Sprague–Grundy theory in bounded arithmetic

On Interpretations of Bounded Arithmetic and Bounded Set Theory

In [5], Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic. Theorem 1 The first-order theories of Peano arithmetic and ZF with the axiom of infinity negated are bi-interpretable: that is, they are mutually interpretable with interpretations that are inverse to each other. In this note, I describe a theory of sets that is bi-interpret...

Cuts and overspill properties in models of bounded arithmetic

In this paper we are concerned with cuts in models of Samuel Buss' theories of bounded arithmetic, i.e. theories like $S_{2}^i$ and $T_{2}^i$. In correspondence with polynomial induction, we consider a rather new notion of cut that we call p-cut. We also consider small cuts, i.e. cuts that are bounded above by a small element. We study the basic properties of p-cuts and small cuts. In particula...

Arithmetic Teichmuller Theory

By Grothedieck's Anabelian conjectures, Galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over number fields encode all arithmetic information of these curves. The goal of this paper is to develope and arithmetic teichmuller theory, by which we mean, introducing arithmetic objects summarizing th...

Finitisation in Bounded Arithmetic

I prove various results concerning undecidability in weak fragments of Arithmetic. All results are concerned with S1 2 ⊆ T 1 2 ⊆ S2 2 ⊆ T 2 2 ⊆ .... a hierarchy of theories which have already been intensively studied in the literature. Ideally one would like to separate these systems. However this is generally expected to be a very deep problem, closely related to some of the most famous open p...

Complexity Theory and Bounded Arithmetic for Truly Feasible Computation