FOM: more comments on ZFC, NF and NFU

[Harvey Friedman]

This is a followup to my message of 4:AM 4/1/99. I wrote:

>6. There is a nice way of considering issues about NF as interesting
>questions about ordinary set theory without the axiom of choice - ZF. Let
>*) be:
>
>there is a set A such that the relational structure with infinitely many
>sorts, (A,SA,SSA,SSSA,...) is elementarily equivalent to (SA,SSA,SSSA,...)?
>
>THEOREM (well known). *) is not consistent with ZFC. If *) is consistent
>with ZF then NF (and various extensions) is consistent.
>
>Let **) be:
>
>for all n there is a set A such that the relational structure with n+1
>sorts, (A,SA,...,S^n(A)) is elementarily equivalent to (SA,...,S^n+1(A)).
>
>THEOREM (well known). **) is not consistent with ZFC. If (() is consistent
>with ZF then NF (and various extensions) is consistent.

It seems as if ZF + **) is quite interesting:

THEOREM (known?). In ZF + **), we can prove the existence of a transitive
model of ZFC + "there is a cardinal that is n-subtle simultaneously for all
n > 0." In fact, we don't need ZF for this. Z would be enough.