FOM: As to a "naivete" of G.Cantor's set theory

[Charles Parsons]

At 3:49 PM +0300 1/24/99, Alexander Zenkin wrote:
>Dear Colleagues,
>
>can anyone to formulate explicite arguments why modern meta-mathematics
>calls  the G.Cantor's set theory by a "naive" theory? What does that
>"naivete" consist in? And what does the "non-naivete" of the modern
>meta-mathematics consist in?
>
>Thanks in advance,
>
>Alexander Zenkin
>

I don't think anyone who has responded to this query so far has got it
quite right, although the elements of the right answer are in previous
postings.

1. One sense in which Cantor's set theory is said to be "naive" is simply
that it was not axiomatized, still less formalized. (Ideas toward an
informal axiomatization are found in Cantor's correspondence with Dedekind
in 1899, but that was at the end of Cantor's own work on the subject.) In
this sense, there's no doubt that Cantor's set theory was naive, except
perhaps for that very last stage.

2. The phrase "naive set theory" is sometimes used for a theory that would
admit the universal comprehension schema, allowing for any predicate a
"set" of the objects true of it. This of course falls victim to Russell's
paradox, at least in the absence of other restrictions that are not
envisaged when it is called naive.

Cantor was widely thought to have practiced naive set theory in _this_
sense. This view has been pretty thoroughly exploded by scholarship on
Cantor in recent years, in particular Michael Hallett's book _Cantorian Set
Theory and Limitation of Size_ (Oxford 1984).

3. A third possible sense of "naive" would be: unaware of the possibility
of paradoxes like the standard ones. I don't think this really fits Cantor
either.

My memory of Halmos is that he uses "naive" primarily in sense 1. But I
don't recall that his book contains arguments that would not be
straightforwardly formalizable in ZFC. So if I'm right it's certainly not
naive in sense 2.

Charles Parsons