[FOM] Improving set theory

[Paul Blain Levy]

On 2020-01-14 17:01, fom-request at cs.nyu.edu wrote:
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> Today's Topics:
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>    1. Re: Improving set theory (martdowd at aol.com)
>    2. Re: Improving Set Theory (Timothy Y. Chow)
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> ----------------------------------------------------------------------
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> Message: 1
> Date: Mon, 13 Jan 2020 21:51:55 +0000 (UTC)
> From: martdowd at aol.com
> To: fom at cs.nyu.edu
> Subject: Re: [FOM] Improving set theory
> Message-ID: <1328060705.10949279.1578952315784 at mail.yahoo.com>
> Content-Type: text/plain; charset="utf-8"
> 
>  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.?

Sorry, Martin, but surely it's a fudge to say "it is necessary to use 
caution". Either we accept ZFC (which allows unrestricted 
quantification) and throw caution to the wind, or we don't, in which 
case we should adopt a different theory such as TOCS (or TOPS, or 
whatever I'm currently calling it).

Paul



> 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
>