[FOM] Platonism and Formalism
Harvey Friedman
friedman at math.ohio-state.edu
Mon Sep 15 09:01:58 EDT 2003
Reply to Steel.
On 9/14/03 5:30 PM, "John Steel" <steel at math.berkeley.edu> wrote:
>
> There has been some discussion as to the likelihood that
>
> #) ZF + "there exists a nontrivial elementary embedding from some V(kappa)
>>> into V(kappa), where kappa is an inaccessible cardinal"
>
>
> is consistent. My personal, not very strong, opinion is that (#)
> is consistent. Woodin's relative consistency results using the
> consistency of # as a hypothesis constitute some evidence.
> It is evidence of the same sort as the many relative consistency
> results using the existence of supercompacts as a hypothesis provide
> for con(supercompacts)--there just isn't as much in this case.
> (The hypothesis need not be necessary; in fact, the use of supercompacts
> in getting con(GCH fails at a singular strong limit) in a way
> provides even stronger evidence of con(supercompacts), now that we know
> this consequence only requires many measurables of high Mitchell
> order.)
I have a problem with this view of evidence. Woodin's relative consistency
results that you refer to have the form
Con(ZF + #) implies Con(ZFC + phi).
Con(ZF + #+) implies Con(ZFC + phi).
Here #+ is similar to #, but stronger.
You are saying that these results provide some evidence that Con(ZF + #) and
Con(ZF + #+).
BACKGROUND: Kunen proved that Con(ZFC + #) is INCONSISTENT in the late
1960's.
Now let us consider the following relative consistency results.
Con(ZFC + #) implies Con(ZFC + phi).
Con(ZFC + #+) implies Con(ZFC + phi).
Do these provide evidence that Con(ZFC + #) and Con(ZFC + #+)?
I think you might want to defend your view of evidence here by saying that
it's not just the relative consistency results, its also HOW they are
proved.
I don't know if I can anticipate what you are going to say, but I could well
imagine that we could perform the following experiment with some real
creativity.
The experiment is to stay well clear of Kunen's inconsistency proof, and try
to use Con(ZFC + #), Con(ZFC + #+) to redo some of these relative
consistency proofs, with perhaps easier arguments, and even where we
casually use choice (even high up).
By the way, my conjecture about equally interesting and satisfying
statements in V(9) = the 9th level of the finite cumulative hierarchy -
applies to #+ and even stronger statements, etc.
> The strongest evidence of consistency for large card. hypos. comes from
> the inner model program. In a general sense, it is like the evidence
> above, in that you are doing something nontrivial with the hypothesis,
> developing a theory based on it. The theory of a canonical inner model
> of P is particularly systematic, thorough, and detailed, and hence
> provides especially good evidence of con(P). At the moment,
> inner model theory can't reach supercompacts, much less #.
I do find inner model constructions somewhat more compelling, as you do,
than the relative consistency situation discussed above. I'm sure I find it
less compelling than you do. But part of my reason for thinking it is
somewhat more compelling may be that it is probably harder to run the kind
of experiment I mentioned above. I.e., to stay clear of Kunen's
inconsistency, and try to build an inner model theory for ZFC + # and/or ZFC
+ #+. Presumably that's quite hard to do.
> Finally, it is interesting to ask what the behavioral evidence that
> X beieves con(P) would be, for P a large card. hypothesis.
> Announcing that you believe it isn't much--you could be lying, or mistaken
> about your own beliefs. Perhaps developing ZFC +P, or supporting others
> who do that, is the most important evidence that you believe it
> consistent. Ironically, that is exactly what one might try to do in
> order to show ZFC + P is inconsistent.
I should mention that there are at least two well known serious attempts to
refute large cardinals by well known set theorists. One is Jensen's
circulated manuscript "refuting" measurable cardinals. The other is an
ongoing program for several decades. Both efforts led to major developments
concerning ZFC + P, for P = "there exists a measurable cardinal".
>Behaviorally, it seems the
> most important divide is between those who think ZFC +P is
> interesting, and those who don't.
This has the ring of truth to it. If my program to definitively link ZFC +
any P of the kind we are talking about, such as #) and #+), etc., were to be
carried out under suitably high standards, then the number of mathematicians
and scientists and philosophers and others interested in ZFC + P would
increase.
Of course, these days it is nothing terribly new for logicians to do
something that mathematicians are at least *more* interested in than has
been normally the case with most even major results in mathematical logic.
This has been the strength of some contemporary model theory.
But here we are talking about something whose interest goes well beyond
mathematicians, and also makes a dramatic foundational/philosophical point.
Incidentally, I have pretty much carried out this program *somewhat* well
for **small** large cardinals. More about this later in the year.
Harvey Friedman
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