FOM: Cantor's Diagonal Argument

friedman@math.ohio-state.edu friedman at math.ohio-state.edu
Sat Jun 29 16:28:53 EDT 2002

```As I said earlier, there has been no objection raised to Cantor's theorem, and
the posting of Buckner is no exception.

Buckner writes:

> To summarise an earlier point.
>
> (*1)  Every A is a B, not every B an A
> (*2)  For every B there is an A
> (*3)  There is no collection of A's such that they are all the A's

There are three assertions in *1 and *2, and two of them are meaningless. So
what is the point in talking about "fallacies in De Morgan".

> Let's interpret this in terms of Friedman's Cantorean argument.

I would like to take credit for it, but Cantor precedes me.

>Let there
> be some list L and let A be "item on the list", and B "natural number set",
> then
>
> (1a) Every item in the list is a natural number set (we assume)
> (1b) Not every natural number set is an item in the list.

Every item in the list is a set of natural numbers (not a natural number).
Recall

*in every list of sets of natural numbers, some set of natural numbers is
omitted.*

In particular,

*{n: n not in the n-th item} is omitted*.

> This is what the Cantorean argument, as supplied by Harvey, actually proves.
> Let's now assume the medieval axiom:
>
> (3) There is no collection of items in the list such that they are all the
> items in the list

Supply a reference to this "medieval axiom". In 2002, this is absurd. Just take
the collection of all items in the list.

If you are going to want to talk about some "medieval" interpretation of Cantor
that is radically different than the normal interpretation of Cantor, you must
supply a clear foundation for what you are doing. Your account of it does not
have the necessary coherence which can be used as a basis for discussion. Such
"medieval" ways of looking at things seem to have been long discarded by the
mathematical and scientific community. Should they have been discarded?

```