FOM: Cantor'sTheorem & Paradoxes & Continuum Hypothesis

[Todd Wilson]

On Sat, 10 Feb 2001, Robert Tragesser wrote:
> The logical moves in the "diagonal argument" in the RAA parts of the
> usual proof of Cantor's Theorem (that the cardinal number of the set
> of all subsets of a set is greater than the cardinal of the set)
> exploit a variation on the trick yielding Russell's Paradox, the
> Barber Paradox, etc.  [...] I'm curious about the provenance and
> evolution of this style of argument.

It may be worth mentioning, in this context, a old paper of Bill
Lawvere:

    F. William Lawvere, "Diagonal arguments and cartesian closed
    categories", in Dold, Eckmann (eds.), Category Theory, Homology
    Theory, and their Applications II, Springer Lecture Notes in
    Mathematics 92, 1969.

The first two sentences of this paper read:

    The similarity between the famous arguments of Cantor, Russell,
    G"odel and Tarski is well-known, and suggests that these arguments
    should all be special cases of a single theorem about a suitable
    kind of abstract structure.  We offer here a fixed-point theorem
    in cartesian closed categories which seems to play this role.

-- 
Todd Wilson                               A smile is not an individual
Computer Science Department               product; it is a co-product.
California State University, Fresno                 -- Thich Nhat Hanh