[FOM] Categories satisfying Schoeder-Bernstein theorem

joeshipman@aol.com joeshipman at aol.com
Sun May 24 02:22:20 EDT 2009

I agree, and none of the other replies I have received address my point 
that the asymmetry in the situation for sets (injections each way give 
a bijection without choice, surjections each way require choice to get 
a bijection) is not properly reflected in categorial proofs that use a 
version of AC. Somehow you have to get your hands dirty with actual 
elements in order to achieve Schroder-Bernstein without choice.

I still think there must be some categorial version which respects the 
asymmetry, involving axioms for a set-like category which are NOT 
self-dual, but no one has pointed to one yet.

-- JS

-----Original Message-----
From: Steven Gubkin <steven.gubkin at case.edu>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Fri, 15 May 2009 9:16 am
Subject: Re: [FOM] Categories satisfying Schoeder-Bernstein theorem

Joe Shipman said:

...For the category of sets I know how to prove
Schroder-Bernstein but I don't know a "categorial" proof...

Which inspired me to spend the better part of a day trying to find one,
and failing. I am leaning now toward thinking that there is no really
nice "categorical" proof of Schroder-Bernstein in the category of sets
without using the axiom of choice.   

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