Re: Cardinality of Real Numbers

I think the issue is that without the Axiom of Choice, you cannot
assume that all cardinals are orderable. That is to say, assuming the
Axiom of Choice and given any two arbitrary sets S1 and S2, we know
that exactly one of the following three holds:

1. |S1| = |S2| (there is a bijection between S1 and a S2)
2. |S1| < |S2| (there is a bijection between S1 and a proper subset of
S2, but not between S1 & S2)
3. |S1| > |S2| (there is a bijection between a proper subset of S1 and
S2, but not between S1 & S2)

Without AC, you cannot assume that exactly one of the above three must
hold. You could have a case in which S1 and S2 are incomparable; that
is to say, none of the above conditions hold.

The wikipedia entry may get you started:
http://en.wikipedia.org/wiki/Cardinal_number

As for the reals, the Axiom of Choice proves that there exists a
well-ordering of them.

Good Luck!

Jonathan Hoyle

.

Relevant Pages

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