FOM: Cantor's Argument

wiman lucas raymond lrwiman at
Fri Jun 28 21:06:13 EDT 2002

>As far as I can tell, not a single issue has been raised about Cantor's 
>Diagonal Argument in all of the many many postings about it on the FOM
>list. The general issue of the evolution of standard set theoretic
>notions has 
>been addressed earlier by me (June 26, 2002) and more recently by Tait,
>and I 
>suggest that the contributors to this list on this topic take these
>into account.

>As I wrote earlier on the FOM, a particularly clean formulation is:

>"every listing of sets of natural numbers omits a set of natural

Yes, exactly.  There is no difficulty with this statement whatever.
There are two (nearly) identical ways to prove it.

(1) Assume that you have a listing of all sets of natural numbers, and
from this listing construct one not in the listing.  Hence the statement
was proved by contradiction.

(2) Let L be a listing of sets of natural numbers.  Use the diagonal
construction to find one not in the list (obviously the preferred

Despite the direct and simple argument in (2), the critics of Cantor
have opted for criticism of (1).  Strangely, even these criticisms have
not revealed anything substantive about the argument.  I'm really
somewhat at a loss to explain the intense animosity towards such a
simple, direct and clear argument.  There is not even some problem with
contradiction:  the proof is not by contradiction!

Wilfred Hodges wrote an interesting article on this phenomenon.  See:
The article starts on page 20 of the document.  In section 4 is the
lovely quote "It was surprising how many of our authors [who were
critical of Cantor] failed to realize that to attack an argument, you
must find something wrong with it."

As regards the present string of criticisms of Cantor, I couldn't have
said it better myself.

-Lucas Wiman

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