# [FOM] Cantor's argument

Giuseppina Ronzitti ronzitti at nous.unige.it
Wed Feb 5 11:50:47 EST 2003

```Andrej Bauer wrote:

> Remark: In intuitionistic logic a set X such that there is a
> surjection from a subset of N onto X is sometimes called
> "subcountable". In intuitionistic logic it _cannot_ be proved that the
> set of reals R is not subcountable (because it is subcountable in the
> effective topos?). Alas, we're working in classical logic here.

Just to remark that Cantor's argument has been criticized by people not
working with  intuitionistic logic: the so called semi-intuitionists,
their
arguments are well known.
And, just to give  few references on the treatment of the  concept of
'countable set' in intuitionism (alas !), the interested person may have
a
look at A. Heyting's survey paper "De Telbaarheidspraedicaten van Prof.
Brouwer" ("The Countability Predicates of Prof. Brouwer"), 1029. There
many
such possible predicates are defined and the cardinality of the
continuum is
given negatively: no countability predicate applies to C. More recent
treatments of related problems include: J. P. Burgess, "Brouwer and
Souslin
on
transfinite cardinals", Zeitschr. f. math. Logic und Grundlagen d. Math.
Bd.
26,
1980 (a notion of  'function of countable character' is defined); W.
Gielen,
H.
de Swart, W. Veldman, "The Continuum hypothesis in intuitionism", JSL
vol
46, 1981 ( a notion of 'weakly enumerable set' is defined).

> P.S. What Giuseppina Ronzitti says about the intuitionstic version of
>      Cantor's argument seems wrong to me.

Well, I do not see that I have given any intuitionistic version of
Cantor's argument. Anyhow it may be useful to know why I am wrong. I am
sure:
I
must be wrong,  but just to be told  "you are wrong", does not help very
much

(also, is not an, even wrong,  *argument*).

G. Ronzitti
Genova, Italy
Europe
```