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:
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1.
Varying the size of : let be a cardinal. Then AC is the statement that every set of non-empty sets, where , has a choice function. If , then we have the axiom of countable choice.
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2.
Varying sizes of members of : let be a cardinal. Then AC is the statement that every set of non-empty sets of cardinality has a choice function. Another variation is called the axiom of choice for finite sets AC: every set of non-empty finite sets has a choice function.
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3.
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 .
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4.
Varying any combination of the above three, for example AC is the statement that every collection of size of non-empty sets of size has a choice function.
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.
Other implications include: AC for all implies the axiom of dependent choices, another weaker version of AC. For finite cardinals, we have the following: AC implies AC, where are finite cardinals.
Proof.
Let be a set of non-empty sets of cardinality . For each , define . Then the set is a set of non-empty sets of cardinality , hence has a choice function by assumption. Then is a choice function for , where is the projection given by , and is the function given by . ∎
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 |
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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 |