Converse extensionality and apartness

03/26/2021

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by   Benno van den Berg, et al.

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In this paper we try to find a computational interpretation for a strong form of extensionality, which we call "converse extensionality". These converse extensionality principles, which arise as the Dialectica interpretation of the axiom of extensionality, were first studied by Howard. In order to give a computational interpretation to these principles, we reconsider Brouwer's apartness relation, a strong constructive form of inequality. Formally, we provide a categorical construction to endow every typed combinatory algebra with an apartness relation. We then exploit certain continuity principles and that functions reflect apartness, as opposed to preserving equality, to prove that the resulting categories of assemblies model some converse extensionality principles.