Hilbert's tenth problem for weak theories of arithmetic

AbstractHilbert's tenth problem for a theory T asks if there is an algorithm which decides for a given polynomial p(x̄) from Z[x̄] whether p(x̄) has a root in some model of T. We examine some of the model-theoretic consequences that an affirmative answer would have in cases such as T = Open Induction and others, and apply these methods by providing a negative answer in the cases when T is some particular finite fragment of the weak theories IE1 (bounded existential induction) or IU-1 (parameter-free bounded universal induction)

The Skolem-Bang Theorems in Ordered Fields with an IPIP

This paper is concerned with the extent to which the Skolem-Bang theorems in Diophantine approximations generalise from the standard setting of $$to structures of the form$$, where $F$ is an ordered field and $I$ is an integer part of $F$. We show that some of these theorems are hold unconditionally in general case (ordered fields with an integer part). The remainder results are based on Dirichlet's and Kronecker's theorems. Finally we extend Dirichlet's theorem to ordered fields with $IE_1$ integer part.Comment: 28 page

How to escape Tennenbaum's theorem

We construct a theory definitionally equivalent to first-order Peano arithmetic PA and a non-standard computable model of this theory. The same technique allows us to construct a theory definitionally equivalent to Zermelo-Fraenkel set theory ZF that has a computable model.Comment: 10 page

The Borel complexity of the class of models of first-order theories

We investigate the descriptive complexity of the set of models of first-order theories. Using classical results of Knight and Solovay, we give a sharp condition for complete theories to have a $\boldsymbol\Pi_\omega^0$-complete set of models. We also give sharp conditions for theories to have a $\boldsymbol\Pi^0_n$-complete set of models. Finally, we determine the Turing degrees needed to witness the completeness

Models of VTC0VTC^0 as exponential integer parts

We prove that (additive) ordered group reducts of nonstandard models of the bounded arithmetical theory $\mathsf{VTC^0}$ are recursively saturated in a rich language with predicates expressing the integers, rationals, and logarithmically bounded numbers. Combined with our previous results on the construction of the real exponential function on completions of models of $\mathsf{VTC^0}$, we show that every countable model of $\mathsf{VTC^0}$ is an exponential integer part of a real-closed exponential field.Comment: 21 page

An Analysis of Tennenbaum's Theorem in Constructive Type Theory

Tennenbaum's theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit, analyze and generalize this result. The chosen framework allows for a synthetic approach to computability theory, exploiting that, externally, all functions definable in constructive type theory can be shown computable. We then build on this viewpoint and furthermore internalize it by assuming a version of Church's thesis, which expresses that any function on natural numbers is representable by a formula in PA. This assumption provides for a conveniently abstract setup to carry out rigorous computability arguments, even in the theorem's mechanization. Concretely, we constructivize several classical proofs and present one inherently constructive rendering of Tennenbaum's theorem, all following arguments from the literature. Concerning the classical proofs in particular, the constructive setting allows us to highlight differences in their assumptions and conclusions which are not visible classically. All versions are accompanied by a unified mechanization in the Coq proof assistant.Comment: 23 pages, extension of conference paper published at FSCD 202

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