variations on axiom of choice
The axiom of choice states that every set 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 , the sizes of the elements in , as well as the choice function itself. Below are some of the variations of AC:
Varying choice function: The most popular is what is known as the axiom of multiple choice (AMC), which states that every set of non-empty sets, there is a multivalued function from to such that is finite non-empty and .
It’s easy to see that all of the variations are provable in ZFC. In addition, some of them are provable in ZF, for example, AC for any finite cardinal, and AC.
Conversely, it can be shown that AMC implies AC in ZF.
For more implications, see the references below.
- 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|
|Date of creation||2013-03-22 18:47:09|
|Last modified on||2013-03-22 18:47:09|
|Last modified by||CWoo (3771)|
|Defines||axiom of choice for finite sets|
|Defines||axiom of multiple choice|