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 . 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 :
:LPO: For any sequence $a_0$, $a_1$, ... 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 $a_0$, $a_1$, ... such that each $a_i$ is either $0$ or $1$, and such that at most one $a_i$ is nonzero, the following holds: either $a_=0$ for all $i$, or $a_=0$ for all $i$, where $a_$ and $a_$ 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 .

Analytic versions

Both principles have analogous properties of the real numbers. The analytic LPO states that every real number satisfies the tritochtomy $x < 0$ or $x = 0$ or $x > 0$ . The analytic LLPO states that every real number satisfies the ditochtomy $x \geq 0$ or $x \leq 0$, while the analytic Markov's principle states that if $x \geq 0$ is false, then $x < 0$.

All three analytic principles if assumed to hold for the Dedekind or Cauchy real numbers imply their arithmetic versions, while the converse is true if we assume (weak) countable choice, as shown in .

References

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External links

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Constructivism (mathematics)

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