[FOM] Countable choice

Bill Taylor W.Taylor at math.canterbury.ac.nz
Tue Jun 10 23:45:46 EDT 2008

Robert Black:

-> I don't know any literature, but one thought might be that in order 
-> to construct a countable choice set all we have to do is perform a 
-> countable supertask

Yes, this is the (strong) intuition behind forms of countable choice,
that they are "one-shot" supertasks - those that can be completed in one
straight sweep, an omega-sweep.  (Possibly with definable pre-codings,
such as mapping N^2 to N, etc)

-> I'm not myself terribly impressed 
-> by this, because as a good platonist I don't think sets have anything 
-> to do with possible constructions in time 

The general thought behind the cumulative hierarchy, and in particular
Dana Scott's landmark paper on intuitions of "stages" of class-building, 
(Proc Symp Pure Math, vol 13 part 2 1974), seems to contradict this thought.

-> and even if they do it 
-> doesn't seem necessary that time should have the structure of R.

Certainly not R!  But very akin to omega, as in both the above mentions.

Of course it is not a matter of "time", as in physical time, 
but a matter of necessary logical priority, which is all to do with omega.
Essentially, that the members of a set must exist "before" the set can.

Thomas Forster and I disussed this last time we met face to face, and
I recall he assured me that "I wouldn't get a fight from anyone" about this.

-- Bill Taylor

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