Source: J. Symbolic Logic
Volume 70, Issue 1
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
We consider also the spectrum problem. We show that the spectrum of
arithmetic with multiplication and arithmetic with exponentiation is
strictly contained in the spectrum of arithmetic with addition and
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