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Infinite sets that Satisfy the Principle of Omniscience in any Variety of Constructive Mathematics

Published online by Cambridge University Press:  12 August 2016

Martín H. Escardó
Martín H. Escardó*
Affiliation:
School of Computer Science, University of Birmingham, Birmingham, B15 2TT, UK, E-mail: [email protected]
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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 εQ, and richer type systems allow us to get higher.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 3 , September 2013 , pp. 764 - 784
Copyright
Copyright © Association for Symbolic Logic 2013

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