[FOM] A Defence of Set Theory as Foundations

Patrick Caldon patc at cse.unsw.edu.au
Thu Oct 6 23:52:52 EDT 2005

On Thu, Oct 06, 2005 at 12:16:04PM +0300, Aatu Koskensilta wrote:
> On Oct 6, 2005, at 6:40 AM, Patrick Caldon wrote:
> >We now know that naive comprehension doesn't work; near the turn
> >of the last century this was obviously true to the point where it was
> >made an axiom of set theory by the leading minds of the day!
> I know only one mind who did this: Frege. Whether or not Frege's work 
> should be considered set theory is open to debate. Do you have some 
> other leading minds in mind?

Cantor's definition of the concept of set reads (translated from
German) "A set is a collection into a whole of definite distinct objects of
our intuition or our thought. ... " The occurance of the antimonies
showed that the naive concept of set appearing in Cantor's
"definition" ... cannot form the basis for set theory ...
-- Fraenkel, Bar-Hillel, Levy, "Foundations of Set Theory". - page 15

Russell's antimony came as a veritable shock to those few thinkers who
occupied themselves with foundational problems at the turn of the
century.  Dedekind, in his profound essay on the nature and purpose of
the numbers had based number theory on the membership relation ... and
had utilized the notion of a set in in its full Cantorian sense for
the proof of the existence of an infinite ... set.  Under the impact
of Russell's antimony, he stopped for some time the publication of his
essay, the fundaments of which he regarded as shattered.
-- Fraenkel, Bar-Hillel, Levy, "Foundations of Set Theory". - page 2


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