FOM: determinate truth values

[Harvey Friedman]

Davis writes Sun, 03 Sep 2000 21:20:56:

>I have no problem understanding arithmetic truth and  agree that
>every arithmetic sentence has a determinate truth value. I even believe
>that CH has a determinate truth value, and would even bet that in twenty
>years that truth value will be known and generally accepted. And if
>challenged, I believe that I could explain why I believe these things,
>proceeding botttom-up, without resorting to Argle's top-down ontology. I
>have not managed to understand the notion of "arbitrary collection" where
>there is no knowledge of what sort of things are being collected. Perhaps
>some day there will be such an understanding, but I do not believe that day
>has arrived.

NOTE: CH is the continuum hypothesis, not the axiom of choice.

Not only will I challenge you to tell me why you believe that CH has a
determinate truth value, but I agree to bet against you that in twenty
years that truth value "will be known and generally accepted."

Actually, one should ask: generally accepted by who? I take it that you
mean "the mathematical community." Then I will bet against you very
strongly.

I should mention that I was at the UCLA 1967 set theory meeting, and John
Addison ran a poll on questions just like this, if not exactly this. You
might want to ask him what happened with the poll. We know what happened to
the question.

My own personal view is that it is unknown whether CH has a determinate
truth value, and that it will remain unknown whether or not it has a
determinate truth value for longer than 20 years. More specifically, I
don't see anything whatsoever in the informal descriptions of sets written
by either the founder, Cantor, or anyone else, that gives me even the
slightest confidence that there are any self evident principles missing
from ZFC that are of such a different character than ZFC that they decide
CH.

Let us consider A = "there is a nonconstructible set of integers." It is
clear to me that the truth value of A is unknown and it is not generally
accepted that A is false. Do you really expect within 20 years for there to
be any significantly greater reason than there is now to disbelieve A? You
can sit here right now and contemplate the constructible hierarchy of sets
and insist that someone show you why this isn't everything - in particular,
why there is a set of integers that is missing. The most obvious way to try
to show that this isn't everything is to produce an example of a
nonconstructible set of integers. But the way this is done now is quite
circular and unconvincing. This is an old story, and it hasn't changed over
many years.

I do believe that the mathematical community can come to adopt new axioms
beyond ZFC, at least in the sense of finding them convenient and useful.
But only if they are very convenient and very useful for a sufficiently
wide range of situations that are sufficiently part of the fabric of what
they regard as normal mathematics. As you know, it appears that I have
succeeded in setting this process into motion.

However, it seems extremely far fetched to think that this process is going
to apply to CH or to any statement that settles CH  - since such statements
appear to be in principle divorced from what the mathematicians view as
normal mathematics. It is extremely likely that any statement put forth
that settles CH will have the same or fewer consequences for absolute - or
normal - mathematical statements as does large cardinal axioms. It is the
large cardinal axioms that are slated to be adopted by this process.

When you write

>I
>have not managed to understand the notion of "arbitrary collection" where
>there is no knowledge of what sort of things are being collected.

I take it to mean that you have not managed to understand the notion of
arbitrary set in the cumulative hierarchy, regardless of its rank? I take
it that you do feel that you understand the notion of set of rank < omega +
omega? Under this view, what do you make of large cardinal axioms? After
all, they make sense only in the context of sets of unrestricted rank.