[FOM] "Mathematician in the street" on AC

Timothy Y. Chow tchow at alum.mit.edu
Mon Aug 17 21:41:46 EDT 2009

Vaughan Pratt <pratt at cs.stanford.edu> wrote:
> Define a countable set to be a pair (X,f) consisting of a set X and an 
> injection f: X --> N.  (If you have some other definition of "countable 
> set," why is yours better?)

I think it was Dan Bernstein from whom I first heard the suggestion that 
the term "counted set" be used for a countable set equipped with a 
bijection to N.

This certainly circumvents the use of AC, but I submit that it is somewhat 
contrary to the mathematical practice of *not* equipping structures with 
non-canonical stuff that is extraneous to their essence.  You could define 
a vector space as something that comes equipped with a basis, or a 
manifold as something that comes equipped with an embedding in R^n, or a 
group as something that comes equipped with a homomorphism to an 
automorphism group of something, etc.  But experience has shown that there 
are great conceptual advantages to *not* saddling our mathematical objects 
with extrinsic structure.  Instead we wait until we need the explicit 
basis or embedding or representation, and introduce it at that point.

Specifically, in the case of counted sets, a counted union of counted sets 
isn't *canonically* counted.  You can construct a counting, of course, but 
"A counted union of counted sets is counted" is not strictly true unless 
you insist that some particular counting is *the* one to use.  Why tie 
yourself to a particular non-canonical counting when it's not important?


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