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arXiv · 2410.01972

Completeness and Well-Definability of a Provability Degree Measure in Sufficiently Powerful Formal Systems, and Finite-Time Effective Knowers

Rohan Bahl
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Abstract

We show that including degrees of a particular kind of provability in the search target for any theorem-prover in sufficiently powerful formal systems over finite-sized statements preserves well-definition and a sufficient consistency while establishing completeness. Moreover, the union of such degrees is isomorphic to such a system's {\aleph_0} statements and permits the construction of a best-possible (up to a quadratic term) finite-time theorem knower, {\phi}', while still subject to limitations in these formal systems. These results, owing to the fact that {\phi}' may arise through the behavior of any unbounded inductive computation, establish results on the eventual behavior of a class of computational processes.

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Rohan Bahl. 2024-10-02. Completeness and Well-Definability of a Provability Degree Measure in Sufficiently Powerful Formal Systems, and Finite-Time Effective Knowers. https://arxiv.org/abs/2410.01972

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