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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.
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.
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argumentAn 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
Perhaps we can collaborate to write a proper research paper on this.
This is why I'm focusing on the first question - how did you intend to get the paper published or made available to others?
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.
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.
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
Math can be shortMath 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.
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.
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 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.
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 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.
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.
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.
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.
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.
That's where the problem arises.
But to Cantor’s immense distress, he couldn’t prove it.
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.
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.
How to "Prove" That 2 = 1Let'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 = bMultiply both sides by a to get: a2 = abSubtract b2 from both sides to get: a2 - b2 = ab - b2This 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 = bSince a = b (that's the assumption we started with), we can substitute b in for a to get: b + b = bCombining the two terms on the left gives us: 2b = bSince b appears on both sides, we can divide through by b to get: 2 = 1Wait, 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
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."
This is a naive understanding of what infinity is.