A new perspective on completeness and finitist consistency.

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. [ABSTRACT FROM AUTHOR]