The predicate K of the current mathematical knowledge non-trivially extends computability theory

Tyszka, Apoloniusz

The predicate K of the current mathematical knowledge non-trivially extends computability theory

Abstract

K denotes both the knowledge predicate satisfied by every currently known theorem and the finite set of all currently known theorems. The set K is time-dependent and publicly available. Any theorem of any mathematician from past or present forever belongs to K. We prove the following Statement 1: there is a limit-computable function β:N→N of unknown computability which eventually dominates every function δ:N→N with a single-fold Diophantine representation. Statement 1 holds when, for every n∈N, β(n) is the smallest b∈N such that if a system of equations S⊆{1=x_k, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k∈{0,...,n}} has a unique solution in N^{n+1}, then this solution belongs to {0,...,b}^{n+1}. Statement 1 does not follow from any widely known mathematical theorem. Ignoring the epistemic condition in Statement 1, Statement 1 is implied by the following result of Royer and Case: There is a limit-computable function β_1:N→N which eventually dominates every computable function δ_1:N→N. The proof of Statement 1 shows that Statement 1 follows from some mathematical theorem and the following conjunction: ((The function β is computable)∉K)∧((The function β is uncomputable)∉K).

Keywords

current mathematical knowledge eventually dominates limit-computable function single-fold Diophantine representation

Citation

Tyszka, Apoloniusz, The predicate K of the current mathematical knowledge non-trivially extends computability theory (June 11, 2024). Available at SSRN: https://ssrn.com/abstract=4710446 or http://dx.doi.org/10.2139/ssrn.4710446