I have seen many proofs of Cantor's theorem that the irrationals (or

reals) are uncountable, and none is at all convincing. [...]

The primary operational fact here is that no matter how many

irrationals you have to count, you will always have an integer

available to count it with. Always. Therefore, the claim that there

are more irrationals or reals than integers or rationals is nonsense.

Supporters of Cantor reply that this chart need not be in any

order. It can be a random chart. That is, a11 can be higher or lower

than a12 or a42. And line 2 can be a higher or lower irrational than

line 1 or 3. But this is not to the point, since whether random or in

series the chart can never be postulated to be complete. This fact is

not proof that the interval is uncountable. It is proof that Cantor's

assumptions are false. [...]

Cantor also claims that the irrationals are infinitely more dense

than the rationals. But the rationals are already infinitely dense.

You cannot get more dense than continuous. You cannot get more dense

than infinitely dense. The rationals, by themselves, are continuous.

They have no space between them on the number line. You cannot get

more dense than that. You may ask, if the rationals are already

continuous, how can you add the irrationals to them? Where do they fit

on the number line? The answer is that they don't fit in that way.

Numbers and number lines are abstractions. You do not "add" the

irrationals to the rationals, like adding three oranges to two

oranges. Rationals and irrationals are relationships between numbers—

therefore they are not the same order of abstraction as the numbers

themselves. Besides, at infinity, the irrationals and the rationals

are the same thing. Just as .9 repeating is the same as 1 at infinity,

the distance between some rational and some irrational goes to zero at

infinity. There is no distance between .9 repeating and 1—that is what

a continuous numberline means. It works the same way with the

rationals and irrationals. You do not need to find room for the

irrationals on an infinitely dense rational numberline. At infinity,

one is an overlay of the other, just as .9 repeating is an overlay of

1.

This means that there is no such thing as an uncountable set.

[Miles Mathis: "Introductory Remarks on Cantor"]

http://milesmathis.com/cant.html
Ich danke Eckard Blumschein für den Hinweis auf diese Quelle.

Gruß, WM

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