Re: Incompleteness of Cantor's enumeration of the rational numbers

On 12/21/24 4:40 PM, WM wrote:

On 21.12.2024 14:19, Richard Damon wrote:

On 12/21/24 6:00 AM, WM wrote:

 

The cursor moves until it hits a unit fraction.

>
Then why did it it not stop till after it has passed one?

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Ask it! My answer is that it stops at the smallest unit fraction. But you deny its existence. Then it can only stop where you allow it. But then many smaller unit fractions show up. So your permission concerns only visible unit fractions.

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But you admit that it didn't because it ended up past it.

 The reason is that after every visible unit fraction there are more visible unit fractions created. That is potential infinity. You can't understand that matter.

No, they are not "created" they have always been there, they just haven't been enumerated/discovered yet.
And, you had been talking about ACTUAL infinity, not POTENTIAL infinity. I guess you are just showing you don't understand either.
Of course, the real problem is that infinity is a concept too big for your brain and its logic to handle, so it just blows up into a big black hole when you try to think about it.

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The numbers didn't just show up, they were always there and we didn't do anything that prohibited it from stopping at any of them.

 They were dark.

No, you just didn't see them because your logic closes its eyes and lies to you.
Your "darkness" is just a figment of your corrupted brain that can't handle the light of facts and truth,.

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The only thing that prevented it from stopping at the "first" unit fraction is the fact that "the first unit fraction" (from the 0 end) just doesn't exist.

 Before any unit fraction where it stops there show up many more - but only afterwards.

Nope, they are always there. YOU just can't see them because you where blinders and insist that this is how the world is, when it is just your own ignorance.

Use the function NUF(x). It shows the smallest unit fraction.

 

You think circular definitions are acceptable.

 It is not circular.

Its definition is.

Sure it does, as NUF(x) doesn't exist

 You can't grasp it. That's not my problem.

No it doesn't. as if has no mapping in the realm of finite to finite (for x > 0) like you claim. There is no finite value of x > 0 where NUF(x) is not infinite.

 

>
E(n) = {n+1, n+2, n+3, ...}
with
E(0) = ℕ
and
∀n ∈ ℕ : E(n+1) = E(n) \ {n+1}.
>
This means the sequence of endsegments can decrease only by one natnumber per step. Therefore the sequence of endsegments cannot become empty (i.e., not all natnumbers can be applied as indices) unless the empty endsegment is reached, and before finite endsegments have been passed. These however, if existing at all, cannot be seen. They are dark. Therefore it is impossible to introduce the corresponding entries in Cantor's list.

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What Natural Number can't be used as an index?

 The true infinite can be exhausted, but only collectively:
ℕ \ {1, 2, 3, ...} = { }. So it gets empty. The  getting empty happens collectively but cannot be observed individually. We cannot construct a list that assigns natural numbers to all natural numbers. Almost all cannot be manipulated individually. But
∀n ∈ ℕ : E(n+1) = E(n) \ {n+1}
proves that the sequence of endsegments can decrease only by one natnumber per step. Therefore the sequence of endsegments cannot become empty (i.e., not all natnumbers can be applied as indices) unless the empty endsegment is reached, and before finite endsegments, endsegments containing only 1, 2, 3, or n ∈ ℕ numbers, have been passed.
 Regards, WM
 

No, all natural numbers can be used individually, as they are finite numbers and thus have a finite expression to define them.
All you are doing is proving that you don't understand what infinite means, The fact that you can't create or destroy the infinite set with finite operations is what makes it infinite, and the fact that finite operations are all your logic allows says that it can't actually deal with the infinite sets, and just blows up when you try to do it.
Sorry, you are just proving your stupidity in that you ignorantly cling to your naive logic and mathematics of the finite when trying to deal with something that isn't finite.