Re: Contradiction of bijections as a measure for infinite sets

On 3/25/24 7:10 AM, WM wrote:

Le 24/03/2024 à 22:02, Dieter Heidorn a écrit :

WM schrieb:

>
Does ℕ = {1, 2, 3, ...} contain all natural numbers such that none
can be added?
>

Sure

 Of course.

>
But you can do the following described by Cantor:

 Irrelevant. Important is only this: You cannot add a natural number to the set ℕ.

>

If so, then the bijection of ℕ with E = {2, 4, 6, ...} would prove
that both sets have the same number of elements.

>
Infinite sets don't have a "number of elements".

 This cannot be denied: A bijection, if really existing, proves that one of both sets has not one element more nor less than the other!

Nope, just that they have the "same size", since for infinite sets, one more or less is still the "same size", even twice the number of elements can be the "same size", or the square of the number of elements is the "same size"
That is just a property of infinite sets,

>
And indeed: there is a bijection from the set of natural numbers ℕ
to the set of even natural numbers 𝔼 = {2, 4, 6, ..}.
>
    f: ℕ → 𝔼 ,  n ↦ 2n
>
This function is both injective (or one-to-one) and surjective (or
onto), thus it is bijective.

 If so, that would result in: The set 𝔼 has not one element more nor less than the set ℕ.

Nope, you are just AGAIN using the wrong definition of "same size" because of your mind being stuck in "finite thinking".

>

Then the completion of 𝔼 resulting in E = {1, 2, 3, 4, 5, 6, ...}
would double the number of its elements. Then there are more natural
numbers than were originally in ℕ.

>
Rubbish. The cardinality of an infinite set is described by an
transfinite cardinal number and not by a finite "number of elements".

 Here we do not use the rubbish of cardinality but the definition of bijection proving that one of both sets has not one element more or less than the other!

Nope, it proves thay have the "same size". "Number of Elements" is not a defined term for infinite sets, in the same way it is defined for finite sets.
You are just stuck in your finite thinking.

 

Your problem is: You try to apply facts,

 I apply logic which is universally valid.

Nope, you think it is, so you break it.

 If the set 𝔼 = {2, 4, 6, ..} has not one element more or less than the set ℕ = {1, 2, 3, ...}, then adding an element to 𝔼 destroys this state.

Nope.

 

that hold for finite sets,
on infinite sets. That doesn't work.

 Your problem is you deny logic which is universally valid.

Nope, your problem is you THINK your logic is universally valid, which makes it go BOOM and destroys itself.

 Regards, WM