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 and 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.

Definitions

The limited principle of omniscience states (Bridges and Richman 1987:3):

LPO: For any sequence a₀, a₁, ... such that each a_i is either 0 or 1, the following holds: either a_i = 0 for all i, or there is a k with a_k = 1.

The lesser limited principle of omniscience states:

LLPO: For any sequence a₀, a₁, ... such that each a_i is either 0 or 1, and such that at most one a_i is nonzero, the following holds: either a_2i = 0 for all i, or a_2i+1 = 0 for all i, where a_2i and a_2i+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 (a_i). Answering the question "is there a k with a_k = 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).

References

Bishop, Errett Foundations of Constructive Analysis, 1967. ISBN 4-87187-714-0
Douglas Bridges and Fred Richman, Varieties of Constructive Mathematics, London Mathematical Society Lecture Notes v. 57, 1987. ISBN 0-521-31802-5

External links

Constructive mathematics", Douglas Bridges, Stanford Encyclopedia of Philosophy

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