[FOM] Barton FOM Survey

[Harvey Friedman]

NEIL BARTON:

> It comprises one question:
>
> 1. Is there a unique, maximal, proper class universe of set theory?
>
> Available answers are:
>
> A1 Yes.
>
> A2 No. There are maximal universes, but they are incompatible (our
> set-concept bifurcates).
>
> A3 No. There are many universes extending each other in height. The powerset
> operation is determinate, and there is an unbounded sequence of universes,
> each of which is a rank initial segment of the next.
>
> A4 No. There are many universes extending each other in width. Privileged
> universes contain all the ordinals, but the powerset operation in not
> determinate.
>
> A5 No. There are many universes, and any universe can be extended in both
> height and width. The (set-theoretic representatives of) natural numbers are
> determinate though.
>
> A6 No. Any first-order model of some set theory is a universe as legitimate
> as any other.
>
> A7 I am strongly agnostic about this question.
>
> A8 I think this is a bad question (say because one thinks that set theory is
> just a bad foundation, or the issue is intractable).
>
> A9 Other (please specify)
>
FRIEDMAN:

A9. I don't know.

The best I can do with this question, and many questions like this, is to

1. Indicate what kinds of results would bear on it, both pro and con.
2. Make guesses as to the chances for finding various kinds of results
- both absolutely and in comparison with with each other.

It should be mentioned that for almost all of the categorical
foundations community, the answer is "obviously not", and also "there
is obviously no matter of factness about set theoretic statements such
as CH". The idea here is that set theory is a not too interesting
special case of a much wider theory of structures in which obviously
some obey CH while others do not. I don't endorse this point of view,
and there are problems with it. And the categorical foundations people
cannot play this game for Concrete Incompleteness, which is coming of
age.

I am under the impression that for most people who think about this
question, the very long term inability of anyone to come up with any
reasonably compelling new axioms that settle a lot of the most set
theoretically basic questions, especially CH, is viewed as evidence
for "no" I do know that for some, that is not regarded as evidence for
"no", but more a comment on our lack of skills, and even for some
completely irrelevant. I.e., in the extreme view, even if it is
impossible to plausibly decide CH, that in no way shape or form casts
any doubt on the matter of factness of CH.

My own view is that there are some new imaginative simple axiom
proposals with an intuitive sense, which cannot be properly viewed as
limitative, and which prove CH is false. They should have the general
shape that "given certain simple kinds of data there are always real
numbers that have any "possible simple behavior" in relation to that
simple data. However, I think it likely that the kind of axiom
proposals I have in mind are going to remain controversial, and there
will be incremental steps made to improve on them. How this process
converges, I have no idea. At the moment nothing really good like this
is on the table as far as I know. As progress like this is made on not
CH, math will progress so that intensely set theoretic questions like
CH become more and more conspicuously irrelevant to what is happening
in mathematics. Nevertheless, such progress on not Ch and surrounding
questions will be reasonably exciting to a shrinking set theory
community.

There is a very attractive obvious limitative axiom that proves CH.
This is the opposite of "any possible simple behavior" mentioned
above. This is V = L, and it has huge advantages. It's principal
disadvantage, and this disadvantage is widely cited, is that it is
indeed limitative. There are ways of viewing V = L as not limitative,
but people tend to be unconvinced by such.

At the moment, by far the most attractive attitude toward the
foundations of set theory, from a general intellectual perspective, is
the following. This doesn't mean that it will REMAIN the most
attractive attitude toward the foundations of set theory. Unexpected
results may tip the balance.

1. The notion of set has a crystal clear robust satisfying meaning in
the finite, and V(omega) denotes a unique model of finite set theory.
2. The general conceptual framework pretty much extends nicely to the
infinite, and more or less generates ZFC.
3. The picture clearly degrades some in the following sense. For
V(omega), extensionality plus empty set plus insertion plus insertion
induction is enough to DERIVE pairing, union, power set, separation,
replacement, foundation, But for V we have ZFC, and the axioms of ZFC
are not going to be derivable from just a few basic things like we
have for the finite case. Hence comparative hodgepoges like ZFC. (I do
have a completeness theorem for ZF surrounding these ideas, that I
have written about on the FOM).
4. So it is immediately clear that the universe of finite sets are not
going to be pinned down nearly as powerfully as the universe of finite
sets.
5. So there is no good reason to believe that we have a priori matter
of factness with the universe of infinite sets.
6. in particular, if you do not specify more clearly than usual how
subsets of an infinite set are formed, then you cannot expect matter
of factness.
7. You would like to specify how they are formed in a particularly
simple and clear way that is sufficient to allow us to determine the
truth values of the backlog of open set theoretic questions particular
those with at least some connection with some mathematics. This
represents an abandonment of the naive matter of factness in the realm
of arbitrary sets in favor of a move to consider only the tangible.
8. Hence one is led to the universe of constructible sets. Either
through claiming that V = l is true, or simply saying that we are
going to just work with constructible sets, as they are tangible, and
non constructible sets are not tangible.
9. It seems much easier to defend the decision to consider only
constructible sets, because of their tangibility, than it is to defend
the decision that every set whatsoever is a constructible set.
10. You can perform the following thought experiment. OK, you think
that all sets of integers are recursive. Can you give me an example of
a nonrecrusive set of integers. Yes, through the halting problem, and
mathematically relevantly, through Hilbert's 10th. Also see Erdelyi,
Friedman. OK, you think that all sets of integers are arithmetical.
Can you give me an example of a non arithmetical set of integers? Yes,
through a diagonalization. This immediately leads to the question of
doing an Erdelyi Friedman for this, giving a specific mathematical
example. Never seen this done.
11. So I can obviously play this game for any family of sets of
integers indexable by integers. E.g., hypoterarithmetic sets of
integers.
12. So in some sense, the "first" natural situation where we run into
an uncountable set of sets of integers is the family of constructible
sets of integers. Only here, we do not know that there are uncountably
many such.  In any case, it is a natural idea to say that you need to
stop once you cannot do any more diagonalizations. Hence the
constructible reals. Or not...

AXIOM PROPOSAL

Let F be a binary function from the reals to the reals. There exist
reals x,y such that some real F generated from x is not F generated
from y, or some real F generated from y is not F generated from x.

THEOREM. The above is equivalent to the negation of the continuum
hypothesis. The same holds for F of any arity >= 2. For F of arity 1,
the statement is provable.

Harvey Friedman