[FOM] "Mathematician in the street" on AC

Vaughan Pratt 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 
argument for

> 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.

Vaughan Pratt

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