Re: Kantor dust

[Vaughan Pratt <pratt@cs.stanford.edu>]

Paul Taylor wrote:
> The first interesting mathematical question is this: if we define
> Cantor space starting from its points, but take these to be given
> by TOTAL RECURSIVE FUNCTIONS  N-->2,  and then put the topology
> that I have just mentioned on it,  what are the properties of this
> topology?   In particular, is it compact?
>
> The answer is no, because of the "Kleene Tree".  This is a computably
> defined infnite tree that has no infinite computable path.  There
> are many descriptions of this, including one by Andrej Bauer:
> math.andrej.com/2006/04/25/konigs-lemma-and-the-kleene-tree/
> [...]
> ABSTRACT STONE DUALITY:  Andrej Bauer and Paul Taylor,  "The Dedekind
>   reals in abstract Stone duality",  www.PaulTaylor.EU/ASD/dedras

In concrete Stone duality, increasing structure on one side is offset by
decreasing structure on the other.  One would hope for a similar
phenomenon in abstract Stone duality.

If we can consider constructivity as part of the structure of an object,
then we should expect that the more constructive some type of object,
the less constructive the "object of all objects of that type."   So for
example if (total) recursive functions are demonstrably more
constructive than partial recursive functions by some criterion, we
should expect the set of all recursive functions to be *less*
constructive than that of partial recursive functions by the same
criterion, rather than more.

The phenomena you're observing here seem entirely consistent with this
principle, and point up the need to be clear, when judging
constructivity in some context, whether it is the collection or the
individuals therein being so judged, with the added complication that
Stone duality makes the roles of collection and individual therein
interchangeable, such as when elements of sets are understood as
ultrafilters of Boolean algebras.

Vaughan Pratt