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

Lasse Rempe l.rempe at liverpool.ac.uk
Mon Aug 3 17:04:25 EDT 2009

> Dunion: Essentially, yes, one could put it that way, at least in the current
> discussion.  That circumstance arises it seems to me because of the
> (quite legitimate) insistence that probability measures satisfy the
> countable additivity property.  Finite random samples (i.e. no subset
> of size N is favored
> over any other subset of size N) can possibly be selected from finite or from
> uncountable sets -- not so, for countable ones!

But that is simply not true - as you well know, e.g. a Vitali set cannot 
be measurable _precisely_ because otherwise there would be a 
contradiction to countable additivity. In other words: pick a Vitali set 
V that is contained in [0,1] (that is, for every real number x, V 
contains exactly one rational translate of x), and randomly throw darts 
at [0,1]. What is the probability that you land in V?

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, 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 paradox. 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. Finally, I would be 
interested to know what has led you to conclude that most "mainstream 
mathematicians" find your arguments convincing.

Best wishes,

Dr. Lasse Rempe
Dept. of Math. Sciences, Univ. of Liverpool, Liverpool L69 7ZL
Office 505; tel. +44 (0)151 794 4058, fax +44 (0)151 794 4061

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