[FOM] Cantor's argument

[Giuseppina Ronzitti]

Alasdair Urquhart wrote:

> 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.

As far as I understand, you also need the assumption that each predicate
C_n be decidable, which intuitionistically needs not to be the case. If
one discharges this assumption,  no contradiction arises.

Just as an example about diagonal arguments and constructivist
perspectives  in general, Lusin 1930, p. 55 note 3 remarks that "M.R.
Baire n'employe jamais cette méthode".

G. Ronzitti
Genova, Italy
Europe