nLab
countable choice

Countable choice

Idea

The axiom of countable choice (CCCC), also called AC ωAC_\omega or AC NAC_N, is a weak form of the axiom of choice; it says that the set of natural numbers is a projective object in Set. (Recall that the full axiom of choice states that every set is projective.)

Definition

More explicitly, let XX be any set and let p:XNp\colon X \to \mathbf{N} be a surjection. Then the axiom of countable choice states that pp has a section. If you phrase the axiom of choice in terms of entire relations, then countable choice states that any entire relation from N\mathbf{N} to any other set contains (in the 2-poset Rel) a functional entire relation.

Variations

COSHEPCOSHEP & DCDC

Unlike the full axiom of choice, countable choice is often considered to be a constructively acceptable principle. In particular, it does not imply the principle of excluded middle. It is a consequence of COSHEP. A stronger version of countable choice, also a consequence of COSHEPCOSHEP, is the axiom of dependent choice (DCDC).

AC 00AC_{00}

Sometimes in foundations it is useful to consider a weaker version of countable choice, called AC 00AC_{00}. This states that any entire relation from N\mathbf{N} to itself contains a functional entire relation. In terms of surjections, this states that any surjection p:XNp\colon X \to \mathbf{N} has a section if XX is a subset of N×N\mathbf{N} \times \mathbf{N} and pp is the restriction to XX of a product projection.

Weak countable choice

The axiom of weak countable choice (WCCWCC) states that a surjection p:XNp\colon X \to \mathbf{N} has a section if, whenever mnm \neq n, at least one of the preimages p *(m)p^*(m) and p *(n)p^*(n) is a singleton. WCCWCC follows (for different reasons) from either CCCC or excluded middle. On the other hand, WCCWCC is enough to prove that every Dedekind real number is a Cauchy real number (the converse is always true); more generally, WCCWCC is enough to justify sequential reasoning in analysis. See Bridges et al (1998).

References

  • Douglas Bridges, Fred Richman, and Peter Schuster (1998). A weak countable choice principle. Available from Fred Richman’s Documents.

Revised on December 28, 2013 01:04:39 by Toby Bartels (75.88.42.104)