[FOM] counted sets

W.Taylor@math.canterbury.ac.nz W.Taylor at math.canterbury.ac.nz
Fri Aug 21 23:07:39 EDT 2009

> the term "counted set" but had no reason to believe it had not been
> independently invented by someone else earlier or later.

Oh well, I'll put in a claim.  I've been using this term on sci.logic,
in such discussions, for a decade or so.

> In general Chow is correct about getting rid of gadgetry in
> mathematical concepts;

Yes; but it has its downsides too, as here.

> 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 guess it depends on what you mean by a "way".

The obvious try is w_1^CK, the set of recursive ordinals.
It (necessarily) has no recursive description of all its members,
so I guess that could be interpreted as there being "no way" in which
it could be counted.  It would thus be a good example of a countable set
of counted sets which was not itself counted.

It is simply proved countable, OC, non-constructively.

-- Bill Taylor

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