Incompleteness of arithmetic and the Diophantine sets theory - Hupal, Anatolii; Hupal, Mykyta (PHYSICO-MATHEMATICAL MODELLING AND INFORMATIONAL TECHNOLOGIES, 2023)
Analysis of Diophantine sets showed that all recursively enumerated sets are Diophantine. Based on the classical results in the theory of computable functions, a simple version of the theorem on the incompleteness of arithmetic can be given: there is a polynomial that does not have positive integer solutions, and for which it is impossible to prove the absence of positive roots