nLab principle of omniscience

Principles of omniscience

Context

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms

Principles of omniscience

Idea

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 (EMEM); to apply p¬pp \vee \neg{p} 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 may be seen as principles that extend classical logic from predicates (where EMEM may happen to be valid, even constructively, for certain predicates) to their quantifications over infinite domains (where EMEM is typically not constructively valid).

Examples

Examples of principles of omniscience include

Bishop’s principles of omniscience take the domain of discourse to be the natural numbers. However, it is possible to use some other domain of discourse to get a different principle of omniscience.

There are also the analytic principles of omniscience, which are versions of Bishop’s principles of omniscience that are used in real analysis, including

In addition, in King 2024 there are more generalizations of the principles of omniscience listed above, which are equivalent to Bishop’s principles of omniscience under the weak countable choice principle AC ,2\mathrm{AC}_{\mathbb{N}, 2}.

There are also generalisations of the principles of omniscience involving inequality spaces:

References

category: disambiguation

Last revised on September 19, 2024 at 21:06:47. See the history of this page for a list of all contributions to it.