[FOM] Improving set theory

[martdowd at aol.com]

 FOM:
I would maintain that the classical position is "yes".  The "set building" axioms, e.g. pairing, are universally quantified.  They are natural and typical of other "formation" axioms, such as the successor function of arithmetic.   It is a distinguishing fact about set theory that the totality of sets is different from tamer universes of discourse, and as a result it is necessary to use caution.  Cantor held this view in later writings (although I'd have to look up a reference - van Heijenoort perhaps).  I suspect many working set theorists do.
- Martin Dowd
 
 
-----Original Message-----
From: Christopher Menzel <cmenzel at tamu.edu>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Mon, Jan 13, 2020 12:19 pm
Subject: Re: [FOM] Improving set theory


For the purposes of ordinary mathematics, though, quantification over the totality of all sets is almost never necessary.

For philosophers, though, the interesting question is whether it is even possible. :-)
-chris
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