Arithmetic of divisibility in finite models (original) (raw)
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Theories of arithmetics in finite models
The Journal of Symbolic Logic, 2005
We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ2-theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ1-theory of multiplication and order is decidable in finite models as well as in the standard model. We show also that the exponentiation function is definable in finite models by a formula of arithmetic with multiplication and that one can define in finite models the arithmetic of addition and multiplication with the concatenation operation.
Fundamenta Informaticae
The paper presents the current state of knowledge in the field of logical investigations of finite arithmetics. This is an attempt to summarize the ideas and results in this area. Some new results are presented - these are mainly generalizations of the earlier results related to properties of sl-theories and some nontrivial cases of FM-representability theorem.
Existentially Closed Models in the Framework of Arithmetic
Journal of Symbolic Logic, 2016
We prove that the standard cut is definable in each existentially closed model of I Δ 0 +exp by a (parameter free) Π 1-formula. This definition is optimal with respect to quantifier complexity and allows us to improve some previously known results on existentially closed models of fragments of arithmetic. 1. Introduction. This work was initially motivated by a gap in the proof of Corollary 1.3 of [2] providing a parameter free Π 1-definition of the standard cut, N, in each existentially closed (e.c.) model of I Δ 0 + exp. Our aim is to provide a correct proof of the above result and, use it to obtain an updated view of the theory of e.c. models of I Δ 0 +exp. Existentially closed models of arithmetic were investigated in the 1970's as a part of the efforts to get a full understanding of the model theory of existentially closed structures (existence of model completions and companion theories, finite and infinite forcing, etc.). The results obtained in the early 1970's by A. Robinson, J. Hirschfeld, D. C. Goldrei, A. Macintyre, and H. Simmons pointed out the most important property of e.c. models of sufficiently strong arithmetic theories: there exist formulas defining N in each such model. These results were not stated in their full generality. In the 1970's a systematic study of fragments of Peano arithmetic PA was still to come and the authors focused essentially on e.c. models of Π 2 (N) (thesetoftrueΠ 2-sentences) or of Π 2 (PA)(thesetofΠ 2 consequences of PA), and more generally on e.c. models of Π 2 (T B), where T B is any extension of Π 2 (PA). Regarding Π 2 (N), Robinson (see [14]) proved N to be Σ 3-definable in every e.c. model of Π 2 (N) and Hirschfeld (see [7]) improved Robinson's result obtaining a Σ 2-definition of N,or even aΠ 1-definition, if parameters are allowed. Hirschfeld also showed that these definitions are optimal (in terms of quantifier complexity) for e.c. models of Π 2 (N). As to Π 2 (T B), in [11] Macintyre and Simmons (see also [5]) extended Hirschfeld's Σ 2-definition of N to all e.c. models of Π 2 (T B) and showed that the parametric Π 1definition can be extended to those e.c. models of Π 2 (T B)inwhichtheΣ 1-definable elements are not cofinal. However, these definitions are not best possible, since there is no general result ruling out the possibility of a parameter free Π 1-definition of N valid in all e.c. models. As a matter of fact, such an optimal definition was Key words and phrases. fragments of Peano arithmetic, existentially closed models, turing degrees of arithmetic theories.
Computable Quotient Presentations of Models of Arithmetic and Set Theory
2017
We prove various extensions of the Tennenbaum phenomenon to the case of computable quotient presentations of models of arithmetic and set theory. Specifically, no nonstandard model of arithmetic has a computable quotient presentation by a c.e. equivalence relation. No \(\Sigma _1\)-sound nonstandard model of arithmetic has a computable quotient presentation by a co-c.e. equivalence relation. No nonstandard model of arithmetic in the language \(\{+,\cdot ,\le \}\) has a computably enumerable quotient presentation by any equivalence relation of any complexity. No model of ZFC or even much weaker set theories has a computable quotient presentation by any equivalence relation of any complexity. And similarly no nonstandard model of finite set theory has a computable quotient presentation.
Completeness theorems, incompleteness theorems and models of arithmetic
Transactions of the American Mathematical Society, 1978
Let & be a consistent extension of Peano arithmetic and let 6EJJ denote the set of TL°" consequences of &. Employing incompleteness theorems to generate independent formulas and completeness theorems to construct models, we build nonstandard models of SP"+2 m which the standard integers are A°+1-definable. We thus pinpoint induction axioms which are not provable in éE¡¡+2; in particular, we show that (parameter free) A?-induction is not provable in Primitive Recursive Arithmetic. Also, we give a solution of a problem of Gaifman on the existence of roots of diophantine equations in end extensions and answer questions about existentially complete models of 3^. Furthermore, it is shown that the proof of the Gödel Completeness Theorem cannot be formalized in 6E § and that the MacDowell-Specker Theorem fails for all truncated theories (£¡¡.
Arithmetical definability and computational complexity
Theoretical Computer Science, 2004
In this paper, we introduce and study some syntactical fragments of monadic second-order and ÿrst-order (PEANO) arithmetic which we will prove the connection to famous complexity classes. Starting from descriptive complexity results, and giving an e ective method for translating formulas between di erent logical structures representing encodings of integers, we give some new arithmetical characterizations of NP, PH, NL, and P.