>I guess the problem people are having with my thesis is that they >are willing to accept (a) the mathematical existence of S, similar >to that of the object 6, and they are willing to accept (b) the >mathematical existence of a bijection f from w_1 to P(N), similar >to that of the object 6, but they are not willing to accept both >(a) and (b), because the object 6 is special -- it really and truly >exists in some sense, and that property of really existing cannot be >shared by contradictory objects like S and f.
I don't think that's right at all. It doesn't have anything to do with 6 being special.
To get it back to a traditional paradox, let's say that there is a strong man, Hercules, who can lift any rock, no matter how heavy. Let's also say that there is a rock, the Heavystone, that is so heavy that no man can lift it.
It's conceivable that there could be a Hercules. It is conceivable that there could be a Heavystone. But certainly it is impossible for *both* to exist. If Heavystone exists, then there is no man who can lift any rock. If Hercules exists, then there is no rock that is too heavy to lift.
Some pairs of defined objects are mutually inconsistent: they can't both exist.
Now, there might be a way to weasel out of it by talking about "possible universes": Hercules exists in one possible universe, and Heavystone exists in a different possible universe. But then you're waffling about the meaning of the word "exists". Do you mean exists within *one* universe, or exists within *any* universe?