Re: CANTOR's theorem

In article <1116235308.499801.262120@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> Virgil wrote:
> > Since that estabishes that there cannot be any surjection from N to
> > P(N), it perfectly establishes that the cardinality of P(N) is
> > greater then that of N, Q.E.D.
>
> Non sequitur.
>
> Consider any mapping P(M) --> P(M).

I think you mean P(N) -> P(N), and I will consider it in that way, otherwise
it makes no sense.

> Let M be the set of natural numbers
> n which, as singletons {n}, are mapped on sets which do not contain
> them.

This is *not* the set M as defined in Cantor's proof for this mapping.

> Introduce the condition, also implicit in Cantor's proof, that a
> singleton {m} be mapped on M. This is impossible.

Yup, which proves that a surjective mapping from {{1},{2},{3},...} to P(N) is
impossible, because for each mapping the M defined for that mapping is not
in the image of the map.

> There is no such m
> although M can even be the empty set. This proves that the proof of
> Cantor's theorem does prove nothing about cardinalities.

You just have shown that there is no map from {{1},{2},{3},...} to P(N)
such that the set M defined for that map is in the image of that map.
So the mapping is not surjective, and so no map can be surjective.
But this proves a bit about cardinalities, because they are defined
with respect to bijections. And a bijection is both surjective and
injective. So you have properly shown that there is no bijective map
from {{1},{2},{3},...} to P(N). And that shows that the cardinality
is not the same.

On the other hand, you have *not* shown in your proof that there is no
surjective map from P(N) to P(N).
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

Quantcast