FOM: comments on ZFC, NF and NFU

[Harvey Friedman]

I want to thank Holmes for providing detailed information about NFU and
extensions in a series of postings in late March, 1999.

I will assume throughout that NFU includes an axiom of infinity - which
needs to be added as an additional axiom. However, NF proves an axiom of
infinity. But we don't know whether or not NF is consistent, even relative
to set theory with large large cardinals.

1. The special status of ZFC in the mathematical community cannot be
overstated. I repeat that it represents the border beyond which explicit
assumptions have to be mentioned in formal publications. Even set theorists
adhere to this standard. Of course, they would not adhere to such a
standard when speaking to each other, or giving talks at set theory
meetings. But they do when formal publications are involved in refereed
mathematics Journals. Furthermore, this standard is entirely appropriate.
The reason is that there is much to be gained by having such a uniform
generally accepted standard; and, given that there is going to be such a
standard, ZFC is obviously the best such standard.

2. I spend a lot of time pushing developments that might eventually put the
issue of the appropriateness of ZFC seriously on the table in the
mathematical community. But we are not there yet. And it is also not clear
what the mathematical community would decide to do even if these
developments reached full fruition.

3. Nobody is going to publish a paper in the Annals of Math in which the
proof of a (an ordinarily stated) Theorem has this in it: by stratified
comprehension, we see that ...

4. When it is said that NFU "can serve as a foundation for mathematics,"
this means that there is a reasonably natural and controllable translation
of mathematics into NFU. Of course, this is not sufficient for being a real
foundation, and I do not believe that NFU constitutes a real foundation for
mathematics.

5. The reason that there is a reasonably natural and controllable
translatation of mathematics into NFU is simply that there is a very
natural and controllable translation of mathematics into the cumulative
theory of types with the natural numbers at the bottom, and that there is a
reaosnably natural and controllable translation of this cumulative theory
of types with the natural numbers at the bottom into NFU. Thus the
characteristic features of NFU are not responsible for any role it has as
foundations.

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.