FOM: determinate truth values

[Martin Davis]

At 11:55 AM 9/4/00 -0400, Harvey Friedman wrote:

>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."

As to the bet, alas it was a rhetorical flourish; being 72 years old, it's 
most unlikely that I'd be around to collect (or pay).

I find the notions: "subset" and "power set" crystal clear. Likewise for 
omega in the sense of the von Neumann finite ordinals. Since CH is a very 
specific assertion involving these notions,  I regard it as having a 
determinate truth value.

My reasons for the bet are much weaker. In principle, I have no problem 
with the possibility that although CH has a determinate truth value, the 
human race may never determine it. After all, there are many such 
propositions. I find myself optimistic about the direction of Woodin's 
recent work on CH. At last an effort is being made to get around the 
plethora of models of set theory the forcing method made it possible to 
produce which seemed to make progress on CH impossible. I agree that at 
this preliminary stage my optimism is hardly warranted by the facts. But so 
be it.

>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.

Yes that's what I mean. Think how much it has changed over the past twenty 
years. And your own work will probably do more than anything else to lead 
them to accept methods going beyond ZFC.


>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.

I agree: no "self evident" principles. I believe (and again your own work 
will play a key role) that mathematicians will become comfortable with the 
view that they must work with principles that are not "self evident".

>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?

As you know very well, the existence of non constructible sets is implied 
by large cardinal axioms (specifically the existence of measurable 
cardinals). Now there is nothing self evident about these axioms. But the 
way in which they line up in linear order with respect to various criteria 
suggests very strongly that they are true. This views them like a 
physicist. Set theorists explore this territory as utterly remote from 
everyday human experience as quarks. They must look for theoretical 
cohesiveness. As Goedel suggested years ago, such axioms can justify 
themselves on the basis of their consequences - and here again it is you 
who are doing so much to bring this about.

>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.

Yes. But they will want more. They will hardly regard them as acceptable, 
however convenient, unless they have reason to believe that their 
arithmetic consequences are true.


>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.

Here you take an inappropriately narrow view. These things change, and we 
know that even the remotest large cardinal axioms have arithmetic consequences.


>  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.

Remains to be seen.


>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.
>

I take it that when "Argle" speaks of arbitrary collections, he does not 
intend them to be restricted to the cumulative hierarchy. Nevertheless, I'm 
glad you asked. Without the availability of long well-orderings, the 
cumulative hierarchy stops short. Methodologically, what the large cardinal 
axioms supply is precisely the well-orderings needed to extend the 
hierarchy. We know (Russell Paradox) that there is no end to the process. 
But there is nothing to stop us from going on and on. And that is the great 
service that the set theorists do.

Martin




                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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                        http://www.eipye.com