Re: SUO: Echos
Joshua Tonkel wrote:
>JT: The appeal of the iterative conception lies in its safe means of
>"forming" collections -- so safe, in fact, as to prohibit the formation of
>any collection that is not a set. That is why some people regard the
>iterative conception as better suited to denying the existence of proper
>classes than to (affirming their existence and) distinguishing them from
>sets. I'm not sure whether Chris really believes in proper classes, but I
>should mention, for what it's worth, that some people who believe in them
>will favor an approach, like that of von Neumann, Bernays, and Godel, in
>which "class" is treated as a primitive notion corresponding to the
>intuitive notion of a collection, a set is defined as a class that is a
>member of some class, and not all classes are sets. Whatever its defects,
>the appeal of such an approach is that it permits the most straightforward
>interpretation of Chris's claim that "The set theoretic paradoxes all arise
>by taking some proper class or other to be a set", i.e. there really are
>proper classes, and the paradoxes arise when they are mistaken for sets.
When it comes to sets, I'm an agnostic. I'm not happy with
the fact that all of the hierarchies of uncountable sets depend
on a single proof by contradiction: Cantor's diagonalization
method. I don't go as far as the intuitionists in ruling out
all proofs by contradiction, but I agree with Whitehead that
it is very dangerous to try to establish a positive result
by such arguments.
Whitehead's point was that when you discover a contradiction,
the only thing you have proved is that somewhere in the great
morass of axioms and definitions, there exists a contradiction.
Since set theory abounds in all sorts of paradoxes, antinomies,
and counterintuitive beasties, there is a very large room for
doubt about "Cantor's paradise", which rests on such a slender
thread. I won't go as far as Wittgenstein in calling it a
"dismal swamp", but I consider it an interesting place to visit,
not somewhere I would like to go for all eternity.
Bottom line: I like the iterative (or cumulative) construction
because it gives me some feeling of confidence. As an agnostic,
I am willing to let other people worship their own gods, but I
don't want to encourage any extension of the pantheon.
John Sowa