Abstract

We present a new constructive proof of the following theorem: there exists a limit-computable function β_1:N→N which eventually dominates every computable function δ_1:N→N. K denotes both the knowledge predicate satisfied by every currently known theorem and the set of all currently known theorems. The set K is time-dependent. Any theorem of any mathematician from past or present forever belongs to K. We prove: (1) there exists a limit-computable function f:N→N of unknown computability which eventually dominates every function δ:N→N with a single-fold Diophantine representation. We present both constructive and non-constructive proof of (1). Statement (1) claims that there exists a function f:N→N such that (f is computable in the limit)∧(¬K(f is computable))∧(¬K(f is uncomputable))∧(f eventually dominates every function δ:N→N with a single-fold Diophantine representation). Since Martin Davis' conjecture on single-fold Diophantine representations disproves statement (1), statement (1) has all properties from the title of the article.