FOM: SOL confusion

[John Mayberry]

Response to Friedman's comments on my previous posting ( 11 Sep)

Friedman makes a fundamental point:

>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. 
It's not just that we can "translate" ordinary mathematical 
definitions and proofs into ZFC, it's rather that if we trace back 
the assumptions that actually underlie our definitions and proofs as 
far as we can, what we finally come to are the very concepts and 
principles of the Zermelo-Fraenkel system which is formalised by ZFC

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.

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

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. 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".

On the matter of the set theoretical machinery need to *establish* 
the Completeness Theorem for FOL, Friedman is perfectly correct

>There is an essential point that is being missed here. One needs an
>extremely weak set of principles about these notions in order to 
>formally derive the completeness theorem. The set of principles 
>needed are essentially well known, and strikingly weak. That is a 
>major 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.

Finally, when Friedman says

>The advantage to using first order logic, as in ZFC, rather than 
>some version of deductive SOL, is that the former is much more 
>robust. In both cases, one needs proper axioms. But it is preferable 
>to add this on top of something that is completely robust. Also, the 
>FOL approach is better because I believe that when you actually do 
>the actual foramlization, it is nicer.

I agree with him entirely. When we formalise an axiomatic theory in 
FOL we make available to ourselves all the marvelous technical 
machinery that has been developed for the study of that logic. 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.

-----------------------------------------
John Mayberry
School of Mathematics
University of Bristol
J.P.Mayberry at Bristol.ac.uk
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