[hamdani yusuf (OP)]

https://en.m.wikipedia.org/wiki/Riemann_sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann,[1] is a model of the extended complex plane, the complex plane plus a point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point "∞" is near to very large numbers, just as the point "0" is near to very small numbers.

The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=∞  well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere.

In Riemann sphere, there's oly one point of infinity. ∞=-∞=∞i=-∞i
But arithmetic operations involving multiple infinities can be messy.
Let's say ∞+∞=∞. 
But ∞=-∞
Hence ∞+∞=∞-∞= undefined.

[Eternal Student]

Hi again.

It's mind boggling that many brilliant mathematicians over a century failed to see this, while a few random forum members can solve it in just a few posts. Perhaps we can collaborate to write a proper research paper on this.

     Since no other random forum members seem to be interested, I suppose I've got to ask.   What did you have in mind?  ...and is there any food or light refreshements provided because it might influence my decision.

Best Wishes.

[hamdani yusuf (OP)]

Since no other random forum members seem to be interested, I suppose I've got to ask.   What did you have in mind?  ...and is there any food or light refreshements provided because it might influence my decision.

I see there are two other members involved in this thread. Their posts here show that they are interested to the topic.

I think that diagonal argument produces more problems and inconsistencies, rather than being useful to solve other problems. So, getting rid of it could help mathematics to move forward and restore its consistency.

https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above.

In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.[1][2]: 20– [3] Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.

The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, which appeared in 1874.[4][5] However, it demonstrates a general technique that has since been used in a wide range of proofs,[6] including the first of Gödel's incompleteness theorems[2] and Turing's answer to the Entscheidungsproblem. Diagonalization arguments are often also the source of contradictions like Russell's paradox[7][8] and Richard's paradox.[2]: 27 

Pragmatically, our work could save a lot of person hours of talented people from fruitlessly  trying to solve problems created by flawed foundation. Hence they would be able to use their time and efforts for some other things that are more useful.

[Eternal Student]

Hi again.

You haven't really answered the first question:

Perhaps we can collaborate to write a proper research paper on this.

    Where and how did you propose it would be published?  Although, I don't suppose that's important, you could call something a paper even if it wasn't published anywhere.    The main thing is what do you want to produce and what do you think will happen to it?
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I'm asking because I'm always wondering what the users hope to get from using this website.   It's set up to be a forum, which is very roughly a place where things can be discussed.  I use it for that purpose.
    You have a noble sentiment about saving other people time and effort, which is great.  However, I don't see how this forum will achieve what you are hoping for.  Most professionals will search through published journals and not forums to find a paper.  This is why I'm focusing on the first question - how did you intend to get the paper published or made available to others?

Best Wishes.

[hamdani yusuf (OP)]

This is why I'm focusing on the first question - how did you intend to get the paper published or made available to others?

When the paper is finished, perhaps we can just send it to ArXiv, or it's alternative like Vixra. Considering the widespread interest on this subject, I think once published, it will reach wider audience in no time.

[Eternal Student]

Hi again.

perhaps we can just send it to ArXiv, or it's alternative like Vixra. Considering the widespread interest on this subject, I think once published, it will reach wider audience in no time.

   Arxiv will not take articles that are submitted by unregistered authors UNLESS these articles are endorsed by others (who are already registered authors).  It's not as simple as sending a copy to Arxiv by mail and expecting them to make it available to the world.  The usual way to obtain endorsement is to be affiliated with a recognised University or research establishment.  To the best of knowledge, the Naked Scientists forum is not one of those recognised establishments.

    Vixra is an alternative but take some time to read what they say about themselves:
ViXra.org is an e-print archive set up as an alternative to the popular arXiv.org service..... It has been founded by scientists who find they are unable to submit their articles to arXiv.org because of Cornell University's policy of endorsements and moderation designed to filter out e-prints that they consider inappropriate....
   ViXra is an open repository for new scientific articles. It does not endorse e-prints accepted on its website, neither does it review them against criteria such as correctness or author's credentials.   -  taken from Vixra website,  foot note on their main page.   (Active link omitted since I can't even recommend that you visit.  No one is entirely sure who pays for or maintains this website).

   It takes only a moment to realise that there are very few quality checks on anything published on Vixra.   There are some people who have gone so far as to say the following:....
Warranted or not, it has a reputation of being an alternative to arXiv for cranks and to host a lot of junk science, fake proofs or even outright nonsense. It is not recommended to submit something to viXra if you wish to be taken seriously, because the reputation of viXra would probably taint your paper's reputation by association.    
- Taken from Academia stackexchange website.  https://academia.stackexchange.com/questions/105518/what-are-the-differences-between-vixra-org-and-arxiv-org

    Even if you did get endorsement from someone and got a paper to Arxiv,   this still doesn't count as any formal publication.  The articles and papers on Arxiv are not peer-reviewed in the same way as more conventional papers published in journals.   More importantly there are thousands of articles on Arxiv and even they (Arxiv) explain that there is no reason to expect anyone else to find or see your article.   You have to actively go out and make contact with others working in the field and invite them to look at the article.  Only after your article has been cited a few times does it stand any chance at all of being found by other academics or the sort of people who you are trying so hard to help.
   Even after this, the only way to really get it into the mainstream of education and further research is to get it properly published and subjected to a peer-review process in some proper journal.   Most professionals do not have the time to study new articles on Arxiv, they barely have enough time to review articles that do get published in journals.
    I can only apologise for writing so much, it's just that publishing a research paper isn't a simple matter.

Considering the widespread interest on this subject, I think once published, it will reach wider audience in no time.

    There is no history that I am aware of where something like a New Theory article presented in a forum has become a shortcut to getting a proper research paper published.  I doubt it will get published to begin with. 
    Forgetting about outside interest, the interest here in this forum doesn't seem to be as "widespread" as you might have hoped.  I'm really sorry but maybe Maths isn't to most people's liking, or maybe there just isn't more than half-a-dozen people here in the forum on a typical day. 
- - - - - -

Anyway, I've already written too much and it's getting boring, sorry.   I'm just asking you to take a moment to evaluate the situation.
You said....

Perhaps we can collaborate to write a proper research paper on this.

   I'm guessing you were talking to me.  Let's say yes, yes I would.  How is it going to be possible?

    I'll answer some of the other subject-related questions or ideas you had in another post,  I've already gone on too long here.  Using the forum to discuss things seems useful and if you have a genuine interest in this subject, great, let's discuss it.

Best Wishes.

[hamdani yusuf (OP)]

There is no history that I am aware of where something like a New Theory article presented in a forum has become a shortcut to getting a proper research paper published.  I doubt it will get published to begin with.
    Forgetting about outside interest, the interest here in this forum doesn't seem to be as "widespread" as you might have hoped.  I'm really sorry but maybe Maths isn't to most people's liking, or maybe there just isn't more than half-a-dozen people here in the forum on a typical day.

I take the discussion here as brainstorming process, which is to collect as many information as possible, and perform initial filter to remove erroneous and irrelevant data from further consideration. It's an early phase of writing a research paper instead of a shortcut. Historically, one of the most influential research paper, Newton's Principia, was published as a result of communication between Newton and Halley. The earlier process itself began when Newton visited an apple garden. Private correspondence between Pascal and Fermat, as well as between Darwin and Wallace, also initiated the publication of important research papers. I don't see any downside of starting the research paper from an open correspondence in a forum, except from someone else stealing the idea for themselves. But at least we would already have left our digital footprint to stake our claim of priority in case the idea is recognised as important in the future.

It is inevitable that viXra will therefore contain e-prints that many scientists will consider clearly wrong and unscientific. However, it will also be a repository for new ideas that the scientific establishment is not currently willing to consider. Other perfectly conventional e-prints will be found here simply because the authors were not able to find a suitable endorser for the arXiv or because they prefer a more open system. It is our belief that anybody who considers themselves to have done scientific work should have the right to place it in an archive in order to communicate the idea to a wide public. They should also be allowed to stake their claim of priority in case the idea is recognised as important in the future.
https://vixra.org/why

[hamdani yusuf (OP)]

https://paperpile.com/blog/shortest-papers/

Math can be short
Math can be hard and tedious resulting in very long papers. The 1995 proof of Fermat's last Theorem was 108 pages long.

But math can also be short.

Lander and Parkin's paper about a conjecture by Euler (related to Fermat's last Theorem), is probably the dream of everyone ever written a paper: It answers an interesting and important question, it's correct beyond any doubt,  it's easy to understand and only two sentences long.

I have a dream.
Perhaps our paper wouldn't be that short, but at least we can try to minimize unnecessary things there.

[Eternal Student]

Hi.

Let's start with a much earlier comment where a proof that 1=2 was presented.

Instead of blaming math, we should do introspection to check on the validity of our method/derivation involving infinity or zero. When used incorrectly, we would be able to "proof" that 1=2, like an example below.

     The proof was trivially incorrect and probably doesn't need much discussion.  There was a division by 0, which I expect you were already aware of.
     There isn't much "introspection" required,  division by 0 isn't supported in the algebraic structures that were being used and assumed in that proof.
     

It's from the title of Veritasium's video, which I quoted. You can click on the hyperlink to go to my post containing the video.

    I've had enough time to watch that Veritasium video.  It's Pop Sci rather than a rigourous derivation of the Godel theorem and I'm afraid it does make at least one mistake.
   When discussing the Godel incompleteness theorem there is no detail provided about the process of "proof" using a formal first order logic and the presenter simplifies the proof provided by Godel to the point where it became incorrect.
    He mentioned a "card" that stated  "there is no proof for the statement with godel number g" and then assumed that card woud have Godel number g.  This is evidently an unreasonable assumption.  The Godel number is determined by the process he described earlier,  it is  2^a . 3^b. 5^c . 7^d.......  = some product of primes determined by the symbols that appear on the card.   Regrettably, having the number g in the statement on that card means that the number g must be treated as a symbol, so it contributes to the overall Godel number you would calculate for that card.   So you would have to guess what the over-all Godel number would be before you calulate it because you have to know what number g you need to include in that statement.  Assuming the number g is quite large it is written as  ssss....ss0   (the g th successor of 0)  and so it influences the overall Godel number by a factor   of   p₁^σp₂^σp₃^σ....p_g^σ    where  σ = the Godel code number assigned to the succesor, s, symbol   and  p_i  are successive primes.   In the general case there would be no solutions to this,  we multiply g (the overall Godel number of the card) by a new prime p_g >> 2    each time we just try to increase g by 1.   You might need to re-watch the video and take a moment to identify this yourself.
    Anyway, Godel's original proof follows the spirit of the method presented in that video but the fine details are different and that's what makes it work.  Godel's test card actually involved a statement more like   "This statement has no proof"  and the over-all Godel number, g, of that statement does NOT appear in the statement.
- - - - - - - -
Summary:    The Veritasium video is of interest to the general public but it's not completely accurate.  It is not a substitute for studying  Set theory, first order Logic and the original work of Godel.   I'm not aware of any quick guides to these subjects, sorry.  We didn't have a "Set theory" course although I know that some Universities do.  Many undergraduate courses will just study and develop bits of set theory as and when required while covering over topics.
I think most of the better resources are going to be old-fashioned books rather than anything online.
    Anyway, I've probably already spent too long here and I need to go and do some work.

Best Wishes.   

[Eternal Student]

Hi again.

I'll try and say a bit more about some of your other posts.

The post you made earlier about the Riemann sphere is interesting although it's only vaguely related to any ideas of cardinality.  This post is already long enough so I'll just move to the next topic.  (Obviously you can raise the issue again if you did want to talk about it).
- - - - - - - - - - -

I think that diagonal argument produces more problems and inconsistencies, rather than being useful to solve other problems. So, getting rid of it could help mathematics to move forward and restore its consistency.

    Firstly, there isn't much reason to believe that our current system of mathematics is inconsistent.  The continuum hypothesis doesn't demonstrate any inconsistency.  It does however, indicate some incompleteness.

About Cantor's diagonalisation method
    The main issue is that the diagonal argument or procedure exists.  Choosing to ignore it or not to use it, doesn't make it go away.  It's a perfectly valid procedure following form the ZFC axiom system and therefore any consequences that follow from it are properties of the mathematical structures that are constructed from this axiom system.  Any attempt to deny or forbid some consequences that follow from the diagonalisation method (e.g. to assume that there actually is a bijection from ℵ to ℜ) will put inconsistencies into the system not remove them.
    The only useful way to "remove" the diagonal argument (and hence also any problems that follow from it) would be to suggest a modification to the fundamental axioms.  This is quite a serious proposition, if you expect your new axiom system to be useful to Mathematicians and Scientists.  The favoured system is ZF(C), it has been developed for good reasons and is believed to be logically consistent.  It isn't expected to be complete (i.e. not all true statements can be proved from the axioms) but so far it's been found to be quite sufficient or complete enough for most purposes.

    Anyway, let's start from the beginning.  You can certainly modify the fundamental axioms of your system of Mathematics and thus prevent the diagonalisation method from being valid, this is quite a trivial procedure.  Your axiom system doesn't even have to be based on set theory if you wish.  However, the system of axioms you have may then be seriously incomplete (i.e. insufficient for the majority of Mathematics that we actually use and therefore of little interest or practical value) or, worst of all, they may be inconsistent
     I'm assuming from the posts you've written so far, that your intention is to have a system of mathematics that is useful for the world in which we live.   For example, there would be a structure that looked and behaved as we would expect the Real numbers to behave.  With this assumption we're really only looking at minor adjustments (if any) to the axiom system we commonly use.

One contentious issue in the diagonalisation argument
    It's worth taking a moment to consider if there is a flaw in the procedure, or just a minor issue where only a small adjustment of the axioms might become relevant.   The main issue that looks contentious is where it is assumed to be possible to construct a number so that its decimal digits will be different to ALL of the enumeration's diagonal decimal digits simultaneously.
    If we look carefully at the procedure,  it is clear that for any finite number, N, of decimal places, we can construct a test number so that it's digits will differ from the diagonal digits of the first N elements in the list.  We can do this for any finite number of digits, N, no matter how large. The contentious issue is whether we can really "speed this up" to make infinitely many selections instantly.
    If we have the axiom of choice in our set theory, then we can and the procedure is valid.
    A weaker form of the axiom of choice called "the axiom of countable choice" would also be sufficient and the procedure would still be valid.
    If we are steadfast and don't admit any version of the axiom of choice into our set theory, so that we are working in the ZF axiom system only then we have to be more careful.  It is still possible to establish the existence of a well defined "choice function" but let's leave such things to one side for a moment and try to investigate exactly what might happen if we just deliberately choose an axiom system where the assumption of being able to identify the infinite set of digits is simply not permitted.

    I've not really spent the time to formally establish exactly what the axiom system might be - but it would be reasonable to assume that some simple results concerning the limits of a sequence of real numbers should hold.   Why is this reasonable? --> If these properties don't hold then the axiom system has become so limited or restrictive that it will be of little value in Mathematics anyway.
    In Cantor's diagonalisation method, instead of declaring the test number, T, to be an infinite decimal with carefully chosen digits, we can declare T to be the limit of a sequence of numbers that have only finite digits.
   
   Let T =  
where each term of the sequence,  t_n,  is determined to be the decimal with only n decimal places and given by the finite version of the usual selection rule.   Specifically,  the  j^th  decimal digit of t_n is one more than the j^th  decimal digit of the j^th entry on the list   (with  digit 9 being mapped to 0).
    It's only a small amount of work to verify that this limit has all the properties we would require (it would be a real number but it will be distinct from any number found on the list).

Summary   In any reasonable axiom system, where the Real numbers would be complete* and finite selections can always be made,  then the diagonalisation method (or the variant of it discussed above) will hold.  This will establish that no bijection from ℵ to ℜ exists.

   * Clarification on terminology:  "Complete" in the sentence just above means that Cauchy sequences converge in ℜ.  This is different to the term "complete" that is used when discussing Logic and axiom systems.
- - - - -
     Now, it's possible to define the size or "cardinality" of sets in some way that doesn't involve finding bijections but this is just changing terminology and not changing the intrinsic properties of the structure.  We can call a tiger by some other name but it will still act like a tiger.  Someone will reasonably want to know if bijections exist between infinite sets (for reasons discussed in an earlier post).  Similarly the diagonalisation method should hold in many other situations and not just for considering bijections from ℵ to ℜ.  (NOTE: I haven't looked at every instance where something like this diagonalisation method is used, there are bound to be some applications that I am unaware of).
- - - - - - -

    Obviously that's a long post, sorry and I should just stop for now.  It's late and there's bound to be spelling and grammatical errors, sorry.

Best Wishes.

[hamdani yusuf (OP)]

The main issue is that the diagonal argument or procedure exists.  Choosing to ignore it or not to use it, doesn't make it go away.  It's a perfectly valid procedure following form the ZFC axiom system and therefore any consequences that follow from it are properties of the mathematical structures that are constructed from this axiom system. 

Many arithmetic procedures break down when infinity is involved.  Otherwise we would prove that 1=2. I see no reason why diagonal procedure is somehow different.
I think it's easier and simpler to analyze it in binary form, since we only need to consider two possible values in each digit. If every possible combination of 0 and 1 is already covered in a set, then asserting that there are other sets containing other combinations not found in the first set is absurd. Hence it should not be used in further analysis.

[Eternal Student]

Hi.

Many arithmetic procedures break down when infinity is involved.  Otherwise we would prove that 1=2.

  Agreed.

I think it's easier and simpler to analyze it in binary form, since we only need to consider two possible values in each digit.

   O.K.

If every possible combination of 0 and 1 is already covered in a set, then asserting that there are other sets containing other combinations not found in the first set is absurd.

  Agreed.  If it is true that the original set did cover all possible combinations of 0 and 1, then there can't be any other set containing other combinations not found in the first set.

    However, the diagonalisation argument is used in a proof by contradiction.   We aren't arbitrarily "asserting" that there other sets containing other combinations, we prove that there would be   AND  we don't really believe that the original set did cover all combinations of 0 and 1.  We are just temporarily assuming that the original set covered all possibilities just to get set-up for a proof by contradiction.  After this we will KNOW that the initial assumption we made had to be false.  So the original set didn't really cover all possible combinations of 0 and 1.

Example
   One of the easiest examples of a proof by contradiction is the proof that √2  is irrational,  this is often seen at school.
Here's one website that presents the proof -  https://www.mathsisfun.com/numbers/euclid-square-root-2-irrational.html

   The typical use of the diagonalisation method is in a proof by contradiction.  For example, this is how it is used to show that there isn't a bijection between ℵ and  the interval [0,1] :      Temporarily assume we have a bijection 
F:  ℵ → [0,1].  Then construct a test number, T,  so that ........etc.......  Ultimately we have T ∈ [0,1] but T ∉ Range(F).  So F was NOT a bijection.     
   This is a contradiction, so our initial (temporary) assumption had to be false.     Now we  KNOW  that there can't be a bijection  from ℵ to the interval [0,1].

- - - - - - - - -
    Sometimes it's best to say things in two different ways.  Here's another approach:

If every possible combination of 0 and 1 is already covered in a set, then asserting that there are other sets containing other combinations not found in the first set is absurd.

    Agreed.   The diagonalisation method can't be used to show the first set didn't cover all combinations of 0 and 1.  If you define the set to be all possible combinations, then that is what it is.    The diagonalisation method would only force the conclusion that this set cannot be "enumerated".   It cannot be put into a 1-to-1 correspondence with the Natural numbers.

Best Wishes.

(Late editing - to try and keep mathematical expressions together on one line and not split over two lines)

[hamdani yusuf (OP)]

However, the diagonalisation argument is used in a proof by contradiction.   We aren't arbitrarily "asserting" that there other sets containing other combinations, we prove that there would be   AND  we don't really believe that the original set did cover all combinations of 0 and 1.  We are just temporarily assuming that the original set covered all possibilities just to get set-up for a proof by contradiction.  After this we will KNOW that the initial assumption we made had to be false.  So the original set didn't really cover all possible combinations of 0 and 1.

My point was, I accept the validity of diagonalisation as long as it doesn't involve infinity.

[Eternal Student]

Hi again.

My point was, I accept the validity of diagonalisation as long as it doesn't involve infinity.

   I'm not sure what you meant by this.
   It applies to sets that are either  finite    or else   are not finite but can be enumerated (put into 1-to-1 correspondence with the natural numbers).   When I say a set is finite or that it has finite size, we mean that it has a finite number of elements.  To be very specific we have the following:
   A set S  is of finite size          if and only if
There exists N∈ℵ  and  a bijection F:  S → {1,2,3,....., N}     
(and we would also allow S to be the empty set and say it has 0 size).   

   Sets of finite size are quite easy to work with.   The only alternative is that a set is NOT of finite size and then, by definition, it is of infinite size.  The diagonal argument does work with some infinite sets, it works with infinite sets that can be enumerated.

     I think we need to discuss some basic history here.   The Natural numbers are just the counting numbers  1,2,3,..... etc.   People understood these.   So defining and understanding the size of a set wasn't difficult when you can just count the elements in the set.   The mathematical definition I gave above looks complicated but it isn't.  It's just trying to set out formally what we mean when we say a set has  N elements in it.   It means we can find a bijection between the elements of the set and the first N counting numbers.   So it turns out that  "a set is of finite size" if and only if   you can identify some Natural number, N  and there is a bijection between the set and the first N counting numbers.
     People wanted to extend this idea,  finite sets can be bijectively mapped to a subset of the Naturals which is just the first N of these Natural numbers.   So it was perfectly sensible to think that if you couldn't really map the elements of the set to the first  N   Natural numbers  (no matter how large N might be)  then  it would still be possible to map the elements of the set to the entire set of Natural numbers.   This is just saying - can we extend the idea of counting so that it never ends and we are in some sense counting all the way up to infinity?   The early Mathematicians (before Cantor) thought that you could always do this.   They believed that there was only one type of infinity,  or that it would always be possible to "count" all the elements of a set, even if the set was of infinite size.
    They were wrong.  However, you need to be very precise about what you even meant by "counting" and "counting to infinity".   These are just words, especially "counting to infinity" since this is not something you can actually do in reality.   I've mentioned in earlier posts that we realised that our instinctive notions of counting things are actually all about finding bijections.
     Anyway, Cantor didn't work with loosely defined phrases from the English language.  He didn't consider some whimsical definition of "counting" or "counting to infinity".   English language is no use in mathematics, it's subject to far too many personal interpretations.  If you're a philosopher then you can still argue that all infinite sets are countable.  However, mathematicians need a precise definition of the things they are working with.  Cantor recognised that "counting" was about finding bijections with subsets of the Natural numbers.   He effectively created the modern definition of what it means for an infinite set to be "countable".
     It turns out that our best definition of what it means to count all the way to infinity is that we can find a bijection from the set of all Natural numbers to the set of things we were trying to count.  Well, even if you don't agree that this is our "best definition" of counting all the way up to infinity, it is the one that is used.  It is the formalisation of an intuitive way to count all the way to infinity as some extension of counting a finite number of things.
 - - - - - - - - -

   So that's the history over.  Now we have the formal definition of a set having a finite size.  The alternative is that the set is not finite and then we say it is of infinite size or that it has an infinite set of elements.   The Natural numbers, for example, is not a finite set.   Given any Natural number, N, we can obviously find more than N elements in the set of Natural numbers.   The  subset  { 1, 2, 3, 4, ......, N,  N+1  }  is contained in the Naturals and it has size N+1  which is clearly greater than N.  So the Natural numbers cannot have a finite size.
    However the Natural numbers are the obvious example of a set that is infinite but can be enumerated.   We can put the set of Naturals into a 1-to1 correspondence with the set of Naturals  --->  The identify mapping will do it (just map  1 → 1;   2 → 2    and   3 → 3   ..... etc....... ).

    So the diagonalisation argument does apply to one category of infinite sets.  It applies to sets that aren't of finite size but can be enumerated.   Rather than saying an infinite set can be enumerated, it is more common just to say that the set is countably infinite.    A set is said to be countable if it is  finite  or  countably infinite.
So if we take the example of the set of Natural Numbers, ℵ, this is infinite but it is countably infinite.   Meanwhile the Real Numbers, ℜ, are also infinite but they are Uncountably infinite.

My point was, I accept the validity of diagonalisation as long as it doesn't involve infinity.

    The diagonalisation argument was developed specifically to analyse and characterise some infinite sets.  It obviously involves infinity.  It's useful for separating infinite sets into two types:   Countably infinite sets   and   Uncountably infinite sets.

Best Wishes.

[hamdani yusuf (OP)]

But Cantor discovered that natural numbers can’t be put into one-to-one correspondence with the continuum of real numbers. For instance, try to pair 1 with 1.00000… and 2 with 1.00001…, and you’ll have skipped over infinitely many real numbers (like 1.000000001…). You can’t possibly count them all; their cardinality is greater than that of the natural numbers.

Let's say that there's infinitely many real number between two consecutive natural numbers. Then there would be infinity times infinity many real number between 0 and infinity. But infinity squared is infinity.

[hamdani yusuf (OP)]

The diagonalisation argument was developed specifically to analyse and characterise some infinite sets.  It obviously involves infinity.  It's useful for separating infinite sets into two types:   Countably infinite sets   and   Uncountably infinite sets.

That's where the problem arises.

Sizes of infinity don’t stop there. Cantor discovered that any infinite set’s power set — the set of all subsets of its elements — has larger cardinality than it does. Every power set itself has a power set, so that cardinal numbers form an infinitely tall tower of infinities.

Standing at the foot of this forbidding edifice, Cantor focused on the first couple of floors. He managed to prove that the set formed from different ways of ordering natural numbers (from smallest to largest, for example, or with all odd numbers first) has cardinality 1, one level up from the natural numbers. Moreover, each of these “order types” encodes a real number.

His continuum hypothesis asserts that this is exactly the size of the continuum — that there are precisely 1 real numbers. In other words, the cardinality of the continuum immediately follow 0, the cardinality of the natural numbers, with no sizes of infinity in between.

But to Cantor’s immense distress, he couldn’t prove it.

[Eternal Student]

Hi.

That's where the problem arises.

   You can't make these problems go away.   In a sensible set theory like ZFC,  you can construct Power sets, it's a perfectly well defined object and then these power sets will have a cardinality that exceeds the original set.

   You highlighted a section : 

But to Cantor’s immense distress, he couldn’t prove it.

    Which does NOT mean that it isn't true, neither does it mean that it is true.   It just means Cantor couldn't prove it.   In fact we now know that it's truth is independent of the ZFC axiom system, so it wasn't Cantor's fault in any way.  He had no chance of proving or disproving it from the ZFC axioms.
  You can add the continuum hyothesis into your set of axioms if you want to,   or you can add the converse of the continuum hypothesis.  Either of those extra axioms wouldn't create any inconsistency.

Let's say that there's infinitely many real number between two consecutive natural numbers. Then there would be infinity times infinity many real number between 0 and infinity. But infinity squared is infinity.

    This is a naive understanding of what infinity is.  However, I'll go along with it.    Where were you going with this?  What was the problem?    There's infinitely many real numbers between 0 and infinity...... and  what's the problem with that?

Best Wishes.

[hamdani yusuf (OP)]

You can't make these problems go away.   In a sensible set theory like ZFC,  you can construct Power sets, it's a perfectly well defined object and then these power sets will have a cardinality that exceeds the original set.

I've posted a similar problem before.

How to "Prove" That 2 = 1
Let's begin our journey into the bizarre world of apparently correct, yet obviously absurd, mathematical proofs by convincing ourselves that 1 + 1 = 1. And therefore that 2 = 1. I know this sounds crazy, but if you follow the logic (and don't already know the trick), I think you'll find that the "proof" is pretty convincing.

Here's how it works:

Assume that we have two variables a and b, and that: a = b

Multiply both sides by a to get: a2 = ab

Subtract b2 from both sides to get: a2 - b2 = ab - b2

This is the tricky part: Factor the left side (using FOIL from algebra) to get (a + b)(a - b) and factor out b from the right side to get b(a - b). If you're not sure how FOIL or factoring works, don't worry—you can check that this all works by multiplying everything out to see that it matches. The end result is that our equation has become: (a + b)(a - b) = b(a - b)

Since (a - b) appears on both sides, we can cancel it to get: a + b = b

Since a = b (that's the assumption we started with), we can substitute b in for a to get: b + b = b

Combining the two terms on the left gives us: 2b = b

Since b appears on both sides, we can divide through by b to get: 2 = 1

Wait, what?! Everything we did there looked totally reasonable. How in the world did we end up proving that 2 = 1?

https://www.quickanddirtytips.com/education/math/how-to-prove-that-1-2

Here is another excerpt from the same article continuing the quote above.

What Are Mathematical Fallacies?
The truth is we didn't actually prove that 2 = 1. Which, good news, means you can relax—we haven't shattered all that you know and love about math. Somewhere buried in that "proof" is a mistake. Actually, "mistake" isn't the right word because it wasn't an error in how we did the arithmetic manipulations, it was a much more subtle kind of whoopsie-daisy known as a "mathematical fallacy."

It's never OK to divide by zero!

What was the fallacy in the famous faux proof we looked at? Like many other mathematical fallacies, our proof relies upon the subtle trick of dividing by zero. And I say subtle because this proof is structured in such a way that you might never even notice that division by zero is happening. Where does it occur? Take a minute and see if you can figure it out…

OK, got it?

It happened when we divided both sides by a - b in the fifth step. But, you say, that's not dividing by zero—it's dividing by a - b. That's true, but we started with the assumption that a is equal to b, which means that a - b is the same thing as zero! And while it's perfectly fine to divide both sides of an equation by the same expression, it's not fine to do that if the expression is zero. Because, as we've been taught forever, it's never OK to divide by zero!

Why Can't You Divide By Zero?
Which might get you wondering: Why exactly is it that we can't divide by zero? We've all been warned about such things since we were little lads and ladies, but have you ever stopped to think about why division by zero is such an offensive thing to do? There are many ways to think about this. We'll talk about two reasons today.

The first has to do with how division is related to multiplication. Let's imagine for a second that division by zero is fine and dandy. In that case, a problem like 10 / 0 would have some value, which we'll call x. We don't know what it is, but we'll just assume that x is some number. So 10 / 0 = x. We can also look at this division problem as a multiplication problem asking what number, x, do we have to multiply by 0 to get 10? Of course, there's no answer to this question since every number multiplied by zero is zero. Which means the operation of dividing by zero is what's dubbed "undefined."

[hamdani yusuf (OP)]

This is a naive understanding of what infinity is.

What is the non-naive understanding of infinity?

[Eternal Student]

Hi.
   I'm not sure what the relevance of the last posts were.   I'm not sure what you were trying to say with it.  I also don't think it would be helpful if I just repeat some of what I've said before.
   Perhaps it would just be better to explain what it is you were or are hoping to do now or where you think there is a problem now.

Best Wishes.