Re: CANTOR's theorem


*** T. Winter wrote:

>
> > Introduce the condition that the set {a,b,c,...} of all
non-generators
> > of the form {n} be the image of a set {m}. This condition which
already
> > is implicite in the mapping N --> P(N) cannot be satisfied.
>
> Sorry. You defined non-generators as elements in a specific mapping.
> You do not define what sets like {n} would be as non-generators for
> sets of the form {a,b,c,...}, in your mapping of P(N) to P(N). I
> would expect {n} to be a generator or non-generator of a set like
> {{a},{b},{c},...}, but that is something quite different. An element
> is *not* the same as a singleton set containing that element.
>
> > Examples:
> >
> > If we have the identical mapping, then the set of non-generators
is
> > empty:
> > { } --> { }
> > {1} --> {1}
> > {1,2} --> {1,2}
>
> Pray define non-generators for mappings from sets to sets. At this
stage
> I can not comment on this assertion.

An element of P(N) is a set like {7}. In the set M the number 7 must
not be contained. This is my condition. But in order to stay with the
same condition which is contained in Cantor's proof, consider the
following simplification:
Map {1} on its power set. Bijectivity is not required. Only condition:
the set of all non-generators must be part of the image. Does this
impossibility prove that bijectivity is impossible? Does it prove that
there is no life in the sun?

Regards, WM

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