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arXiv · 2312.13634

Peano Arithmetic and $\mu$MALL

Matteo Manighetti·Dale Miller
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Abstract

Formal theories of arithmetic have traditionally been based on either classical or intuitionistic logic, leading to the development of Peano and Heyting arithmetic, respectively. We propose to use $\mu$MALL as a formal theory of arithmetic based on linear logic. This formal system is presented as a sequent calculus proof system that extends the standard proof system for multiplicative-additive linear logic (MALL) with the addition of the logical connectives universal and existential quantifiers (first-order quantifiers), term equality and non-equality, and the least and greatest fixed point operators. We first demonstrate how functions defined using $\mu$MALL relational specifications can be computed using a simple proof search algorithm. By incorporating weakening and contraction into $\mu$MALL, we obtain $\mu$LK+, a natural candidate for a classical sequent calculus for arithmetic. While important proof theory results are still lacking for $\mu$LK+ (including cut-elimination and the completeness of focusing), we prove that $\mu$LK+ is consistent and that it contains Peano arithmetic. We also prove some conservativity results regarding $\mu$LK+ over $\mu$MALL.

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Matteo Manighetti, Dale Miller. 2023-12-21. Peano Arithmetic and $\mu$MALL. https://doi.org/10.46298/fi.12736

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