FOM: Proper Names and the Diagonal Proof

[John Goodrick]

On Mon, 24 Jun 2002, Dean Buckner wrote:

> I'm happy to assume that, given that a meaningful expression of English can
> formed using a finite sequence of expressions, using words drawn from a
> finite vocabulary, we can signify a different thing with each different
> expression.  I'm not happy to assume any more than that, unless I
> know what I'm meant to assume.
>
> I'm not arguing against Cantor's proof, I'm saying I don't understand what
> it's meant to prove.
>

I'm not quite sure how to answer this question (or if it's meant to be
answered), but here's an attempt anyways:

First, I haven't read much of Cantor's actual writing, but from what I
have read, it seems he didn't reason axiomatically and that he made a lot
of logically unfounded metaphysical assumptions about sets.  So you
probably wouldn't be too happy with Cantor's *actual* proof.

But the modern version of Cantor's argument, which is formalized in
Zermelo-Fraenkel set theory (and can be found in any basic book on set
theory), can be interpreted as proving that if one accepts a short list of
assumptions about sets (namely, the axioms of ZF -- well, actually you
don't even need all of ZF), then one is logically forced to accept that
there are infinite sets which cannot be put in 1-to-1 correspondence with
the set of all natural numbers.  Roughly speaking, if one assumes that
there are infinite sets and that for every set, the collection of all its
subsets is also a set, plus maybe a few other basic things, then one must
conclude that there are infinite sets whose elements cannot all be listed
by the set of natural numbers.  (Well, at least if you think like a
mathematician, you will conclude this; I suppose there's no actual
coersion going on here.)

That's the standard line.  Another interpretation is that we have a
recursive set of finite strings of symbols, called "ZF," and that using a
certain standard and well-known set of transformation rules -- "rules of
logical deduction" -- one can produce, in a finite number of steps,
that finite string of symbols which mathematicians generally interpret as
signifying the proposion "there exists an uncountable set."  This maybe
sounds less interesting than the standard interpretation, but at least it
seems to carry a minimum of metaphysical baggage.

-John Goodrick