Re: arithmetic in ZF

On 2 May 2005 00:26:51 -0700, Paul Holbach <paulholbachSPAMBAN@xxxxxxxxxx> said:
> [Quoting Priest:]
> "For any claim of the form 'all sets are so and so' to have
> determinate sense there must be a determinate totality over which the
> quantifier ranges.

Well, if the idea of "determinate totality" of things of a certain sort
is meant to imply that there is an *object* of some sort --- the
"totality" --- in addition to the things in question, this is just a
non-sequitur. All we need to make sense of quantification is for the
things quantified over to exist. There doesn't need to be a further
thing that contains them (though of course there does in our usual
mathematical *models* of quantification, which are themselves set
theoretic objects).

This is not to say there is no conceptual difficulty in Cantorian set
theory at all -- if we agree in some cases that all the things of some
sort (e.g., finite sets) form a further set, why not in this case of
"all sets" as well? But that seems to me to be an orthogonal issue.

Chris Menzel

.

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