Algebraic combinatorics in bounded induction

In this paper, new methods for analyzing models of weak subsystems of Peano Arithmetic are proposed. The focus will be on the study of algebro-combinatoric properties of certain definable cuts. Their relationship with segments that satisfy more induction, with those limited by the standard powers/roots of an element, and also with definable sets in Bounded Induction is studied. As a consequence, some considerations on the Π1-interpretability of IΔ0 in weak theories, as well as some alternative axiomatizations, are reviewed. Some of the results of the paper are obtained by immersing Bounded Induction models in its Stone-Cech Compactification, once it is endowed with a topology.Ministerio de Ciencia, Innovación y Universidades PID2019-109152GB-I0

Pell equations and exponentiation in fragments of arithmetic

AbstractWe study the relative strength of the two axioms (P) Every Pell equation has a nontrivial solution (exp) Exponentiation is total over weak fragments, and we show they are equivalent over IE1.We then define the graph of the exponential function using only existentially bounded quantifiers in the language of arithmetic expanded with the symbol #, where # (x, y) = x[log2y]. We prove the recursion laws of exponentiation in the corresponding fragment