FOM: Set theoretic realism

[Harvey Friedman]

Reply to John Steel 10:14AM 1/30/98.

>2. I don't claim to be working on the Continuum Problem.

I was surprised to see you say this. I thought that from the point of view
of any set theoretic realist, this was the major open problem in set
theory. Are you saying that you are not working on the major open problem
in set theory? And why not?

>3. I and many others do think "there are measurable cardinals" is an
>intriguing assertion. That's why the possibility has been investigated
>so thoroughly in the last 40 years!

I also think it is intriguing assertion, especially in the context of the
whole complex of f.o.m. systems ranging from around exponential function
arithmetic (EFA) to ZF + "there exists an elementary embedding from a rank
into itself" (and more). This whole complex seems to hang together in a
fundamental way. However, there are rather extreme changes in the "quality"
or "clarity" of the pictures involved; so much so that it suggests the
liklihood that there are more fundamental ways of expressing these "logical
levels" than we currently know. And as you know, I have been struggling to
find new ways of "generating" some of these higher levels - through work on
transfer principles, and also on reaxiomatizations involving multiple
universes. Also, most recently, how some of the higher logical levels
correspond to combinatorial principles involving finite trees. I don't
think that the usual investigations of the last 40 years are likely to get
to the heart of the matter.

>4. I would subscribe to "either there is a measurable cardinal or there is
>no measurable cardinal". (I don't see what the word "definitely" adds
>to this assertion.) Harvey, do you accept all instances of the law of
>the excluded middle? If you were able to prove some sentence phi of
>the language of set theory from principles you accept by taking cases
>on whether or not there is a measurable cardinal, would you regard
>phi as having been proved?

You are trying to frame the issues with higher set theory in very
conventional terms. It seems more productive to carefully state the unease
most people have with higher set theory in hopefully more productive terms.

The problem people have is this. The natural numbers, and iteration of the
cumulative hierarchy along the natural numbers, have a special kind of
clarity, and one can give extremely clear axioms that go a very long way.
E.g., one can derive separation, replacement, foundation, choice, power
set, etc. People are very comfortable in this setting. And it is also very
seductive to extend the cumulative hierarchy a little bit further, say all
the way along two copies of the natural numbers, w+w. There is a certain
clarity to this, too; just ignore the fact that infinite sets are involved,
and treat it just like the first case, along w, but starting with the
result of iteration along w.

But then one is immediately thrust into the completely fundamental set
theoretic problem of the continuum hypothesis, which has no apparent
analogy with the first case (iteration along w). Even before this, an
analogy breaks down - one isn't able to derive power set from something
trivial, like you can in the length w case. However, that is not so
critical in that power set is explicitly part of the picture - i.e., is
what one is iterating. Nevertheless, it still seems of (possibly great)
significance that power set is derivable in the w case (using just emptyset
and x union {y} and induction with respect to these operations), especially
in retrospect. And also choice is no longer derivable either. But it is in
the length w case. Very strange.

And then CH just sits there with only the following story. First one is
asked to imagine the extension of the cumulative hierarchy "indefinitely."
This is a huge step beyond iteration along w and along w+w. The picture is
already very fuzzy. The best picture I think most people can conjure up is
to look at the picture of iteration of the cumulative hierarchy along w,
and imagine just how "inaccessible" w is from below, and then make a kind
of "transfer" of the picture, starting at the result of iteration along w.
With this way of looking at things, the axiom scheme of collection is very
natural. (This is all connected with my "transfer principles" program).

But these considerations are well known to do nothing about CH. So then one
brings in, say, measurable cardinals. E.g., in the form that there is a
place in the cumulative hierarchy which supports a certain kind of measure
on sets, or a certain kind of elementary embedding, or whatever. But the
nature of this assertion is so radically different than anything one has
come across in this development that one loses any sense of confidence that
one is describing an objective reality. The existence of measurable
cardinals looks like it should be a theorem from something much more
fundamental. But so far we don't have any hint of what that is. (Well, this
is connected with my "multiple unverses" program, involving "reducibility
principles").

The story continues as follows. The existence of measurable cardinals
formally implies various regularity conditions on analytic, coanalytic, and
PCA sets of real numbers. This is regarded as one of the main pieces of
"evidence" that there are measurable cardinals. However, here is where most
people drag their feet. First of all, most people do not think that they
have any independent intuition of how analytic, coanalytic, and PCA sets of
real numbers behave with regard to descriptive set theory. And there is not
any substantial tradition in actual mathematical practice here since
mathematicians avoid such issues both by design and naturally; naturally,
because the focus of mathematics is on more concrete matters.

And the analogies with this situation and that of physical science, where
there are "confirming experiments", is considered by most people to be
strained at best. The results of confirming experiments are, first of all,
(relatively) irrefutable, which is one difference. But there are other
differences in the two situations that are hard to pin down, but are
generally felt. It would be interesting to see a throrough philosophical
analysis of the differences here that are felt but perhaps not fully
formulated.

And to top matters off, even this kind of axiom is well known to do nothing
with CH.

The whole situation leaves most people with the feeling that higher set
theory is "make believe." One postulates coherent systems that appear OK,
and make some sort of sense - although it is not clear what kind of sense.
It is very natural to move into the mode of - these are nice systems, but
they are wholly man made, based on analogies with things we understand
better (e.g., iteration along w). The thing that drives the interest in
these systems is that a little bit goes such a long way. Of course, as you
go into the farther reaches of abstract set theory, a little bit goes less
of a long way. Yet enough of a long way so that it is all worthwhile,
intriguing, and interesting to work out details and relationships. And a
major goal is to make things less mysterious. But one loses confidence in
an objective reality.

>5. I would say that there is  good evidence that there are measurable
>cardinals. "Know" is too all-or-nothing.

In my interpretation of the present situation, one says "there is a good
evidence that ZFC + measurable cardinals is an interesting system in which
a little bit goes a fairly long way. And there is evidence that nobody is
going to find an inconsistency in our lifetime."

>           On approaches to deciding the CH, I had written earlier:
>
>One man's technical jargon is another's deep insight.Immediacy is not
>necessarily the same thing as being fundamental. We are probably past the
>days when new axioms for set theory will be as easily understandable to
>the layman as the Zermelo axioms.

Translation: "There does not appear to be any kind of 'simple' principle of
any 'appropriate' form that settles the continuum hypothesis." (In my
pictorial set theory program, this should become a theorem). And: "Some set
theorists want to see how 'simple" and 'appropriate' a principle they can
find that formally implies CH or not CH, within this limitation."

Steel quotes me as saying:

>>I think you underestimate just how bad this is for your point of view. You
>>may have to eventually compete with things being done in a variety of
>>subjects inside and outside f.o.m., and inside and outside mathematics,
>>which are understandable to the layman compared to what you have in mind,
>>and you may suffer greatly in the inevitable comparisons. It's a serious
>>problem for your approach.

>     This seems to me just public relations. We should follow the terrain
>where it takes us. How easily understood are the basic principles of
>Physics?  Should physicists add to their requirements on a "theory of
>everything" the demand that it be easily understood by the layman?

OK, people complain about Physics, too. I'm not convinced that the latest
"explanation" of the existence of matter via 11 dimensional bubbles is an
"explanation." And when some say that we are beyond the level at which we
can experimentally confirm our theories, many people get very uneasy.

I don't agree that this is "just public relations." On the other hand, I
don't see clearly what the rules of debate should be with regard to our
disagreement here. We could look at the history of science, which I think
is riddled with incredibly simple insights which replace complex hocus
pocus.

>      Harvey continues with a description of "pictorial set theory" with
>which I need some help. So here are some questions:
>
>1. Should we think of a picture as an interpretation (in the informal
>sense of meaning-assignment) of the language of set theory?

If one is doing pictures of set theory, yes. There are also pictures of
other things, such as the continuum.

>2. Is the set of sentences true in a given picture sometimes/always/never
>an r.e. set?

There is no notion of true in a given picture. There is only "formally
derivable from a given picture." And this is always r.e.

>3. Does every picture have all instances of the law of the excluded middle
>in the language of set theory?

All instances of the law of excluded middle are derivable from every picture.

>4. Is it anyone's job to relate different pictures to each other, and in
>particular to put them into a common framework?

The big theorem will be the classification of all pictures of a general
type, and a theorem to the effect that for certain classes of sentences,
the derivable sentences are linearly ordered under inclusion. In fact, well
ordered under inclusion. This may break down for pictures of overly general
type.

I hope to get the opportunity to comment on the remainder of Steel's
posting concerning generic absoluteness and the CH.