[FOM] Platonism and Formalism

Harvey Friedman friedman at math.ohio-state.edu
Mon Sep 15 15:25:16 EDT 2003

Reply to Steel.

On 9/15/03 2:14 PM, "John Steel" <steel at Math.Berkeley.EDU> wrote:

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.

In this case, one simple story would go like this. We don't have Kunen's
inconsistency results, and so, as usual, one uses ZFC as the base theory.

Then one proves the above implications - or ones like it - without bumping
up against the inconsistency.

I would assume that this can plausibly be done, if one tried the experiment.

>> 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.

It's a nice idea - that if something of a fundamentally set theoretic nature
isn't very quickly seen to be inconsistent, then it is in fact consistent.

The trouble is that the number of data points that seem relevant to this is
very small. 

The applicability of the concept of evidence to mathematical contexts is
well known to be a highly contentious matter.

It is much better understood in ordinary mathematics, than it is in the
context we are talking about, but even there it is not clear what is going
on. It is not clear how to make sense of it. I don't even know if the
appropriate experiments have been conducted.

For that matter, it is not clear what is going on in a purely finite context
such as chess. Top grandmasters make evaluations of positions all the time.
When the positions are highly tactical, we know that their evaluations are
often quite poor. We know this mainly because a higher being exists to
consult on tactical positions - called the computer. The computer can't
prove that its evaluations are correct either, in such positions. But when
the two play against each other, the top grandmaster is often reduced to a
crying baby. 

It may well be that our ability to find inconsistencies in axiomatic
theories lying far beyond our innate intuitions are just as poor as top
chess grandmasters in tactical chess positions.

I DO place some stock in what humans feel about axiomatic theories that are
much more down to earth, much closer to innate conceptions, such as
(fragments of) PA = Peano arithmetic. Certainly ZFC is an intermediate case.

NOTE: I have been caught proposing as an axiom

"anything simple that is inconsistent has a simple inconsistency",

or in a more productive form,

"if a simple system has no simple inconsistency, then it is consistent".

Of course, if one can make appropriate sense of this, it forms a potentially
enormously powerful system!

>> 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.

You make an important point in the last paragraph that I wasn't aware of. It
might be interesting to see if Jensen has the same view of it all.

But I call your attention to a distinction.

If many of the other leading set theorists came up with an inconsistency
from ZFC + measurable, then the ones I am thinking about would almost
certainly NOT have written it up, and if they did, would CERTAINLY NOT have
circulated it. They would have thought about it until he found the mistake,
which they "KNEW" was there, no matter how long that would take. Jensen
acted differently, and that is of some significance.
>> 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?

The FOM has a rule of not quoting uncirculated work of others without their
permission, which I abide by. I'll contact you privately.

By the way, I left out a phrase in what I wrote. I wrote:

link ZFC +
any P of the kind we are talking about, such as #) and #+), etc.,

I meant to write:

... link ZFC +
any P of the kind we are talking about, such as #) and #+), etc., with V(9),
the ninth level of the finite cumulative hierarchy, ...

Harvey Friedman

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