2005
DOI: 10.2178/jsl/1107298508
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Theories of arithmetics in finite models

Abstract: 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 Σ 2 -theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ 1 -theory of multiplication and order is decidable in finite models as well as in the standard mo… Show more

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Cited by 7 publications
(3 citation statements)
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“…However passing to finite models the complexity of the theory of divisibility appears to be the same as that of the theory of addition and multiplication. A similar result has been obtained by Michał Krynicki and Konrad Zdanowski for multiplication in [3]. We consider also finite arithmetic of divisibility from the point of view of FM-representability (see [5]).…”
Section: Introductionmentioning
confidence: 54%
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“…However passing to finite models the complexity of the theory of divisibility appears to be the same as that of the theory of addition and multiplication. A similar result has been obtained by Michał Krynicki and Konrad Zdanowski for multiplication in [3]. We consider also finite arithmetic of divisibility from the point of view of FM-representability (see [5]).…”
Section: Introductionmentioning
confidence: 54%
“…However it is known that multiplication is sufficient here. This follows from the following Theorem 3.5 (Krynicki-Zdanowski [3]…”
Section: Definition 33 (Fm-representability)mentioning
confidence: 90%
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