[FOM] From theorems of infinity to axioms of infinity

Timothy Y. Chow tchow at alum.mit.edu
Wed Mar 20 21:41:30 EDT 2013

Nik Weaver wrote:

> Monroe Eskew wrote:
>> Set theory is the only branch of mathematics currently capable of
>> addressing classical questions which turned out to be independent such
>> as:
>> 1) Is the continuum hypothesis true?
>> 2) Is the axiom of choice needed to build a nonmeasurable set of reals?
>> 3) Can there be a probability measure on R which measures all subsets?
> You seem to be confusing working within ZFC and reasoning about ZFC.

Given Monroe Eskew's unfortunate phrasing of his point, Nik Weaver's 
criticism is accurate.  However, I think that Eskew's point is that the 
only way to even *state* questions 1, 2, and 3 is by using the language of 
set theory.  Those questions, as stated, were of central interest to the 
general mathematical community at the time.  It was not until later that 
the related finitary questions (about their provability from the axioms of 
set theory) captured people's attention.

To put it another way, set theory played, and continues to play, an 
important role in the unification of mathematics under a single umbrella, 
providing a single language and a single set of axioms to which all 
mathematics can be reduced.  That is plenty of justification for set 
theory, or at least some version of set theory.

This argument, though, isn't sharp enough to justify a specific version of 
set theory, and there's still plenty of room to argue that this or that 
umbrella is too large or too small.


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