Re: finding a bijection from [0,1]^2 to [0,1]

From: Dave Seaman (dseaman_at_no.such.host)
Date: 03/05/05 

Date: Sat, 5 Mar 2005 14:16:53 +0000 (UTC)

On Sat, 05 Mar 2005 01:46:54 -0600, Mike Oliver wrote:
> Dave Seaman wrote:

>> I have heard it called Cantor-Bernstein, Schroeder-Bernstein, and
>> Cantor-Schroeder-Bernstein.
>>
>> I believe Cantor was the first to state the result, but his proof
>> depended on the unproved assertion that every set can be well ordered.
>> This was before the axiom of choice had been formulated as a part of set
>> theory.
>>
>> Schroeder and Bernstein independently showed how to remove that
>> assumption.

> Well, if that's the history (I'm terrible at history) then I'd
> argue that Schroeder-Bernstein is the right name, whereas the
> previous result is the Cantor trichotomy law. Schroeder-Bernstein
> is a conceptually different result from trichotomy, even though
> they're equivalent if you believe AC. S-B (or at least its
> proof) takes injections in both directions and gives you a bijection
> that's *definable* from those injections; trichotomy doesn't.

I won't disagree. It's just that "Cantor-Bernstein" is the name by which
I originally learned it.

> Lots of times people ask questions like "is there a
> Schroeder-Berstein theorem for the ----- category?".

> Obviously this is not intended to take anything away
> from Cantor, who's one of my great heroes, but I don't
> think he's in any danger of being forgotten.

Agreed. 

-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>

Relevant Pages

  • Re: On the cardinal of the Cantor set
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  • Re: Distinct linear orderings on Z
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  • Re: finding a bijection from [0,1]^2 to [0,1]
    ... Dave Seaman wrote: ... >> When I studied Set Theory some forty years this theorem was known ... >> farm? ... > I believe Cantor was the first to state the result, ...
    (sci.math)
  • Re: finding a bijection from [0,1]^2 to [0,1]
    ... > argue that Schroeder-Bernstein is the right name, ... > previous result is the Cantor trichotomy law. ... Oh, I'd also add, S-B doesn't get you trichotomy either, because ...
    (sci.math)
  • Re: finding a bijection from [0,1]^2 to [0,1]
    ... argue that Schroeder-Bernstein is the right name, ... previous result is the Cantor trichotomy law. ... that's *definable* from those injections; ...
    (sci.math)
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