[FOM] throwing darts at natural numbers (rejoinder to Arnon Avron's reply)
Timothy Y. Chow
tchow at alum.mit.edu
Sun Aug 9 21:10:15 EDT 2009
Joe Shipman wrote:
> That can't be right, if you define "mathematician in the streeet" as a
> random person with a Ph.D. in mathematics.
[...]
> I agree that applied mathematicians have no use for AC
I think Joe Shipman is right here. In fact I would go further and say
that many applied mathematicians have at least encountered AC: If they've
taken a course in real analysis that involves measure theory, then
non-measurable sets would certainly have been mentioned. That might be
the extent of their exposure to AC (plus perhaps the Hahn-Banach theorem
if they go on to a course in functional analysis), but I think it's enough
for them to develop a mental link between AC and pathological sets of real
numbers.
By the way, I want to revise something I said before. I said something
like, "In that case, AC sounds pretty dubious." On further reflection, I
would rephrase that, because the word "dubious" suggests that one is
considering rejecting AC. I didn't mean to connote that. What I would
argue is that the pathological sets associated with Freiling's argument
can be, and perhaps are (by many people at least), thought of as a
consequence of AC + CH together rather than of CH alone.
The distinction I'm making is that we can regard a counterintuitive result
as being a *consequence* of X without necessarily *doubting* X.
Note that intuitions about what is a consequence of what do not always
line up neatly with what f.o.m.ers are accustomed to thinking. For
example, I think that many mathematicians think of Brouwer's fixed-point
theorem and Sperner's lemma as being essentially equivalent, despite the
unprovability of Brouwer's fixed-point theorem in RCA_0. The fact that
Freiling's argument involves weird measurability phenomena, and can't be
carried out without assuming AC or something like it, strongly suggests to
me that AC is partly to "blame" for the pathology.
Tim
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