# FOM: Power Towers/Clarification

Harvey Friedman friedman at math.ohio-state.edu
Thu May 27 11:57:54 EDT 1999

```In my message of 4/1/99 11:19AM, more comments on ZFC, NF and NFU, I forgot
to define ***) for the last Theorem, and wrongly used **) instead for the
last Theorem there. Here is a restatement of that posting. Also see my
postings 4/5/99 10:33AM "power shift axiom/correction", which supercedes
earlier postings on the power shift axioms.
**********

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

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,...)?

Here S stands for the power set operation, and we use the epsilon relation
only between successive sorts.

THEOREM 1 (well known). *) is not consistent with ZFC. If *) is
consistent>with ZF then NF (and various extensions) is consistent.

THEOREM 2 (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.
>
>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 3 (well known). **) is not consistent with ZFC. If **) is consistent
>with ZF then NF (and various extensions) is consistent.

We now define a strengthening. ***) for all n >= 1 there is a set A such
that for all k >= 1, the relational structure with n+1 sorts,
(S^k(A),...,S^k+n-1(A)) is elementarily equivalent to
(S^k+1(A),...,S^k+n(A)).

THEOREM 4 (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.

One can squeeze a bit more out of *) which is stronger than ***), and also
out of ***).

```