The Axiom of Multiple Choice and Models for Constructive Set Theory

Details The Axiom of Multiple Choice and Models for Constructive Set Theory The Axiom of Multiple Choice and Models for Constructive Set Theory

2012-04-18

arxiv.org/abs/1204.4045v1

Mathematics

Logic

Scientific paper

We propose an extension of Aczel's constructive set theory CZF which is acceptable from a generalised-predicative point of view, because interpretable in Martin-Lof's type theory; strong enough to prove the Set Compactness Theorem and capture a substantial part of formal topology; and preserved by the model constructions of exact completion, realizability and sheaves. More concretely, our proposal is to extend CZF with an axiom postulating the existence of W-types and a choice principle we call the Axiom of Multiple Choice (and which is a slightly simpler and weaker version of the axiom as originally introduced by Palmgren and the second author). We will also show that this extension has a certain robustness about it in that these additional axioms are also reflected by the model constructions we mentioned.

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