[FOM] Alternative foundations?

[Dustin Wehr]

I've been reading "objects not in general representable as sets" as
"objects not in general representable as sets in a way that's
satisfactory to ME". Either that, or I expect the authors of such
phrases are assuming a too-restrictive notion of "set", like ZFC-set.
Obviously one sometimes needs to use notions of "set" with weaker
closure properties, which receive names like "class" or "collection",
and this was known long before category theory.

There was mention of intentional types. Want something weaker than
extensional set equality? Then you are probably just reasoning about
descriptions of sets (e.g. programs, which are obviously representable
as sets).

The type theory people are smart and know all this, so in short it
sounds like rhetoric to me. I may be terribly wrong, but I know I'm
not the only one on this list who is thinking this, so a direct
rebuttal to the above would be a welcome contribution to the
conversation.

Dustin