On Dec 1, 4:17 pm, Virgil <Vir...@home.esc> wrote: > In article > <5fa54469-e270-4070-b2bc-90f35bb8c...@p8g2000yqb.googlegroups.com>, > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > > > > On Dec 1, 2:17 pm, Virgil <Vir...@home.esc> wrote: > > > In article > > > <f37ab501-a998-446b-8aaf-e88059d16...@z41g2000yqz.googlegroups.com>, > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > > > On 1 Dez., 20:46, Virgil <Vir...@home.esc> wrote: > > > > > > What makes you think that the elements of a limit set can be counted? > > > > > If the set exists, its elements exist and can be counted. > > > > That depends on what one means by "counting". Outside of > > > Wolkenmuekenheim, there are sets which are not images of the naturals > > > under any function, and such sets are uncountable. > > > No, then you would have proven ZFC consistent. > > Not outside of Wolkenmuekenheim, I wouldn't. > > Ross seems to think that there are no uncountable sets in any set theory > unless that set theory is embedded in ZFC.
That's accurate. I just don't imagine you'd be using some other set theory (ZFC <=> NBG). It's also accurate that adherents of regular (well-founded) set theories get no absolutes, everything qualified by incompleteness.
Then, where any of the other set theories so described contain the naturals thus encoding Peano arithmetic, purporting trans-finites, they can be lumped together with ZFC in never being provable, as a consequence of those incommensurable trans-finites, in the Goedelian incompleteness which is structurally defined regardless of whether, for example, there are sets too big for the theory that don't observe unmappable powersets (eg, NFU).
Set theory: only sets. Class theory: non-sets theory. Large cardinals: non-trichotomous.