Re: An uncountable countable set


Virgil schrieb:

Cantor's diagonal proof does not prove that all numbers of the list are
included.

All number of the list are certainly included in the list. What else are
they supposed to be included in???

Cantor's diagonal proof does not prove that all numbers of the list are

included in that sense that all are covered by the diagonal. Only to
give you an example: If the list was longer than wide, the diagonal
could not cover all list numbers. Further there is no limit
consideration at all for the infinite.

Cantor's diagonal proof does not prove that all numbers of the list are
included.

Perhaps not to you, but it does to most.

That is not an argument for truth, rather the opposite.

The society of mathematicians is limited of those who agree to accept
whatever is proved according to certain generally accepted standards of
logical proof and not to insist on things whose proofs cannot meet that
standard.

The diagonal proof is not valid for an infinite list. There is no
definition at all by Cantor. He only stated continuation.

You reject things that have been proved within that standard and insist
on things whose proof does not meet that standard.

Same did Cantor 130 years ago. Therefore this arguing is invalid.

So that what you claim is not acceptable as mathematics.

Either equal rights for all or for none.

Standards of proof are not democratic.

Correct. And your arguing few lines above about the society of
mathematicans is nonsense. But you seem not even to be able to
recognize that you contradict yourself within few words.

There are not infinitely many values between two values, because the
set is and remains well-ordered, i.e. each element is indexed by a
natural number.

Then the set of rationals never becomes naturally ordered, which would
require exactly what you say never occurs.

Hence there must be one assumption which is wrong, isn't it?

The assumption that any sequence of transpostions can convert a well
ordered set into a dense set, or vice versa.

That is the result. Which is the wrong assumption?

Hence there must be one assumption which is wrong, isn't it?
The only assumption I made is the existence of infinite sets.

You also assumed that you could convert between dense ordering and well
ordering by a sequence to transpostions.

That is the result. Which is the wrong assumption?


Do you have an idea why? (What is wrong with my assumptions?)

You assumed, in contradiction to what Cantor said,

In *agreement* with what Cantor said.

that you could
convert between dense ordering and well ordering by a sequence to
transpostions.

That is the result. Which is the wrong assumption?

Regards, WM

.

Relevant Pages

  • Re: An uncountable countable set
    ... The society of mathematicians is limited of those who agree to accept ... On the contrary, Cantor, and others, have specifically defined valid ... Standards of proof are not democratic. ... You also assumed that you could convert between dense ordering and well ...
    (sci.math)
  • Re: First diagnal proof for real numbers
    ... already in 1932 (G. Cantor, Gesammelte Werke, p. 280-281) ... diagonal proof for real numbers. ... Cantor formulated cardinal exponentiation ... Whitehead and Russel in Principia Mathematica, however, ...
    (sci.math.research)
  • Re: First diagnal proof for real numbers
    ... already in 1932 (G. Cantor, Gesammelte Werke, p. 280-281) ... diagonal proof for real numbers. ... simple proof that uncountable sets exist, ... Cantor formulated cardinal exponentiation ...
    (sci.math.research)
  • First diagnal proof for real numbers
    ... In his original paper of 1891 G. Cantor does not consider the fact ... that dual representation of real numbers can spoil his diagonal proof. ... (In fact Cantor treats sequences in general.) ...
    (sci.math.research)
  • Re: Number theory
    ... >> I am an electrical engineering student.I am learning number theory by ... >> myself.I want to know if there is a separate number theory sci.math ... > Post your doubts (especially the Cantor doubts) here! ... > diagonal proof doubts). ...
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