Arithmetics in finite but potentially infinite worlds

Let FM(A), for A = (ω, R̄), be the family of finite models being initial segments of A. The thesis investigates logical properties of families of the form FM(A) for various arithmetics like arithmetic of addition and multiplication, Skolem arithmetic of multiplication, arithmetic of coprimality, of exponentiation and arithmetic of concatenation. We concentrate on questions such as decidability of various theories of FM(A); definability and interpretability of one arithmetic, FM(A) in another, FM(B); the problem of representing infinite relations in such families of models and on the spectrum problem for such arithmetics. Following M. Mostowski ([31, 32]), we define some methods which one can use to represent infinite relations in finite models and some natural theories of families FM(A), such as the set of sentences true in almost all finite models from FM(A), sl(FM(A)), or the set of sentences which are almost surely true in FM(A) (in a probabilistic sense). We show that for A = (ω,+,×), the first set is Σ2–complete and the second one is Π3–complete. We also characterize relations which can be represented in both theories as exactly ∆2 relations (for the first theory such a characterization was obtained in [31]). We show that the above remains true even in the relatively weak arithmetic of multiplication. We also consider various notions of definability and interpretability between arithmetics of finite models. We give the definition of FM((ω,+,×)) in the finite models of arithmetic of concatenation. This is an analogous to the situation in the infinite models for these arithmetics but one should use a different method to give a suitable definition. We show that, contrary to the infinite case, arithmetic of exponentiation, FM((ω, exp)), is definable from arithmetic of multiplication only, FM((ω,×)). We also give interpretations of FM((ω,+,×)) in arithmetic of coprimality, FM((ω,⊥)), and in FM((ω, exp)). The interpretations reveal that in finite models coprimality or exponentiation are as hard as the full arithmetic of addition and multiplication, which is especially surprising in the case of coprimality. We also describe the decidability border for finite model arithmetic of multiplication showing that the Σ1–theory of FM((ω,×,≤)) is decidable while the Σ2–theory of FM((ω,×)) is undecidable. We close the thesis with a partial characterization of families of spectra for FM((ω,×)) and FM((ω, exp)).