variations on axiom of choice


The axiom of choiceMathworldPlanetmath states that every set C of non-empty sets has a choice function. There are a number of ways to modify the statement so as to produce something that is similar but perhaps weaker version of the axiom. For example, we can play with the size (cardinality) of the set C, the sizes of the elements in C, as well as the choice function itself. Below are some of the variations of AC:

  1. 1.

    Varying the size of C: let λ be a cardinal. Then AC(λ,) is the statement that every set C of non-empty sets, where |C|=λ, has a choice function. If λ=0, then we have the axiom of countable choice.

  2. 2.

    Varying sizes of members of C: let λ be a cardinal. Then AC(,κ) is the statement that every set C of non-empty sets of cardinality κ has a choice function. Another variation is called the axiom of choice for finite sets AC(,<0): every set C of non-empty finite setsMathworldPlanetmath has a choice function.

  3. 3.

    Varying choice function: The most popular is what is known as the axiom of multiple choice (AMC), which states that every set C of non-empty sets, there is a multivalued function from C to C such that f(A) is finite non-empty and f(A)A.

  4. 4.

    Varying any combinationMathworldPlanetmathPlanetmath of the above three, for example AC(λ,κ) is the statement that every collectionMathworldPlanetmath C of size λ of non-empty sets of size κ has a choice function.

It’s easy to see that all of the variations are provable in ZFC. In additionPlanetmathPlanetmath, some of them are provable in ZF, for example, AC(n,) for any finite cardinal, and AC(,1).

Conversely, it can be shown that AMC implies AC in ZF.

Other implicationsMathworldPlanetmath include: AC(,κ) for all κ implies the axiom of dependent choices, another weaker version of AC. For finite cardinals, we have the following: AC(,mn) implies AC(,n), where m,n are finite cardinals.

Proof.

Let C be a set of non-empty sets of cardinality n. For each aC, define a*:=a×m:={(x,i)xa, and im}. Then the set D={a*aC} is a set of non-empty sets of cardinality mn, hence has a choice function f by assumptionPlanetmathPlanetmath. Then pfg is a choice function for C, where p:DC is the projection given by p(x,i)=x, and g:CD is the function given by g(a)=a*. ∎

For more implications, see the references below.

References

  • 1 H. Herrlich, Axiom of Choice, Springer, (2006).
  • 2 T. J. Jech, The Axiom of Choice, North-Holland Pub. Co., Amsterdam, (1973).
Title variations on axiom of choice
Canonical name VariationsOnAxiomOfChoice
Date of creation 2013-03-22 18:47:09
Last modified on 2013-03-22 18:47:09
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 03E30
Classification msc 03E25
Synonym AMC
Synonym MC
Defines axiom of choice for finite sets
Defines axiom of multiple choice