A new perspective on completeness and finitist consistency

Abstract

In this paper, we study the metamathematics of consistent arithmetical theories $T$ (containing $\textsf {I}\varSigma _{1}$); we investigate numerical properties based on proof predicates that depend on numerations of the axioms. Numeral Completeness. For every true (in $\mathbb {N}$) sentence $\vec {Q}\vec {x}.\varphi (\vec {x})$, with $\varphi (\vec {x})$ a $\varSigma _{1}(\textsf {I}\varSigma _1)$-formula, there is a numeration $\tau $ of the axioms of $T$ such that $\textsf {I}\varSigma _1\vdash \vec {Q}\vec {x}. \texttt {Pr}_{\tau }(\ulcorner \varphi (\overset {\text{.} }{\vec {x}})\urcorner )$, where $\texttt {Pr}_{\tau }$ is the provability predicate for the numeration $\tau $. Numeral Consistency. If $T$ is consistent, there is a $\varSigma _{1}(\textsf {I}\varSigma _1)$-numeration $\tau $ of the axioms of $\textsf {I}\varSigma _{1}$ such that $\textsf {I}\varSigma _1\vdash \forall\, x. \texttt {Pr}_{\tau }(\ulcorner \neg \textit {Prf}(\ulcorner \perp \urcorner , \overset {\text{.}}{x})\urcorner )$, where $\textit {Prf}(x,y)$ denotes a $\varDelta _{1}(\textsf {I}\varSigma _1)$-definition of ‘$y$ is a $T$-proof of $x$’. Finitist consistency is addressed by generalizing a result of Artemov: Partial finitism. If $T$ is consistent, there is a primitive recursive function $f$ such that, for all $n\in \mathbb {N}$, $f(n)$ is the code of an $\textsf {I}\varSigma _{1}$-proof of $\neg\, \textit{Prf}(\ulcorner \perp \urcorner ,\overline {n})$. These results are not in conflict with Gödel’s Incompleteness Theorems. Rather, they allow to extend their usual interpretation and show a deep connection to reflections in Hilbert’s last papers of 1931.