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August 9th, 2011, 12:01 AM   #1
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a question about the Axiom of choice

how to proof the following conclusion:
for any infinite set S,there exists a bijection f:S--->S*S
implies
the Axiom of choice.

Can you give a proof without the theory of ordinal numbers.
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August 11th, 2011, 04:22 AM   #2
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Re: a question about the Axiom of choice

For existential proof, you only need one example. Of all the functions in , the identity function is the easiest.

Let be an indexed set , so .
It follows where
Since is a function, where denotes an arbitrary ordered pair. This implies one of the six Axiom of Choice.

Conversely,
, where is a relation.
Let .
Then .
Hence at least an identity function exists.
This implies one of the other six Axiom of Choice.
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