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September 2013 Infinite sets that satisfy the principle of omniscience in any variety of constructive mathematics
Martín Escardó
J. Symbolic Logic 78(3): 764-784 (September 2013). DOI: 10.2178/jsl.7803040

Abstract

We show that there are plenty of infinite sets that satisfy the omniscience principle, in a minimalistic setting for constructive mathematics that is compatible with classical mathematics. A first example of an omniscient set is the one-point compactification of the natural numbers, also known as the generic convergent sequence. We relate this to Grilliot's and Ishihara's Tricks. We generalize this example to many infinite subsets of the Cantor space. These subsets turn out to be ordinals in a constructive sense, with respect to the lexicographic order, satisfying both a well-foundedness condition with respect to decidable subsets, and transfinite induction restricted to decidable predicates. The use of simple types allows us to reach any ordinal below $\epsilon_0$, and richer type systems allow us to get higher.

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Martín Escardó. "Infinite sets that satisfy the principle of omniscience in any variety of constructive mathematics." J. Symbolic Logic 78 (3) 764 - 784, September 2013. https://doi.org/10.2178/jsl.7803040

Information

Published: September 2013
First available in Project Euclid: 6 January 2014

zbMATH: 1308.03060
MathSciNet: MR3135497
Digital Object Identifier: 10.2178/jsl.7803040

Rights: Copyright © 2013 Association for Symbolic Logic

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Vol.78 • No. 3 • September 2013
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