Abstract

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 ɛ0, and richer type systems allow us to get higher. §1. Introduction. We show that there are plenty of infinite sets X that satisfy the omniscience principle for every function p: X → 2, ∃x ∈ X(p(x) = 0) ∨ ∀x ∈ X(p(x) = 1). For X finite this is trivial, and for X = N, this is LPO, the limited principle of omniscience, which of course is and will remain a taboo in any variety of

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