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