[FOM] Improving Set Theory

[Christopher Menzel]

I've argued here <https://www.jstor.org/stable/43820940> for a
generalization of ZFCU that permits an absolutely infinite *set* of
urelements, motivated chiefly (a) by philosophical views (notably, those of
David Lewis and Tim Williamson) that seem to require an absolute infinity
of urelements (Harvey has provided mathematical reasons for postulating an
absolute infinity of urelements here
<https://cs.nyu.edu/pipermail/fom/2004-January/007845.html>) and (b) simply
by the iterative conception of set — urelements are all of rank 0, so the
usual arguments against "class-sized" sets don't apply; the set of
urelements is of rank 1 so by the iterative conception it seems it ought to
exist. It's not clear whether this generalization has any significant
consequences for the foundations of mathematics (though Guillermo Badia and
I are exploring some interesting consequences of the 2nd-order version of
the theory) but it arguably yields a philosophically more inclusive theory
of sets.

-chris
--
Christopher Menzel
Professor
Department of Philosophy
Texas A&M University
College Station, TX 77843-4237

On Thu, Jan 9, 2020 at 6:52 PM Harvey Friedman <hmflogic at gmail.com> wrote:

> ZFC has become the standard foundation for mathematics since about
> 1920. Alternatives have been proposed but not widely endorsed at least
> not yet.
>
> I am particularly interested in what people think is lacking or is
> flawed about ZFC.
>
> What needs exploring is to what extent and how can ZFC can be
> adjusted, modified, improved to meet such objections or requirements.
>
> My general thesis is that ZFC is extremely flexible and supports many
> modifications in many different directions for may different purposes,
> and that such modifications or adjustments of ZFC are far preferable
> to any kind of overhaul in f.o.m.
>
> Harvey Friedman
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