[FOM] More Friedman/Baez

[Harvey Friedman]

More from https://golem.ph.utexas.edu/category/2016/03/foundations_of_mathematics.html#c050427

Baez:

I think this stuff is fascinating. I’m perfectly happy to “believe, or
wear the hat, that (V,∈)V has an objective reality”, and try to
explore this magnificent structure… or, more realistically, cheer from
the sidelines as others do so.

You mention some empirically observed, apparently profound, but I
guess still rather mysterious phenomena in set theory, like this:

FOR ANY TWO SET THEORIES ANYBODY THINKS ARE NATURAL, WE (GENERALLY)
KNOW THAT EITHER EVERY ARITHMETIC SENTENCE PROVABLE IN THE FIRST IS
PROVABLE IN THE SECOND, OR EVERY ARITHMETIC SENTENCE PROVABLE IN THE
SECOND IS PROVABLE IN THE FIRST.

and this:

In fact, there is a stronger observed fundamental phenom. For any two
natural set theories, either the first is interpretable in the second,
or the first proves the consistency of the second.

I’ve heard people say these things before, but I don’t know enough
concrete details. Where can I read a nice overview of the results
you’re alluding to?

Ideally someday we’d have some theorems that let us understand why —
and under what precise conditions — the phenomena you describe occur.
The objective reality of (V,∈)V might count as some sort of
explanation of these phenomena, but I’d prefer an explanation in terms
of theorems, like:

Given set theories TT and T′T such that TT is not interpretable in T′T
and the TT does not proves the consistency of T′T, then at least one
of them is ‘badly behaved’ (in some sense to be specified.)

Is there any hope of a theorem like this within our lifetimes?

Friedman:

I always think about this issue. An example of what I mean by a
crucial issue in f.o.m. The best prospect is at the moment
quantitative, that is to look only at “simple” formal systems cast in
a certain way, show that all of the interesting systems are mutually
interpretable in a “simple” one, and then show that the phenom is true
for “simple” systems. But for crucial issues anything like this, the
only tools I am aware of are from heavy duty standard f.o.m.
developments. I would be impressed and excited if any of the
alternative foundations people could bring anything to the table for
issues like this.

Shulman:

Thank you for that extensive and very interesting discussion!

Personally, though, my question was about your point (2) (the
objective reality of (V,∈)V), which you specifically declined to talk
about. Can you say anything about how this belief is justified?

Friedman:

The objective reality people on (V,epsilon) are currently on the
defensive, at least somewhat. The most unequivocal of the diehards are
Robert Solovay, Hugh Woodin, and Peter Koellner. Donald Martin also
has a more nuanced unequivocal view.

I don’t find such positions convincing. In fact, I am rather agnostic
on such issues, finding no position convincing. The late Hilary Putnam
had similar sentiments in that he published a famous piece called
“Philosophy of Mathematics: Why Nothing Words”.

I am a philosophical and foundational whore. I am interested in
obtaining substantial results that are suggested by the relevant
foundational debates. They are designed to attack or defend any even
unreasonable point of view that lends itself to such results.

In a way, this is very Goedelian. His famous Incompleteness Theorems
have been used to attack or defend a huge variety of views in the
philosophy and foundations of mathematics.

So I would recommend that you ask Solovay, Woodin, Koellner, Martin
(in no order). However, I should warn you that you probably won’t be
satisfied, but that is just my personal biased opinion.

Friedman:

I have some comments about nonstandard models of arithmetic and other
theories. It ideally should go on your Azimuth, but that has gotten so
involved structurally, that I just don’t know a good place to put it.
Perhaps you can copy this and stick it in there somewhere.

The “standard” view of nonstandard models has not been very well
represented on the Azimuth thread. Here is the hard line version.

As far as PA (first order Peano arithmetic) is concerned. The
“standard” view is that there is absolutely no such thing as a
nonstandard integer. The phrase arose for purely technical reasons
connected with PA Incompleteness and technical tools and technical
questions and technical dreams that have been unrealized.

A nonstandard integer is not anything at all, but a point in a model
of PA that is of order type omega. There is only one model of PA of
order type omega up to isomorphism.

To date, there has not been any reasonable preferred nonstandard model
of PA ever presented - but I am not quite sure about something decades
ago, which I will mention below. Hopefully somebody reading this can
clarify.

There is a reasonably preferred and interesting category of closely
related such. These are the ultra powers of the standard model via any
non principal ultrafilter on omega.

However, this construction is not very satisfactory for two reasons.
One is that these ultra powers are not isomorphic to each other (I
don’t know how much the isomorphism types of these have been studied).
Secondly, all of these ultra powers satisfy the same sentences as the
standard model of PA.

So this renders these nonstandard models prima facie useless for
independence results from PA. In contrast, in set theory, one does
have a lot of interesting ways of constructing models which differ
greatly in what sentences hold and fail, thereby getting lots of
interesting Incompleteness from ZFC.

There have been some people, including Nelson and later Sanders, and
others, who are attracted to the idea that the standard/nonstandard
distinction should be viewed as fundamental, and can provide an
alternative f.o.m. that has some degree of serious radicality.

As I said before, I am a foundational whore, and have played with this
also. As usual, whores don’t like to endorse anything.

However, this approach of taking the standard/nonstandard integer idea
seriously, has the major problem of losing one of the great impressive
features of standard f.o.m., adding greatly to its coherence and
robustness. And that is the strong internal categoricity properties of
standard f.o.m.

There was a development I think attributed to Kochen and Kripke, maybe
late 1970s, which purported to give an at least somewhat preferred
model of PA plus the negation of the 1-consistency of PA. Thus it
would satisfy a false Pi-0-2 sentence. I think it was by a restricted
ultra power construction, where the ultra filter only applies to some
sets. I’m not sure how preferred this model was, and I don’t remember
the details.

In any case, it doesn’t come close to fulfilling the technical dream
of being able to manipulate models of PA to get independence results
with anything like the flexibility and power that one has for ZFC.

As a technical tool, I have been using models of ZFC with nonstandard
integers for almost 50 years. I have to, in order to get perfectly
natural arithmetic independence results from ZFC.

Sanders also emphasizes his use of nonstandardism as a technical tool
in addition to being something allegedly foundational.

Harvey Friedman