FOM: Re:precursors of Cantor
John Goodrick
goodrick at math.berkeley.edu
Fri Jun 14 15:38:06 EDT 2002
I'm afraid I still don't understand your question. The proof of
Schroder-Bernstein that I've seen in textbooks begins with the assumption
that there are 1-1 functions f: A --> B and g: B --> A and then
constructs a bijection. So it shows that if A and B are EquiSize, then
they have the same power; and the converse is trivial.
-John
On Fri, 14 Jun 2002, charles silver wrote:
> Sorry, let me clarify my question. I know Schroder-Bernstein. Call
> two sets A and B EquiSize (ES) iff there's a 1-1 f'n from A to B and vice
> versa. Suppose set theory does not have Cantor's usual def'n for sets
> being "the same size." Now, one would have to prove that there's also a
> 1-1 *onto* f'n from A to B to prove Cantor's original notion. What I'm
> really wondering about is whether the proofs of *this* direction (from ES)
> would be better ("more intuitive," "more natural," etc.) or worse (etc.)
> than the usual proofs of Schroder-Bernstein (when starting out with Cantor's
> usual def'n). Furthermore, I'm wondering whether "adjusting" set theory
> this way would have other desirable/undesirable effects.
>
> Charlie Silver
>
>
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