## Theories of arithmetics in finite models.(English)Zbl 1094.03022

Let $$A$$ be a first-order structure whose universe is $$\omega$$. For each $$n<\omega$$, let $$A_n$$ denote the (naturally defined) restriction of $$A$$ to $$\{0,\dots, n\}$$. Let FM$$(A)$$ be the family of all $$A_n$$’s. Marcin Mostowski introduced the notion of satisfiability in sufficiently large finite models in FM$$(A)$$. This is the central notion of the paper.
The focus is on decidability and other properties of the suitably defined theory Th(FM$$(A))$$, and the theory of sufficiently large models sl(FM$$(A))$$, where $$A$$ is one of the standard models of the form $$(\omega, +)$$, $$(\omega, \times)$$, $$(\omega, \times, \leq)$$, or $$(\omega, \exp)$$. It is a substantial paper, I only list some of the main results.
In Section 4, the authors prove that the set of $$\Sigma_2$$ sentences of arithmetic of multiplication which are satisfiable in finite models is $$\Sigma^0_1$$-complete, and the set of those $$\Sigma_2$$ sentences which are true in all sufficiently large models is $$\Sigma^0_1$$-hard. Also, for $$A$$ being either $$(\omega, \times)$$ or $$(\omega, \exp)$$, Th(FM$$(A))$$ is $$\Pi^0_1$$-complete, and sl(FM$$(A))$$ is $$\Sigma^0_2$$-complete.
The main result of Section 5 is that the theory of sufficiently large models in FM$$(\omega, \times,\leq)$$ is decidable and so is the existential theory of the standard model of multiplication and order.
In Section 6 the authors consider the arithmetic of concatenation and show that in finite models it has the same strength as the arithmetic of addition and multiplication.
The FM$$(A)$$-spectrum of a sentence $$\varphi$$ is the set $$\{n+1: A_n\models \varphi\}$$. The spectrum of FM$$(A)$$ is the set of all FM$$(A)$$-spectra of all sentences of the language of FM$$(A)$$. Section 7 contains several interesting results concerning the spectra of FM$$(A)$$ for various $$A$$. In particular, it is shown that the spectrum of exponentiation is strictly contained in the spectrum of addition and multiplication.

### MSC:

 03C13 Model theory of finite structures 03C68 Other classical first-order model theory 03F30 First-order arithmetic and fragments 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 03B25 Decidability of theories and sets of sentences
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