[FOM] "Mathematician in the street" on AC

joeshipman@aol.com joeshipman at aol.com
Sun Aug 16 14:29:38 EDT 2009

Your definition of countable set insists that a particular injection to 
N be provided.  But I regard countability as a property of a set "there 
exists an injection of S into N" even if it is hard to specify a 
particular such injection.

You have not shown that ZF (without choice) is inadequate, you have 
just shown that a particular definition that is standard in many 
presentations of ZF is inadequate for constructive mathematics; however 
your alternative definition can be formalized in ZF with no problems 
and then you can prove your version of the theorem "a countable union 
of countable sets is countable" in ZF. Not all mathematicians do 
mathematics in the style you do, and those who do not proceed 
constructively will be unable to prove their version of the theorem 
without using AC.

The "benefit" of not requiring a particular injection into N to be 
provided in the definition of countable set is simply that SOME 
"mathematicians in the street" see no need to proceed constructively as 
a matter of general principle and this unconcern allows them to prove 
theorems of interest to themselves more easily. One must make clains 
about "the mathematician in the street" carefully because so many 
styles of doing mathematics exist; I do not think I have seen a good 
argument yet for the claim that mathematicians GENERALLY have no use 
for AC.

In other words, I think that the set of mathematicians who really use 
AC is not restricted to people in the subfield of foundations, but 
rather that the set of mathematicians who really use AC comprises a 
substantial portion of the set of all mathematicians.

That statement depends on a definition of "mathematican" which may be 
more restricted than yours; I regard a concern for providing a proof of 
one's assertions as being essential, and am not, for the purposes of 
this discussion, including in the class "mathematician" those who do 
not construct "proofs".

I have very recently expressed the opinion that the truth-value of 
assertions about infinite sets should not be of great concern to 
mathematicians; since AC is irrelevant to the truth of arithmetical 
statements, this is a normative statement that AC *ought* to be 
unimportant. However, I continue to assert the descriptive statement 
that AC *is* important in current mathematical practice.

-- JS

-----Original Message-----
From: Vaughan Pratt <pratt at cs.stanford.edu>
  If the problem is with your definition of "countable set" then you 
need to
convince me (and I imagine Wachsmut, Orr, and presumably many other
mathematicians) that your definition confers some benefit.

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