[FOM] "Mathematician in the street" on AC
pratt at cs.stanford.edu
Fri Aug 28 01:20:27 EDT 2009
In view of the 1963 Feferman-Levy positive answer to
>> For that matter, could the real numbers be the union of a countable
>> family of countable sets?
that Harvey mentioned I have to withdraw what I thought was an obvious
> No, because the latter is a countable union of null sets and hence
> itself null.
So where exactly must Choice be used here? To show that countable sets
have measure zero, or that the measure of the union of two measurable
sets is at most the sum of their measures, or that the limit (countable
union) of an increasing omega-sequence of sets each of measure zero must
itself have measure zero? (I'm replacing the sequence X_1, X_2, X_3,...
by X_1, X_1 U X_2, X_1 U X_2 U X_3, ... for the sake of reducing to the
special case of an increasing sequence.)
Intuitively the last of these seems qualitatively different from a
sequence of countable sets, which obliges one to choose a counting for
each element of the sequence in order to produce a counting of the
elements of the union of the sequence. Here the measure is always zero
so there is no need to choose from among alternative measures at each
set in the sequence. Is the problem that the language of ZF does not
permit abstracting away from the definition of null set but must leave
the existentials in its definition "exposed" in some sense?
I'm finding it hard to get an intuitive handle on the necessity of
Choice here, even after being told that it's necessary. Usually it
isn't too hard to see where Choice is needed once one has been tipped
off to its necessity, here I can't see what I'm overlooking.
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