🚀 go-pugleaf

In article <1992Nov2.040843.4524@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>That Russell felt obliged to give a more broadly applicable argument
>suggests that FA was not held to be an evident property of sets by
>mathematicians in 1905.

Most certainly true.  Well-foundedness wasn't really an issue until
the cumulative hierarchy started to take hold.  Mirimanoff explored
the axiom a bit around 1918, and sensed an important distinction
between well-founded and non-well-founded ("extraodinary," in his
terms) sets.  But it wasn't really until Zermelo's great 1931 paper
(still untranslated, for some reason) "Uber Grenzzahlen und
Mengenbereiche" in which he described the hierarchy in great detail
that foundation became an unquestioned part of set theory.

>That it is self-evident to you now seems more
>an accident of culture---your training---than of something inherent in
>the notion of set....

I think that's too strong if, like me, you were introduced to set
theory via the iterative hierarchy.  If that's your picture of sets,
then the axiom *is* self-evident, and well-foundedness *is* an
inherent property of sets.  The question is rather whether the
iterative hierarchy encompasses *all* the sets, or whether we've just
been looking at a rather limited portion of the universe, as is the
case if we take the universe to be nonwellfounded ala Aczel.