Re: CANTOR's theorem

In article <1116088861.173262.228860@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> All those naturals n, which are mapped upon sets S(n) which do not
> contain n, form a set M. Call them non-generators. If e.g. 2 --> {1,3}
> then 2 is a non-generator.

What mapping are you talking about? Ah, I begin to understand. You are
talking about Cantor's proof that P(X) is strictly greater than X.

> It is in M. M is in P(M). Hence there must
> be a natural m which is mapped upon M. M contains only the
> non-generators. If m is not in M, then it is a non-generator, and m
> must be included into M. But if m is in M, then it is not a
> non-generator and it must be removed. This dilemma cannot be solved.
> Therefore, no surjective mapping N --> P(N) is possible.

Right.

> That is the reasoning of the Cantorians. But it is wrong. Not the
> mapping is proven impossible but the set M cannot exist. It is an
> impossible set.

Oh. What is the difference? You state above that we state that the
set can not exist, and so the mapping can not exist.

> Here is he proof. Consider the bijection P(N) --> P(N) which certainly
> does not suffer from too small a cardinality on either side. For
> instance consider the identical mapping id: P(N) --> P(N).

Ok, so considered.

> 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.

I think a thorough confusion between elements and singleton sets.
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