[FOM] From theorems of infinity to axioms of infinity

Nik Weaver nweaver at math.wustl.edu
Thu Mar 21 06:31:52 EDT 2013

Monroe Eskew wrote:

> These days, independence results are the main product manufactured by
> set theorists.  No one else has tools that can manufacture the same
> products.  It is, by objective standards, a successful area of 
> mathematics, not one worthy of "suspicion that something is wrong with
> it."

The phrase I used was "suspicions that something is wrong with set
theory as a foundational system".  Your removal of that context alters
the meaning.  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.

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.

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.

> Many classical questions of analysis and topology have been answered or 
> shown independent by set theory ... You'd have to be a revisionist to
> say set theory is irrelevant.

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.


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