automatically have a certain amount of saturation: Any encodable partial type whose formulae all fall into the domain of applicability of a truth definition must, by finite satisfiability and Overspill, be nonstandard-finitely satisfiable—whence realized. This fact was first exploited by A. Robinson, who in Robinson [1963] cited the unrealizability in a given model of a certain encodable partial type to prove Tarski's Theorem on the Undefinability of Truth. A decade later, H. Friedman brought this phenomenon to the public's attention by using it to establish impressive embeddability criteria for countable nonstandard models of arithmetic. Subsequently, Wilkie considered models expandable to “strong theories” and, among such models, complemented Friedman's embeddability criteria with elementary embeddability and isomorphism criteria. Oddly enough, the fact that some kind of saturation property was being employed was not explicitly acknowledged in any of this work.It is in the unpublished dissertation of Wilmers that these submerged saturation properties first surfaced. [As I have only seen accounts of it (most notably Murawski [1976/1977]) and not the dissertation itself, what I have to say about it will not quite be accurate. (Indeed, the referee has refuted, without providing alternate information, every conjecture I have made about the contents of this thesis.)

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.

Omega-inconsistency without cuts and nonstandard models

2016 ◽  
Vol 13 (5) ◽  
Author(s):  
Andreas Fjellstad
Keyword(s):  
Deontic Logic ◽  
Sequent Calculus ◽  
Standard Deontic Logic ◽  
Nonstandard Models ◽  
Deontic Modality ◽  
The Relationship ◽  
Labelled Sequent Calculus ◽  
Models Of Arithmetic

This paper concerns the relationship between transitivity of entailment, omega-inconsistency and nonstandard models of arithmetic. First, it provides a cut-free sequent calculus for non-transitive logic of truth STT based on Robinson Arithmetic and shows that this logic is omega-inconsistent. It then identifies the conditions in McGee (1985) for an omega-inconsistent logic as quantified standard deontic logic, presents a cut-free labelled sequent calculus for quantified standard deontic logic based on Robinson Arithmetic where the deontic modality is treated as a predicate, proves omega-inconsistency and shows thus, pace Cobreros et al.(2013), that the result in McGee (1985) does not rely on transitivity. Finally, it also explains why the omega-inconsistent logics of truth in question do not require nonstandard models of arithmetic.

Jon Barwise and John Schlipf. On recursively saturated models of arithmetic. Model theory and algebra, A memorial tribute to Abraham Robinson, edited by D. H. Saracino and V. B. Weispfenning, Lecture notes in mathematics, vol. 498, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, pp. 42–55. - Patrick Cegielski, Kenneth McAloon, and George Wilmers. Modèles récursivement saturés de l'addition et de la multiplication des entiers naturels. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and London, 1982, pp. 57–68. - Julia F. Knight. Theories whose resplendent models are homogeneous. Israel journal of mathematics, vol. 42 (1982), pp. 151–161. - Julia Knight and Mark Nadel. Expansions of models and Turing degrees. The journal of symbolic logic, vol. 47 (1982), pp. 587–604. - Julia Knight and Mark Nadel. Models of arithmetic and closed ideals. The journal of symbolic logic, vol. 47 no. 4 (for 1982, pub. 1983), pp. 833–840. - Henryk Kotlarski. On elementary cuts in models of arithmetic. Fundamenta mathematicae, vol. 115 (1983), pp. 27–31. - H. Kotlarski, S. Krajewski, and A. H. Lachlan. Construction of satisfaction classes for nonstandard models. Canadian mathematical bulletin—Bulletin canadien de mathématiques, vol. 24 (1981), pp. 283–293. - A. H. Lachlan. Full satisfaction classes and recursive saturation. Canadian mathematical bulletin—Bulletin canadien de mathématiques, pp. 295–297. - Leonard Lipshitz and Mark Nadel. The additive structure of models of arithmetic. Proceedings of the American Mathematical Society, vol. 68 (1978), pp. 331–336. - Mark Nadel. On a problem of MacDowell and Specker. The journal of symbolic logic, vol. 45 (1980), pp. 612–622. - C. Smoryński. Back-and-forth inside a recursively saturated model of arithmetic. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and London, 1982, pp. 273–278. - C. Smoryński and J. Stavi. Cofinal extension preserves recursive saturation. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7,1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 338–345. - George Wilmers. Minimally saturated models. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 370–380.

1987 ◽  
Vol 52 (1) ◽  
pp. 279-284
Author(s):  
J.-P. Ressayre
Keyword(s):  
New York ◽  
Model Theory ◽  
Foundations Of Mathematics ◽  
Symbolic Logic ◽  
Recursive Saturation ◽  
North Holland Publishing ◽  
Lecture Notes ◽  
Recursively Saturated ◽  
Satisfaction Classes ◽  
Models Of Arithmetic

scholarly journals The Complexity of Classification Problems for Models of Arithmetic

2010 ◽  
Vol 16 (3) ◽  
pp. 345-358 ◽  
Author(s):  
Samuel Coskey ◽  
Roman Kossak
Keyword(s):  
Classification Problem ◽  
The Other ◽  
Saturated Model ◽  
Classification Problems ◽  
Finitely Generated ◽  
Saturated Models ◽  
Other Hand ◽  
Recursively Saturated ◽  
Recursively Saturated Model ◽  
Models Of Arithmetic

AbstractWe 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.

scholarly journals Interpreting the compositional truth predicate in models of arithmetic

2021 ◽  
Author(s):  
Cezary Cieśliński
Keyword(s):  
Proof Theory ◽  
Formal Proof ◽  
Truth Predicate ◽  
Saturated Model ◽  
First Order ◽  
Order Arithmetic ◽  
Recursively Saturated ◽  
Recursively Saturated Model ◽  
Models Of Arithmetic

AbstractWe present a construction of a truth class (an interpretation of a compositional truth predicate) in an arbitrary countable recursively saturated model of first-order arithmetic. The construction is fully classical in that it employs nothing more than the classical techniques of formal proof theory.

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