[FOM] From theorems of infinity to axioms of infinity

Monroe Eskew meskew at math.uci.edu
Thu Mar 21 15:56:25 EDT 2013

On Mar 21, 2013, at 3:31 AM, Nik Weaver <nweaver at math.wustl.edu> wrote:

> Again, my point is that number theoretic systems provide
> a far more comfortable fit for core mathematics than does set theory, which postulates a vast realm of nonseparable pathology about which
> it is unable to answer even the simplest questions.

On the other hand, there is a large collection of number theoretic questions, namely consistency questions, which a low-strength number theoretic system is unable to answer.  At this time it seems only the methods of set theory are capable of providing some answers.

> I will stand on my assertion that this phenomenon raises serious
> doubt about the value of set theory as a foundational system, as
> the ideal arena for situating mainstream mathematics.

Why is a less powerful system better?  Certainly set theory can serve as a foundation, and it also brings with it some extra power to answer even number theoretic questions.  What's so bad about pathologies anyway?  They easily coexist with non-pathological things.

> It sounds like your position is that we should be interested in set
> theory because we can use it to prove independence results, and the
> independence results are important because they answer questions
> about set theory.  My position is that core mathematics can be
> detached from this feedback loop with no essential loss to it.

I tried to argue that it's not entirely a feedback loop, that many questions addressed by set theory come from mainstream or classical mathematics.  Of course some feedback is inevitable, as a successful theory or method will generate questions internal to it.

> Again, you subtly misquote me to strengthen your response.  The phrase
> I used was "little or no relevance", not a flat "irrelevant".  You don't
> win this argument by producing examples where set theory has some very
> small, marginal relevance to mainstream mathematics.  You have to find
> examples where the relevance is central.
> Not to put too fine a point on it, but I'm fairly familiar with some
> of the examples you cite.  These are not central questions in
> mainstream areas.  They are very marginal.

You'll have to explain why your use of "small" and "marginal" is objective.  Was George Elliot's classification program for C^* algebras marginal?  Results, including anti-classification results, were found by descriptive set theorists.  (I am stealing this example from Matt Foreman.) It seems like you will have to claim that in some cases, the hard work of esteemed mathematicians (some of whom are not set theorists) is marginal and unimportant because set theory has something to say about it.  Perhaps it is your viewpoint that is marginal?  We need some objective standards here.

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