# 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
>
>

```