[FOM] throwing darts at natural numbers (reply to Lasse Rempe)
tom.dunion at gmail.com
Thu Aug 6 15:17:43 EDT 2009
On Aug 3, Lasse Rempe wrote
>.. as you well know, e.g. a Vitali set cannot
>be measurable _precisely_ because otherwise there would be a
>contradiction to countable additivity.
I agree completely, and regret if anything in my previous postings
was unclear on this point.
>I don't think anyone would argue that the existence of nonmeasurable
>sets and their properties do not seem counterintuitive when they are
>first encountered. However, you seem to be saying that the phenomenon
>you described is less intuitive than those that simply arise from the
>existence of a well-ordering of the reals
Yes, I am saying that.
>and that seems to require some further justification. It would be
>useful to provide some rationale why the continuum having cardinality
>aleph_1 leads to more unusual results than, say, the Banach-Tarski
Please refer back to Joe Shipman's post of Aug. 4, especially the
third paragraph. He has already concisely expressed the essence of
what I would want to say in response to this.
>Furthermore, I would like to know why you think these results should
>lead us to reject the continuum hypothesis but not the axiom of choice
Please see my post of July 13, which was actually an attempt to do two things,
(a) to encourage some in the f.o.m. community in repairing the "bridge
out" between those concerned with axiomatics, and those concerned
with so-called "practical" issues, in particular statisticians and
those who integrate for a living, to borrow a phrase; but also
(b) to demonstrate that Freiling's argument need not undermine AC,
while it does militate against the CH.
>Finally, I would be interested to know what has led you to conclude
>that most "mainstream mathematicians" find your arguments convincing.
A combination of seeing thoughtful viewpoints on the Web (not just
on FOM), as well as running Freiling's argument by various colleagues.
Their responses have been in accord with Shipman's remark that the
Freiling paradox may be seen as even more counterintuitive than
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