Church's thesis and related axioms in Coq's type theory
| Autor: | Forster, Yannick |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
FOS: Computer and information sciences
Mathematics::Logic Computer Science - Logic in Computer Science constructive type theory Computer Science::Logic in Computer Science Church’s thesis constructive reverse mathematics Coq synthetic computability theory Theory of computation → Constructive mathematics Theory of computation → Type theory Logic in Computer Science (cs.LO) |
| Popis: | "Church’s thesis" (CT) as an axiom in constructive logic states that every total function of type ℕ → ℕ is computable, i.e. definable in a model of computation. CT is inconsistent both in classical mathematics and in Brouwer’s intuitionism since it contradicts weak Kőnig’s lemma and the fan theorem, respectively. Recently, CT was proved consistent for (univalent) constructive type theory. Since neither weak Kőnig’s lemma nor the fan theorem is a consequence of just logical axioms or just choice-like axioms assumed in constructive logic, it seems likely that CT is inconsistent only with a combination of classical logic and choice axioms. We study consequences of CT and its relation to several classes of axioms in Coq’s type theory, a constructive type theory with a universe of propositions which proves neither classical logical axioms nor strong choice axioms. We thereby provide a partial answer to the question as to which axioms may preserve computational intuitions inherent to type theory, and which certainly do not. The paper can also be read as a broad survey of axioms in type theory, with all results mechanised in the Coq proof assistant. LIPIcs, Vol. 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021), pages 21:1-21:19 |
| Databáze: | OpenAIRE |
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