In logic and foundations, a principle of omniscience is any principle of classical mathematics that is not valid in constructive mathematics. The idea behind the name (which is due to Bishop (1967)) is that, if we attempt to extend the computational interpretation of constructive mathematics to incorporate one of these principles, we would have to know something that we cannot compute. The main example is the law of excluded middle (); to apply computationally, we must know which of these disjuncts hold; to apply this in all situations, we would have to know everything (hence ‘omniscience’).
Bishop's principles of omniscience (stated below) may be seen as principles that extend classical logic from predicates (where may happen to be valid, even constructively, for certain predicates) to their quantifications over infinite domains (where is typically not constructively valid).
We have not stated the domain of quantification of the variable . If you take it to be the subsingleton corresponding to a given truth value and apply this principle to the constantly true proposition, then follows; conversely, implies (over any domain). However, Bishop's takes the domain to be the set of natural numbers, giving a weaker principle than . (It appears that a realizability topos based on infinite-time Turing machine?s validates but not ; see Bauer (2011).) Note that propositions of the form when is decidable are the open truth values in the Rosolini dominance.
We can also state the principle set-theoretically, with explicit reference to the domain of quantification. Given a set , the limited principle of omniscience for () states that, given any function from to the boolean domain , either is the constant map to or belongs to the image of . Then Bishop's is , which applies to any infinite sequence of bits.
While for is a classic example of the difference between constructive and classical mathematics, holds for the set of extended natural numbers; this is related to the fact that may constructively be given a compact topology. See Escardó (2011) for this and much more.
The lesser limited principle of omniscience () states that, if the existential quantification of the conjunction of two decidable propositions is false, then one of their separate existential quantifications is false. That is,
If, as before, you take the domains of quantification to be subsingletons, you get de Morgan's law (), which is weaker than ; conversely, implies (over any domain). Again, Bishop's takes the domain to be , giving a principle weaker than (and also weaker than ).
Stated set-theoretically, the lesser limited principle of omniscience for () states that, given functions , if , then or . So Bishop's is . Note that follows from , but not conversely.
Bishop introduced the above principles of omniscience to show that certain results in pointwise analysis could not be constructive, by showing that these results implied a principle of omniscience. The principles of omniscience that appear directly in analysis are these:
The analytic (L)LPO implies the (L)LPO for natural numbers; the converses hold if we assume weak countable choice (as Bishop did). In any case, if we use the Cauchy real numbers (sequential real numbers), then the sequential-analytic (L)LPO is the same as the (L)LPO for natural numbers.
(Note that we need not accept to see that an analytic result implies the (L)LPO and so cannot be constructively valid.)
That the real numbers have decidable equality is weaker than the analytic LPO (and decidable equality for the Cauchy reals in weaker than ), unless we also assume Markov's principle (over for the Cauchy reals, an analytic version for the Dedekind reals).
There are various other results that are equivalent to or related to the principles of omniscience. Here are a few:
Every Cauchy real number has a radix expansion? in any base (e.g., a decimal expansion or binary expansion) iff holds; every Dedekind real number has a radix expansion iff the analytic holds. At least in the presence of countable choice (which also implies that Cauchy and Dedekind reals agree), this is equivalent to the claim that the rings of radix expansions in any two bases are isomorphic. See Daniel Mehkeri's answer to Feldman (2010).
Let be the quotient of the unit interval that identifies the endpoints, and let be the quotient ring; both are classically isomorphic to the circle . (Constructively, we take to be , although can also be constructed as a completion of .) Constructively, there is an injection , which is a bijection if and only if the holds (for the appropriate kind of real number).
The LPO for natural numbers fails in Johnstone’s topological topos, due to its internal continuity principle. Hence, the analytic LPO also fails, since the Cauchy and Dedekind reals coincide in this topos. However, the (analytic) LLPO holds, as a consequence of the preservation of finite closed unions by the inclusion of sequential spaces.
Martín Escardó (2011) via constructivenews, Infinite sets that satisfy the principle of omniscience in all varieties of constructive mathematics (pdf)
David Feldman (2010) on Math.Overflow, Radix notation and toposes