[FOM] Countable choice

Robert Black Mongre at gmx.de
Fri Jun 13 17:55:19 EDT 2008

>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.

Well, Tom's colleague Michael Potter in his book on set theory which 
uses a Scott-style axiomatization remarks that to be told that sets 
are subject to a time-like structure that is not time is not to be 
told very much and I'm inclined to agree. The intuition supporting 
the cumulative hierarchy is that sets presuppose their members (I 
would prefer to say metaphysically rather than logically, but let's 
not argue about words). What has that got to do with omega? (Scott of 
course has a truly stunning argument which gets out of apparently 
nothing that the stages are well-ordered so you can't have infinite 
downwards sequences, but I don't see the relevance of that here, and 
anyway it shows that going downwards you always have *less* than 
omega, whereas going upwards there is no limit.)


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