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Implicative algebras: a new foundation for realizability and forcing

Published online by Cambridge University Press:  03 July 2020

Alexandre Miquel Open the ORCID record for Alexandre Miquel [Opens in a new window]
Alexandre Miquel*
Affiliation:
Instituto de Matemática y Estadística Prof. Ing. Rafael Laguardia, Facultad de Ingeniería, Universidad de la República, Julio Herrera y Reissig 565, MontevideoC.P.11300, Uruguay
*
*Corresponding author. Email: amiquel@fing.edu.uy
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Abstract

We introduce the notion of implicative algebra, a simple algebraic structure intended to factorize the model-theoretic constructions underlying forcing and realizability (both in intuitionistic and classical logic). The salient feature of this structure is that its elements can be seen both as truth values and as (generalized) realizers, thus blurring the frontier between proofs and types. We show that each implicative algebra induces a (Set-based) tripos, using a construction that is reminiscent from the construction of a realizability tripos from a partial combinatory algebra. Relating this construction with the corresponding constructions in forcing and realizability, we conclude that the class of implicative triposes encompasses all forcing triposes (both intuitionistic and classical), all classical realizability triposes (in the sense of Krivine), and all intuitionistic realizability triposes built from partial combinatory algebras.

Keywords

RealizabilityForcingCategorical Logic
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 5 , May 2020 , pp. 458 - 510
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

This work was partly supported by the Uruguayan National Research & Innovation Agency (ANII) under the project “Realizability, Forcing and Quantum Computing,” FCE_1_2014_1_104800.

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