Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the (...) potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover. (shrink)

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. 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 multiplication. (shrink)

The paper discusses the notion of finite model truth definitions (or FM-truth definitions), introduced by M. Mostowski as a finite model analogue of Tarski's classical notion of truth definition. We compare FM-truth definitions with Vardi's concept of the combined complexity of logics, noting an important difference: the difficulty of defining FM-truth for a logic ᵍ does not depend on the syntax of L, as long as it is decidable. It follows that for a natural ᵍ there exist FM-truth definitions whose (...) evaluation is much easier than the combined complexiy of ᵍ would suggest. We apply the general theory to give a complexity-theoretical characterization of the logics for which the $\Sigma_{m}^{d}$ classes (prenex classes of higher order logics) define FM-truth. For any $d \geq 2.m \geq 1$ we construct a family $\{[\Sigma_{m}^{d}]^{\leg\kappa}\}\kappa\in\omega$ of syntactically defined fragments of $\Sigma_{m}^{d}$ which satisfy this characterization. We also use the $[\Sigma_{m}^{d}]\leg\kappa$ classes to give a refinement of known results on the complexity classes captured by $\Sigma_{m}^{d}$ . We close with a few simple corollaries, one of which gives a sufficient condition for the existence, given a vocabulary $\sigma$ , of a fixed number $\kappa$ such that model checking for all first order sentences over $\sigma$ can be done in deterministic time $n^{\kappa}$. (shrink)