Is the Aleph0-1 Conjecture Correct?

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Discussion Overview

The discussion revolves around the "Aleph0-1 conjecture," which proposes a relationship between the completeness of a list of rational numbers and the outputs of Cantor's function. Participants explore the implications of this conjecture, particularly regarding the representation of rational numbers and the nature of the outputs produced by Cantor's function.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Organic presents the Aleph0-1 conjecture, suggesting that if even one rational number is missing from a complete list, Cantor's function will yield a rational number, while a complete list will only yield an irrational number.
  • Some participants express difficulty in engaging with Organic's conjecture due to the use of non-standard terminology and lack of definitions for key terms like "repetition over scales" and "base^power representation."
  • HallsofIvy and others emphasize the importance of defining terms to facilitate understanding and discussion, arguing that without clear definitions, meaningful dialogue is hindered.
  • Organic requests assistance from other participants, indicating a desire for help in articulating and proving the conjecture despite the criticisms received.
  • Several participants assert that the conjecture lacks clarity and rigor, with some stating it is "not even wrong," suggesting that it fails to meet basic mathematical standards.
  • There is a recurring theme of frustration regarding the perceived unwillingness to engage with foundational concepts necessary for discussing the conjecture effectively.

Areas of Agreement / Disagreement

Participants generally disagree on the validity and clarity of the Aleph0-1 conjecture. While some express a willingness to help if the terms are defined, others maintain that the conjecture is fundamentally flawed or nonsensical as presented.

Contextual Notes

The discussion highlights limitations in communication and understanding due to the use of non-standard terminology. Participants express the need for a clearer framework to evaluate the conjecture, indicating that foundational knowledge is essential for meaningful contributions.

Organic
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An edited post:

Please can you show a proof that contradicts my conjecture, saying:

"By using base^power representation method we can represent a list of rational numbers (repetitions over scales, for example: 0.123123123...) where the missing rational number is based on the diagonal rational number used as an input for Cantor's function (the function that defines the rational number, which is not in the list), where the result (the rational number, which is not in the list) depends on some arbitrary order of the list and some rule, which is used by Cantor's function.

For example:

0 . 1 7 1 1 3 1 7 1 1 3 1 7 ...
1 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 4 2 1 3 4 2 1 3 4 2 1 3 ...
0 . 1 0 1 0 1 0 1 0 1 0 1 0 ...
0 . 3 3 3 3 3 3 3 3 3 3 3 3 ...
2 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 3 5 4 9 5 5 1 3 5 4 9 5 ...
3 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 6 4 1 6 4 1 6 4 1 6 4 1 ...
0 . 3 0 2 0 3 0 2 0 3 0 2 0 ...
0 . 6 1 3 6 1 3 6 1 3 6 1 3 ...
0 . 2 7 1 0 2 7 1 0 2 7 1 0 ...
...

In this case Cantor's function result is 0.0101010101010101... which is not in the list.


The rule is:

a) Every 0 in the original diagonal number is turned to 1 in Cantor's new number.

b) Every non-zero in the original diagonal number is turned to 0 in Cantor's new number.

We can add 0.0101010101010101... to the list, and then rearrenge the list in such a way that give us another rational number as cantor's function result, which means that our list is still not complete.


my conjecture is:

When we have the Aleph0 complete list of rational numbers, represented by base^power representation method, then and only then we can find only some irrational diagonal number (no repetitions over scales, for example: 0.123005497762...) as an input for Cantor's function."

I call this conjecture "Aleph0-1 conjecture" because i clime that if even 1 rational number is not in the list, we shall find it as the result of Cantor's function, but when the list is the complete list of all rational numbers (an aleph0 list), we can find only some irrational number as Contor function result.

Please try to contradict this conjecture.


Sincerely yours,

Organic
 
Last edited:
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As long as you refuse to define your terms:

"repetition over scales"
"aleph0-1"
"represented by base value expansion"
"input for Cantor's function"

it is impossible to answer your question.
 
HallsofIvy

Two things i know:

1) I am a bad formalist, and need some help to take my idea and write it in a rigorous way.

2) Your attitude leads to nothing, if you don't want to help.
 
It's not a matter of "writing it in a rigorous way".

No one CAN help when you use non-standard terms and don't define them because they can't understand what you are saying!
 
To use an analogy, it's like you're asking us how to win the world cup, but you refuse to learn how to kick a soccer ball.


Based on the past three months, it seems the only way you're going to make progress is if you learn the basics, and you don't seem to grasp the importance of learning the basics... like it or not, HallsofIvy's response is very likely just what you need.
 
There's a reason I stopped responding to your posts when you started lashing out at people who are telling you that you need to learn the basics; it made me realize that you have no interest in learning them.

For instance, you're still using some of the same nonstandard terminology "repetition over scales" you've been using since day one, and you still haven't made much of an effort to define it. Also you're still trying to prove things about aleph0 - 1.

Until you're interested in learning the basics, any help I try to give you with your ideas and conjectures will be a waste of my time and yours.
 
I have said it before, and I will say it again. You cannot think outside of the box if you have no idea where the box is. In this case the box is defined by the basics. You lack of knowledge of the basics invalidates any effort you make to argue against them. First learn the basics then, from a position of understanding, you can attempt to rewrite the rules.


Of course, once you learn the basics you will not feel a need to redo that which works.
 
  • #11
To quote a famous mathematician it "not even wrong"!
It simply makes no sense. You don't seem to recognize the difference between writing mathematics and writing lots of big words.

By the way, contrary to
When you have a new idea that casting a doubt of some basics, than it can't be done by standard formalism, built on these basics
When you are challenging a statement you must challenge it on its own terms or you have done nothing.
 
  • #12
When you are challenging a statement you must challenge it on its own terms or you have done nothing.

And when my idea change the terms, then what shall i do?
 
Last edited:
  • #13
Originally posted by HallsofIvy
To quote a famous mathematician it "not even wrong"!
It simply makes no sense. You don't seem to recognize the difference between writing mathematics and writing lots of big words.
"Not even wrong. Thats pretty good.
And when my idea changes[ed] the terms, then what shall i do?
DEFINE THEM! (echo, echo, echo, echo, echo).

Do you not understand what we are saying when we tell you you need to define the terms you use if you aren't going to use the standard definition?
 
  • #14
OK, I'll write my idea in more detailed way.


An edited post:

Please can you show a proof that contradicts my conjecture, saying:

"By using base^power representation method we can represent a list of rational numbers (repetitions over scales, for example: 0.123123123...) where the missing rational number is based on the diagonal rational number used as an input for Cantor's function (the function that defines the rational number, which is not in the list), where the result (the rational number, which is not in the list) depends on some arbitrary order of the list and some rule, which is used by Cantor's function.

For example:

0 . 1 7 1 1 3 1 7 1 1 3 1 7 ...
1 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 4 2 1 3 4 2 1 3 4 2 1 3 ...
0 . 1 0 1 0 1 0 1 0 1 0 1 0 ...
0 . 3 3 3 3 3 3 3 3 3 3 3 3 ...
2 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 3 5 4 9 5 5 1 3 5 4 9 5 ...
3 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 6 4 1 6 4 1 6 4 1 6 4 1 ...
0 . 3 0 2 0 3 0 2 0 3 0 2 0 ...
0 . 6 1 3 6 1 3 6 1 3 6 1 3 ...
0 . 2 7 1 0 2 7 1 0 2 7 1 0 ...
...

In this case Cantor's function result is 0.0101010101010101... which is not in the list.


The rule is:

a) Every 0 in the original diagonal number is turned to 1 in Cantor's new number.

b) Every non-zero in the original diagonal number is turned to 0 in Cantor's new number.

We can add 0.0101010101010101... to the list, and then rearrenge the list in such a way that give us another rational number as cantor's function result, which means that our list is still not complete.

my conjecture is:

When we have the Aleph0 complete list of rational numbers, represented by base^power representation method, then and only then we can find only some irrational diagonal number (no repetitions over scales, for example: 0.123005497762...) as an input for Cantor's function."

I call this conjecture "Aleph0-1 conjecture" because i clime that if even 1 rational number is not in the list, we shall find it as the result of Cantor's function, but when the list is the complete list of all rational numbers (an aleph0 list), we can find only some irrational number as Contor function result.

Please try to contradict this conjecture.


Sincerely yours,

Organic
 
Last edited:
  • #15
Ok here's a braindead simple contradiction:

Consider the following set of rational numbers:

0.2
0.23
0.232
0.2322
0.23223
0.232232
0.2322322
0.23223222
0.232232223
.
.
.
Clearly this leads to a diagonal of
0.23223222322223222223...

which is definitely irrational.

The list of numbers is obviously incomplete. It doesn't , for example, contain 0.5.
 
  • #16
"By using base^power representation

By "base^power representation" do you mean decimal representation?

E.G.

[itex]0.2[/itex] is the answer to the questions: "What is the decimal representation of one-fifth?", "What is one-fifth written as a decimal?", and "What is the decimal expansion of one-fifth?"

for example: 0.123123123...

This is called a repeating decimal.


... where the missing rational number ...

What "missing rational number"?



Your post becomes difficult to penetrate beyond this point... though I think I understand your conjecture. Tell me, have you spent any effort looking for a counterexample?
 
  • #17
Hi Hurkyl,

By writing base^power representation i mean that 0.2 = 5^-1 where the used digits are limitad by some base number, for example:
'0','1' (number 2)
'0','1','2' (number 3)
...

The missing rational number is some rational number (an output of Cantor's function) which is not in the list.

By writing 'Counterexample' do you mean that if i tried to contradict my conjecture?

Yes i tried, and i need your help to find it.
 
Last edited:
  • #18
Hi NateTG,

You did not understand my argument.

Please read it again.

Thank you.
 

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