🚀 go-pugleaf

In article <1992Nov5.004725.8252@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.
Stanford.EDU (Vaughan R. Pratt) writes:

   >Again, I'm not in the market to be talked out of Replacement, or talk
   >anyone out of it.  Foundation on the other hand seems as pointless as
   >the cumulative hierarchy: each needs the other and neither seems to be
   >much use to the rest of mathematics.

  The point of the cumulative hierarchy is that it gives an interpretation of
ZF, as Zermelo lucidly explains in his 1930 article. The whole business
of "Down with foundation!", "Oppose unnatural self-membership!",
"Give me replacement or give me death!" as it has appeared here is
completely bogus. If you want to introduce a new set theory or a new
interpretation of existing set theory, by all means do so if you can.
What I am interested in is whether or not you can explain what
you are talking about, as has been done with the "standard" theory of
non-well-founded sets (in that case by giving a model of the theory in ordinary
mathematics).