[FOM] "Mathematician in the street" on AC
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
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.
From: Vaughan Pratt <pratt at cs.stanford.edu>
If the problem is with your definition of "countable set" then you
convince me (and I imagine Wachsmut, Orr, and presumably many other
mathematicians) that your definition confers some benefit.
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