Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity

From: Timothy Little (tim-via-n.i.net_at_little-possums.net)
Date: 02/18/05 

Date: 18 Feb 2005 00:22:29 GMT

Piotr Sawuk wrote:
> theory T with T|-(ZF-infinity) and T|-"all infinite sets are the
> same" with "2 sets are the same iff some invertible mapping does
> exist" (and of course this theory would need to have its own notion
> of "infinite" and maybe even "mapping" would be defined
> differently)...

I think the same definitions of "infinite set" and "mapping" could
apply. (ZF - Infinity + "No set is infinite") would be a perfectly
good example of such a theory.

I think Ross mentioned a theory including ZF without regularity. Such
a theory would need a redefinition of ordered pair. The standard one
uses regularity to prove that {{a},{a,b}} = {{c},{c,d}} iff a=c and
b=d. Something else would need to be found.

- Tim

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