[FOM] Cantor's argument

[Alasdair Urquhart]

Dean Buckner presents a "non-mathematical"
application of Cantor's diagonal argument.
He seems to think it shows that there is 
a problem with the diagonal method.

For clarity, let me restate the argument.
We assume that we have a listing of concepts
C_1, C_2, C_3, ... .  Let us assume that
by "concept" we mean "predicate applying to
the natural numbers."  Now consider the diagonal
concept defined by:

	D(n) <--> ~C_n(n).

If this concept is in the list, say D = C_k,
then we have D(k) <--> C_k(k), but also
D(k) <--> ~C_k(k), a contradiction.  

About this argument, we can say the following.
First, it is completely constructive, and quite
unproblematic from the intuitionistic point of
view.  Second, it depends only on one simple
assumption, i.e. that we can list all numerical
concepts, and hence that D must be in the list.

What conclusion should we draw?  Since there is nothing
wrong with the argument itself, we are forced to the
conclusion that there is something wrong with the
idea of making a list of all "numerical concepts."
This is in fact the generally accepted conclusion.

Arguments of this sort have been discussed since the
early 1900s, and form the basis of the Richard paradox,
the Berry paradox and the Grelling paradox.
It is unproductive to discuss such arguments
in the early 21st century as if nothing had been done
in foundations of mathematics in the last 100 years.