[FOM] counted sets

[Timothy Y. Chow]

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