Limited principle of omniscience

In constructive mathematics, the limited principle of omniscience (LPO) and the lesser limited principle of omniscience (LLPO) are axioms that are nonconstructive but are weaker than the full law of the excluded middle. They are used to gauge the amount of nonconstructivity required for an argument, as in constructive reverse mathematics. These principles are also related to weak counterexamples in the sense of Brouwer. The limited principle of omniscience states : LPO: For any sequence , , ... such that each is either or , the following holds: either for all , or there is a with . The second disjunct can be expressed as and is constructively stronger than the negation of the first, . The weak schema in which the former is replaced with the latter is called WLPO and represents particular instances of excluded middle. The lesser limited principle of omniscience states: LLPO: For any sequence , , ... such that each is either or , and such that at most one is nonzero, the following holds: either for all , or for all . Here and are entries with even and odd index respectively. It can be proved constructively that the law of the excluded middle implies LPO, and LPO implies LLPO. However, none of these implications can be reversed in typical systems of constructive mathematics. The term "omniscience" comes from a thought experiment regarding how a mathematician might tell which of the two cases in the conclusion of LPO holds for a given sequence . Answering the question "is there a with ?" negatively, assuming the answer is negative, seems to require surveying the entire sequence. Because this would require the examination of infinitely many terms, the axiom stating it is possible to make this determination was dubbed an "omniscience principle" by . The two principles can be expressed as purely logical principles, by casting it in terms of decidable predicates on the naturals. I.e. for which does hold. The lesser principle corresponds to a predicate version of that De Morgan's law that does not hold intuitionistically, i.