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 (Bridges & Richman 1987). The LPO and LLPO axioms are used to gauge the amount of nonconstructivity required for an argument, as in constructive reverse mathematics. They are also related to weak counterexamples in the sense of Brouwer.
The limited principle of omniscience states (Bridges & Richman 1987, p. 3):
- LPO: For any sequence a0, a1, ... such that each ai is either 0 or 1, the following holds: either ai = 0 for all i, or there is a k with ak = 1.
The lesser limited principle of omniscience states:
- LLPO: For any sequence a0, a1, ... such that each ai is either 0 or 1, and such that at most one ai is nonzero, the following holds: either a2i = 0 for all i, or a2i+1 = 0 for all i, where a2i and a2i+1 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 (ai). Answering the question "is there a k with ak = 1?" 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 Bishop (1967).