[FOM] throwing darts at natural numbers (rejoinder to Arnon Avron's reply)

[Timothy Y. Chow]

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