FOM: SOL confusion

[Harvey Friedman]

Reply to Mayberry 11:42AM 9/12/00:

>>Set theory originated as a branch of mathematics. There is an
>>important axiomatization of it that serves as our best scientific
>>model of mathematical practice.
>
>This is especially important, because you cannot grasp the
>significance of independence results in set theory unless you
>understand that any proof acceptable to the mathematical profession
>as a whole, is almost certain to be formalisable in ZFC: certainly
>any proof of a result in general set theory must have this property.

>But it is not enough to discover what are the *de facto* foundational
>principles underlying mathematical practice: we must investigate what
>is required to *justify* those principles, and, indeed, provide such
>a justification, if we can. Even if ZFC does provide "the best
>scientific model of mathematical practice", there still remains the
>further question whether that practice can be justified.

I have always been interested in justification of ZFC, although many regard
it as self evident. Very substantial weakenings such as ZC and much weaker
suffice in various senses. Still even here, whereas the self evident nature
of it appears stronger and clearer, one can also pursue issues of
justification. As you know, I have been pushing a conjecture that ZFC
contains all simple, self evident set theoretic statements. I believe I
know how to turn this into a viable research project.

>So when Friedman says
>
>>The fact that ZFC is incomplete is totally irrelevant to whether it
>>is a model of mathematical practice.
>
>he is right, but there is more to foundations than scientifically
>modeling mathematical practice. He goes on to say
>
>>It would be relevant if one of the features
>>of mathematical practice one is seeking to scientifically model is
>>completeness - that mathematicians are in the practice of answering
>>all possible questions that can be raised. But obviously that is not
>>a feature of mathematical practice, and so there is no need to
>>scientifically model it.
>
>Here I have to disagree. The incompleteness of ZFC is a serious
>problem, especially in the light of the fact that ZFC *does* model
>mathematical practice.

I think you have misunderstood me. What I meant is that incompleteness is a
known fact of life for any reasonable axiomatization. We've known this for
almost 70 years. When I said "no need" what I really meant was that

there is no need to reject ZFC as a model of mathematical pratice merely on
the basis of its incompleteness.

>It's not just the incompleteness per se, but
>the nature of the *particular* propositions that are undecidable -
>CH, for example. If CH is in some sense absolutely undecidable, then
>surely that is a sign that something is seriously wrong with our
>foundational assumptions, and we had better try to figure out what it
>is.

I believe that CH is absolutely undecidable in the limited sense that ZFC
captures all of the simple, self evident set theoretic statements. Putting
it differently, CH is absolutely undecidable in the limited sense that CH
is not going to be settled by what is called "intrinsic" axioms. On the
other hand, there may be other ways that mathematicians come to accept CH
in some sense. For example, mathematicians may, for various reasons, find
themselves compelled to accept the existence of a nonatomic probability
measure on all subsets of the unit interval. This implies not CH. Actually,
I am skeptical that this sort of thing will happen, but it is a
possibility. I am much more skeptical that any concept of "truth" drive the
general mathematical community to accept new axioms that settle the CH.

But the main point I am making is that I know of no kind of imaginable
strong undecidability of CH that would lead me to think that there is
anything seriously wrong with our foundational assumptions. I don't know
why anybody would jump to that conclusion, since there is no reason to
believe that the usual informal description of sets should be sufficient to
decide something like the CH. It is truly unexpected that that usual
informal description of sets is enough to set up ZC, ZFC, and derive so
much!

>The fact that those assumptions underlie the practice of a
>prestigious science like mathematics is, in the final analysis, no
>argument for their correctness.

If by prestige, you mean that the people doing the development are
prestigious, then maybe that is an argument for correctness, after all!

>I can't understand why Friedman, of all people, would make this
>point. After all, what is disturbing about incompleteness phenomena
>is the gap they disclose between the means required to *state* a
>problem, and the means required to *settle* it.

All I was saying was that mere incompleteness is not a reason to reject ZFC
as a model of mathematical practice.

No one now working in
>this field has done more than he has to drive this point home - from
>his beautiful work on Borel Determinacy to the recent series of quite
>remarkable results that he has posted on this list. All this goes
>right to the heart of the matter and *ought* to disturb the
>complacency of that much referred to figure, the "ordinary
>mathematician".

I agree that such incompleteness results - which tell us far more than mere
incompleteness - "ought to disturb complacency of ordinary mathematicians"
not because it shows weakness in formal systems, but because it shows the
limitations of their methods and points the way to the productive use of
new methods.

>On the matter of the set theoretical machinery need to *establish*
>the Completeness Theorem for FOL, Friedman is perfectly correct
ajor point that you must consider.
>
>In fact a version of the Completeness Theorem for recursive sets of
>axioms can be proved in Peano Arithmetic. But this misses the
>essential point. It is not what machinery is required to *prove* the
>Completeness Theorem but what concepts are required to *formulate*
>the key notions on which the statement of the theorem depends, -
>validity, satisfiability, and logical consequence - in a natural and
>plausible manner. The essential problem here is *conceptual analysis*
>not proof. And there, it seems to me, set theoretical notions are
>indispensable.

But since so little is needed, one can use conceptual frameworks other than
set theory to interpret words like "domain", "relation", "operation",
etcetera.
>
>Finally, when Friedman says

blah blah blah

>I agree with him entirely.

Good!

> I
>wouldn't recommend formalising set theory with some deductive system
>for SOL as the underlying logic. I was trying to make the simple
>point that it is not the *incompleteness* of such a deductive sytem
>for SOL *per se* that makes it unsuitable, because the incompleteness
>is going to manifest itself in any case.

But you missed my point that it is better to divide foundations neatly into
a complete part (FOL) and an incomplete part (proper axioms that can and
perhaps should and perhaps eventually will be extended), rather than into
an incomplete part (SOL) and an incomplete part (proper axioms that can and
perhaps should and perhaps eventually will be extended).

It is gratifying to see someone use my scientific work to argue against my
philosophical positions! (even if I don't agree with how they are being
used).