[FOM] Platonism and Formalism

John Steel steel at math.berkeley.edu
Mon Sep 15 14:14:14 EDT 2003

On Mon, 15 Sep 2003, Harvey Friedman wrote:

> 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.
    How they are proved is important---after all, the implications
could be proved by showing the hypotheses false, which wouldn't provide
much evidence that they are true.

> 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).

    I have heard the story that Kunen discovered his contradiction
in the course of trying to strengthen Solovay's theorem that the GCH
holds at singular strong limits above a strongly compact. Perhaps
Bob Solovay knows whether this story is true.
    If so, isn't this a tad of evidence against your conjecture? One
of the very earliest attempts to do something substantial with
ZFC + # uncovered its inconsistency.
> 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 th
at 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.
     In fact, the Kunen contradiction comes out immediately if one
thinks about representing embeddings by extenders, as is done at the
very beginning in inner model theory. (The "deus-ex-machina" of the
Erdos-Hajnal function is not needed.)

> 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.

     I think this mis-represents the history. Dodd and Jensen
were working on getting an inner model with a measurable from
the failure of the singular cardinals hypothesis--extending the work
using covering which Jensen had done for L. In the midst of this
ultimately successful and very fruitful project, Jensen thought he
saw an inconsistency in measurables. This happens to everybody who
works with any theory--there are always times when you think you have
two proofs, one of P and one of -P. Jensen's mistake lasted long
enough that he wrote it up and circulated it.
     In the end, it wasn't Jensen's mistake that was fruitful, it was
the original project. His mistake was a detour to a dead-end.

 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".
    What is the 2nd effort, and major developments coming from it?

John Steel

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