[FOM] counted sets
Timothy Y. Chow
tchow at alum.mit.edu
Sat Aug 22 17:46:00 EDT 2009
Joe Shipman wrote:
> I saw Conway yesterday and he confirmed that he remembered inventing
> the term "counted set" but had no reason to believe it had not been
> independently invented by someone else earlier or later.
Dan Bernstein did come up with it independently in the early 1990s, but
that was certainly later than Conway.
> In general Chow is correct about getting rid of gadgetry in
> mathematical concepts; but neither Conway nor I could think of an easy
> example of a set which was countable but which had no obvious way of
> counting it. I suppose if there were a set of real numbers which was
> not obviously countable from its definition, but for which there was a
> nontrivial proof that its points were all isolated, one would have a
> countable set that was not "counted".
Along those lines, perhaps the set of discontinuities of a monotonic
function from R to R is an example. Such a set is always countable, but
depending on what the function is and how it is presented, it may not be
immediately obvious how to count/list the discontinuities explicitly.
Tim
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