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Consistent Ultrafinitist Logic

Author Michał J. Gajda

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Michał J. Gajda
  • Migamake Pte Ltd, Singapore

Acknowledgements

We thank Vendran Čačić, Seth Chaiken, Bhupinder Singh Anand, Orestis Melkonian, Anton Setzer for their invaluable feedback. Thanks to Daniel Schwartz for sparkling my interest in Kleene normal form.

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Michał J. Gajda. Consistent Ultrafinitist Logic. In 29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.TYPES.2023.5

Abstract

Ultrafinitism postulates that we can only compute on relatively short objects, and numbers beyond a certain value are not available. This approach would also forbid many forms of infinitary reasoning and allow removing certain paradoxes stemming from enumeration theorems. For a computational application of ultrafinitist logic, we need more than a proof system, but a logical framework to express both proofs, programs, and theorems in a single framework. We present its inference rules, reduction relation, and self-encoding to allow direct proving of the properties of ultrafinitist logic within itself. We also provide a justification why it can express all bounded Turing programs, and thus serve as a "logic of computability".

Subject Classification

ACM Subject Classification
  • Theory of computation → Constructive mathematics
Keywords
  • ultrafinitism
  • bounded Turing completeness
  • logic of computability
  • decidable logic
  • explicit complexity
  • strict finitism

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