Re: "Kantor dust"

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

Galchin, Vasili wrote:
> i.e. a well-defined algorithm exists to construct Cantor dust but the Cantor
> dust cannot be constructed/built from the algorithm in a finite number of
> steps. Hence, Cantor dust represents potential infinity rather than actual
> infinity. This problem has nagged at me for a while.


Bill, if you mean this literally then you don't accept the existence of
the set N of natural numbers either.  If that's the case then for you it
is very reasonable to reject the Cantor set K as well, e.g. because
you're a finitist as Bas Spitters suggests.

However if you're ok with the idea of a natural numbers object N in a
topos, defined as an initial algebra for the functor F(X) = X+1, then
you would need to draw a fairly fine line to reject as nonconstructive a
Cantor set object K in a topos, defined as a final coalgebra for the
functor F(X) = 2X (~ X+X, 2 being 1+1 in a topos).

 From this standpoint the existence of a Cantor set object is more
plausible than a continuum object rather than less because more is
needed.  If you go with the double-induction approach of Pavlovic and
Pratt, where the functor is F(X) = NX (~ X+X+X+...) then the topos needs
a natural numbers object.  If instead you go with Freyd's
single-induction approach of connecting up (eliminating the gap between)
the two halves of K+K, as preferred e.g. by Tom Leinster, then the topos
needs structure sufficient tfor such gluing.

I'm not aware of any reason why a topos with a Cantor set object K has
to also have a natural number object N, though I'm not enough of a topos
hacker myself to know how to produce one with K but without N (but would
be happy to learn).  Does such a topos exist in nature?  And what can be
said of the free topos with Cantor set object?

Vaughan Pratt