Cantors diagonal - Of course, this follows immediately from Cantor's diagonal argument. But what I find striking is that, in this form, the diagonal argument does not involve the notion of equality. This prompts the question: (A) Are there other interesting examples of mathematical reasonings which don't involve the notion of equality?

 
The idea behind the proof of this theorem, due to G. Cantor (1878), is called "Cantor's diagonal process" and plays a significant role in set theory (and elsewhere). Cantor's theorem implies that no two of the sets $$2^A,2^{2^A},2^{2^{2^A}},\dots,$$ are equipotent.. Can you make it rain in terraria

What you should realize is that each such function is also a sequence. The diagonal arguments works as you assume an enumeration of elements and thereby create an element from the diagonal, different in every position and conclude that that element hasn't been in the enumeration.No question, or deep answers to be found here! I just wanted to share with you a pretty formulation of Cantor's diagonal argument that there is no bijection between a set X and its power set P(X). (the power set is the set of all subsets of X) It's based on the idea of a characteristic function: a function whose values are only 0 and 1.Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is also called the diagonalization argument or the diagonal slash argument or the diagonal method .) The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, but was published ...The argument Georg Cantor presented was in binary. And I don't mean the binary representation of real numbers. Cantor did not apply the diagonal argument to real numbers at all; he used infinite-length binary strings (quote: "there is a proof of this proposition that ... does not depend on considering the irrational numbers.")How to Create an Image for Cantor's *Diagonal Argument* with a Diagonal Oval. Ask Question Asked 4 years, 2 months ago. Modified 4 years, 2 months ago. Viewed 1k times 4 I would like to ...This paper will argue that Cantor's diagonal argument too shares some features of the mahāvidyā inference. A diagonal argument has a counterbalanced statement. Its main defect is its counterbalancing inference. Apart from presenting an epistemological perspective that explains the disquiet over Cantor's proof, this paper would show that ...Using Cantor's Diagonal Argument to compare the cardinality of the natural numbers with the cardinality of the real numbers we end up with a function f: N → ( 0, 1) and a point a ∈ ( 0, 1) such that a ∉ f ( ( 0, 1)); that is, f is not bijective. My question is: can't we find a function g: N → ( 0, 1) such that g ( 1) = a and g ( x) = f ...Search titles only By: Search Advanced search…Cantor's diagonal argument shows that any attempted bijection between the natural numbers and the real numbers will necessarily miss some real numbers, and therefore cannot be a valid bijection. While there may be other ways to approach this problem, the diagonal argument is a well-established and widely used technique in mathematics for ...Cantors diagonal argument is a technique used by Georg Cantor to show that the integers and reals cannot be put into a one-to-one correspondence (i.e., the uncountably infinite set of real numbers is "larger" than the countably infinite set of integers).How to Create an Image for Cantor's *Diagonal Argument* with a Diagonal Oval. Ask Question Asked 4 years, 2 months ago. Modified 4 years, 2 months ago.The idea behind the proof of this theorem, due to G. Cantor (1878), is called "Cantor's diagonal process" and plays a significant role in set theory (and elsewhere). Cantor's theorem implies that no two of the sets $$2^A,2^{2^A},2^{2^{2^A}},\dots,$$ are equipotent.In order for Cantor's construction to work, his array of countably infinite binary sequences has to be square. If si and sj are two binary sequences in the...About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features NFL Sunday Ticket Press Copyright ...Yes, in that case, we would have shown that the set of rational numbers is "uncountable". Since you are the one claiming that you could apply Cantor's argument to the rational numbers, and get the same result, you would have to show that it is possible for this process to result in a rational...Explanation of Cantor's diagonal argument.This topic has great significance in the field of Engineering & Mathematics field.The diagonal lemma applies to theories capable of representing all primitive recursive functions. Such theories include first-order Peano arithmetic and the weaker Robinson arithmetic, and even to a much weaker theory known as R. A common statement of the lemma (as given below) makes the stronger assumption that the theory can represent all ...I have looked into Cantor's diagonal argument, but I am not entirely convinced. Instead of starting with 1 for the natural numbers and working our way up, we could instead try and pair random, infinitely long natural numbers with irrational real numbers, like follows: 97249871263434289... 0.12834798234890899... 29347192834769812...5 ກ.ພ. 2021 ... Cantor's diagonal argument is neat because it provides us with a clever way to confront infinities which can't be avoided. Infinities are ...1. The Cantor's diagonal argument works only to prove that N and R are not equinumerous, and that X and P ( X) are not equinumerous for every set X. There are variants of the same idea that will help you prove other things, but "the same idea" is a pretty informal measure. The best one can really say is that the idea works when it works, and if ...I have recently been given a new and different perspective about Cantor's diagonal proof using bit strings. The new perspective does make much more intuitive, in my opinion, the proof that there is at least one transfinite number greater then the number of natural numbers. First to establish...Georg Ferdinand Ludwig Philipp Cantor (/ ˈ k æ n t ɔːr / KAN-tor, German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantɔʁ]; 3 March [O.S. 19 February] 1845 – 6 January 1918) was a mathematician.He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one …Cantor's Diagonal Argument: The maps are elements in N N = R. The diagonalization is done by changing an element in every diagonal entry. Halting Problem: The maps are partial recursive functions. The killer K program encodes the diagonalization. Diagonal Lemma / Fixed Point Lemma: The maps are formulas, with input being the codes of sentences.In particular, there is no objection to Cantor's argument here which is valid in any of the commonly-used mathematical frameworks. The response to the OP's title question is "Because it doesn't follow the standard rules of logic" - the OP can argue that those rules should be different, but that's a separate issue.Cantor's Diagonal Argument - Rational. 0. Cantor's diagonalization- why we must add $2 \pmod {10}$ to each digit rather than $1 \pmod {10}$? Hot Network Questions Questions on reading the prologue of Aesopus Latinus via LLPSI Are there examples of mutual loanwords in French and in English? Do fighter pilots have to manually input the ordnance ...Cantor's diagonal argument provides a convenient proof that the set of subsets of the natural numbers (also known as its power set) is not countable.More generally, it is a recurring theme in computability theory, where perhaps its most well known application is the negative solution to the halting problem.. Informal description. The original Cantor's idea was to show that the family of 0-1 ...I have found that Cantor’s diagonalization argument doesn’t sit well with some people. It feels like sleight of hand, some kind of trick. Let me try to outline some of the ways it could be a trick. You can’t list all integers One argument against Cantor is that you can never finish writing z because you can never list all of the integers ...We examine Cantor’s Diagonal Argument (CDA). If the same basic assumptions and theorems found in many accounts of set theory are applied with a standard combinatorial formula a contradiction is ...I have found that Cantor's diagonalization argument doesn't sit well with some people. It feels like sleight of hand, some kind of trick. Let me try to outline some of the ways it could be a trick. You can't list all integers One argument against Cantor is that you can never finish writing z because you can never list all of the integers ...Question about Cantor's Diagonalization Proof. My discrete class acquainted me with me Cantor's proof that the real numbers between 0 and 1 are uncountable. I understand it in broad strokes - Cantor was able to show that in a list of all real numbers between 0 and 1, if you look at the list diagonally you find real numbers that are not included ...Applying Cantor's diagonal argument. 0. Is the Digit-Matrix in Cantors' Diagonal Argument square-shaped? Hot Network Questions What is the proper way to remove a receptacle from a wall? How to discourage toddler from pulling out chairs when he loves to be picked up Why ...ÐÏ à¡± á> þÿ C E ...W e are now ready to consider Cantor's Diagonal Argument. It is a reductio It is a reductio argument, set in axiomatic set theory with use of the set of natural numbers.An octagon has 20 diagonals. A shape’s diagonals are determined by counting its number of sides, subtracting three and multiplying that number by the original number of sides. This number is then divided by two to equal the number of diagon...In this video, we prove that set of real numbers is uncountable.The canonical proof that the Cantor set is uncountable does not use Cantor's diagonal argument directly. It uses the fact that there exists a bijection with an uncountable set (usually the interval $[0,1]$). Now, to prove that $[0,1]$ is uncountable, one does use the diagonal argument. I'm personally not aware of a proof that doesn't use it.Georg Ferdinand Ludwig Philipp Cantor ( / ˈkæntɔːr / KAN-tor, German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantɔʁ]; 3 March [ O.S. 19 February] 1845 – 6 January 1918 [1]) was a mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established ...2. Cantor's diagonal argument is one of contradiction. You start with the assumption that your set is countable and then show that the assumption isn't consistent with the conclusion you draw from it, where the conclusion is that you produce a number from your set but isn't on your countable list. Then you show that for any.Applying Cantor's diagonal method (for simplicity let's do it from right to left), a number that does not appear in enumeration can be constructed, thus proving that set of all natural numbers ...To provide a counterexample in the exact format that the "proof" requires, consider the set (numbers written in binary), with diagonal digits bolded: x[1] = 0. 0 00000... x[2] = 0.0 1 1111...So, I understand how Cantor's diagonal argument works for infinite sequences of binary digits. I also know it doesn't apply to natural numbers since they "zero out". However, what if we treated each sequence of binary digits in the original argument, as an integer in base-2? In that case, the newly produced sequence is just another integer, and ...Cantor's theorem implies that there are infinitely many infinite cardinal numbers, and that there is no largest cardinal number. It also has the following interesting consequence: There is no such thing as the "set of all sets''. Suppose A A were the set of all sets. Since every element of P(A) P ( A) is a set, we would have P(A) ⊆ A P ( A ...This you prove by using cantors diagonal argument via a proof by contradiction. Also it is worth noting that (I think you need the continuum hypothesis for this). Interestingly it is the transcendental numbers (i.e numbers that aren't a root of a polynomial with rational coefficients) like pi and e.Cantor's diagonal argument. GitHub Gist: instantly share code, notes, and snippets.I came across Cantors Diagonal Argument and the uncountability of the interval $(0,1)$.The proof makes sense to me except for one specific detail, which is the following.You can iterate over each character, and if the character is part of a word, then each possibility (vertical, horizontal, right-diag, left-diag) can be checked:To make sense of how the diagonal method applied to real numbers show their uncountability while not when applied to rational numbers, you need the concept of real numbers being infinitely unique in two dimensions while rational numbers are only infinitely unique in one dimension, which shows that any "new" number created is same as a rational number already in the list.The proof was later called as "Cantor's diagonal argument". It is in fact quite simple, and there is an excellent animation on that in [1]. That is the reason why starting from mentioned date, mathematicians use to differentiate between those 2 types of infinities. 3. Recalling Cantor's diagonal argumentMolyneux, P. (2022) Some Critical Notes on the Cantor Diagonal Argument. Open Journal of Philosophy, 12, 255-265. doi: 10.4236/ojpp.2022.123017 . 1. Introduction. 1) The concept of infinity is evidently of fundamental importance in number theory, but it is one that at the same time has many contentious and paradoxical aspects.1. The Cantor's diagonal argument works only to prove that N and R are not equinumerous, and that X and P ( X) are not equinumerous for every set X. There are variants of the same idea that will help you prove other things, but "the same idea" is a pretty informal measure. The best one can really say is that the idea works when it works, and if ...The diagonal lemma applies to theories capable of representing all primitive recursive functions. Such theories include first-order Peano arithmetic and the weaker Robinson arithmetic, and even to a much weaker theory known as R. A common statement of the lemma (as given below) makes the stronger assumption that the theory can represent all ...A triangle has zero diagonals. Diagonals must be created across vertices in a polygon, but the vertices must not be adjacent to one another. A triangle has only adjacent vertices. A triangle is made up of three lines and three vertex points...Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is also called the diagonalization argument or the diagonal slash argument or the diagonal method .) The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, but was published ...The original "Cantor's Diagonal Argument" was to show that the set of all real numbers is not "countable". It was an "indirect proof" or "proof by contradiction", starting by saying "suppose we could associate every real number with a natural number", which is the same as saying we can list all real numbers, the shows that this leads to a ...Cantor’s diagonal argument answers that question, loosely, like this: Line up an infinite number of infinite sequences of numbers. Label these sequences with whole numbers, 1, 2, 3, etc. Then, make a new sequence by going along the diagonal and choosing the numbers along the diagonal to be a part of this new sequence — which is …So far my conclusion is that either my textbooks are not being rigid enough in their proofs or the only thing cantors diagonal proof really proves is that it's absurd to talk about a complete list of even a countable set. A "list" means to have a "first", a "second" etc. A list is precisely a one-to-one correspondence with the natural numbers.Let S be the subset of T that is mapped by f (n). (By the assumption, it is an improper subset and S = T .) Diagonalization constructs a new string t0 that is in T, but not in S. Step 3 contradicts the assumption in step 1, so that assumption is proven false. This is an invalid proof, but most people don’t seem to see what is wrong with it.24 ກ.ພ. 2012 ... Theorem (Cantor): The set of real numbers between 0 and 1 is not countable. Proof: This will be a proof by contradiction. That means, we will ...Abstract. We examine Cantor’s Diagonal Argument (CDA). If the same basic assumptions and theorems found in many accounts of set theory are applied with a standard combinatorial formula a ...Cantor's diagonal argument question . I'm by no means a mathematician so this is a layman's confusion after watching Youtube videos. I understand why the (new) real number couldn't be at any position (i.e. if it were, its [integer index] digit would be different, so it contradicts the assumption).Cantor's diagonal argument states that if you make a list of every natural number, and pair each number with a real number between 0 and 1, then go down the list one by one, diagonally adding one to the real number or subtracting one in the case of a nine (ie, the tenths place in the first number, the hundredths place in the second, etc), until ...Georg Cantor's diagonal argument, what exactly does it prove? (This is the question in the title as of the time I write this.) It proves that the set of real numbers is strictly larger than the set of positive integers. In other words, there are more real numbers than there are positive integers. (There are various other equivalent ways of ...$\begingroup$ @Gary In the argument there are infinite rows, and each number contains infinite digits. These plus changing a number in each row creates a "new" number not on the "list." This assumes one could somehow "freeze" the infinite rows and columns to a certain state to change the digits, instead of infinity being a process that never ends.However, it's obviously not all the real numbers in (0,1), it's not even all the real numbers in (0.1, 0.2)! Cantor's argument starts with assuming temporarily that it's possible to list all the reals in (0,1), and then proceeds to generate a contradiction (finding a number which is clearly not on the list, but we assumed the list contains ...$\begingroup$ This seems to be more of a quibble about what should be properly called "Cantor's argument". Certainly the diagonal argument is often presented as one big proof by contradiction, though it is also possible to separate the meat of it out in a direct proof that every function $\mathbb N\to\mathbb R$ is non-surjective, as you do, …Independent of Cantor's diagonal we know all cauchy sequences (and every decimal expansion is a limit of a cauchy sequence) converge to a real number. And we know that for every real number we can find a decimal expansion converging to it. And, other than trailing nines and trailing zeros, each decimal expansions are unique.Cantor's diagonal argument has never sat right with me. I have been trying to get to the bottom of my issue with the argument and a thought occurred to me recently. It is my understanding of Cantor's diagonal argument that it proves that the uncountable numbers are more numerous than the countable numbers via proof via contradiction. If it is ...In any event, Cantor's diagonal argument is about the uncountability of infinite strings, not finite ones. Each row of the table has countably many columns and there are countably many rows. That is, for any positive integers n, m, the table element table(n, m) is defined.Molyneux, P. (2022) Some Critical Notes on the Cantor Diagonal Argument. Open Journal of Philosophy, 12, 255-265. doi: 10.4236/ojpp.2022.123017 . 1. Introduction. 1) The concept of infinity is evidently of fundamental importance in number theory, but it is one that at the same time has many contentious and paradoxical aspects.Cantor's diagonalisation can be rephrased as a selection of elements from the power set of a set (essentially part of Cantor's Theorem). If we consider the set ... But it works only when the impossible characteristic halting function is built from the diagonal of the list of Turing permitted characteristic halting functions, by ...This analysis shows Cantor's diagonal argument published in 1891 cannot form a new sequence that is not a member of a complete list. The proof is based on the pairing of complementary sequences forming a binary tree model. 1. the argument Assume a complete list L of random infinite sequences. Each sequence S is a uniqueApply Cantor’s Diagonalization argument to get an ID for a 4th player that is different from the three IDs already used. I can't wrap my head around this problem. ... This is a good way to understand Cantor's diagonal process but a terrible way to assign IDs.Yet Cantor's diagonal argument demands that the list must be square. And he demands that he has created a COMPLETED list. That's impossible. Cantor's denationalization proof is bogus. It should be removed from all math text books and tossed out as being totally logically flawed. It's a false proof.• Cantor's diagonal argument. • Uncountable sets - R, the cardinality of R (c or 2N0, ]1 - beth-one) is called cardinality of the continuum. ]2 beth-two cardinality of more uncountable numbers. - Cantor set that is an uncountable subset of R and has Hausdorff dimension number between 0 and 1. (Fact: Any subset of R of Hausdorff dimensionGeorg Cantor's first uncountability proof demonstrates that the set of all real numbers is uncountable. This proof differs from the more familiar proof that uses his diagonal argument. Cantor's first uncountability proof was published in 1874, in an article that also contains a proof that the set of real algebraic numbers is countable, and a ...The properties and implications of Cantor's diagonal argument and their later uses by Gödel, Turing and Kleene are outlined more technically in the paper: Gaifman, H. (2006). Naming and Diagonalization, from Cantor to Gödel to Kleene. Logic Journal of the IGPL 14 (5). pp. 709-728.Cantor's diagonal argument is a very simple argument with profound implications. It shows that there are sets which are, in some sense, larger than the set of natural numbers. To understand what this statement even means, we need to say a few words about what sets are and how their sizes are compared. Preliminaries Naively, we…We would like to show you a description here but the site won't allow us.This you prove by using cantors diagonal argument via a proof by contradiction. Also it is worth noting that (I think you need the continuum hypothesis for this). Interestingly it is the transcendental numbers (i.e numbers that aren't a root of a polynomial with rational coefficients) like pi and e.The underlying function is the Cantor pairing function. Yesterday I was writing codes to hash two integers and using the Cantor pairing function turns out to be a neat way. Formally, the Cantor pairing function π is defined as: π: N × N → N π ( k 1, k 2) := 1 2 ( k 1 + k 2) ( k 1 + k 2 + 1) + k 2. It can also be easily extended to ...First, Cantor’s celebrated theorem (1891) demonstrates that there is no surjection from any set X onto the family of its subsets, the power set P(X). The proof is straight forward. Take I = X, and consider the two families {x x : x ∈ X} and {Y x : x ∈ X}, where each Y x is a subset of X.In fact, they all involve the same idea, called "Cantor's Diagonal Argument." Share. Cite. Follow answered Apr 10, 2012 at 1:20. Arturo Magidin Arturo Magidin. 384k 55 55 gold badges 803 803 silver badges 1113 1113 bronze badges $\endgroup$ 6If you find our videos helpful you can support us by buying something from amazon.https://www.amazon.com/?tag=wiki-audio-20Cantor's diagonal argument In set ...I'm joining a global contest where I have to explain about a difficult topic (university level) and I thought Cantor's Diagonal Argument was quite interesting. But after watching a few videos, I thought the proof was quite easy to understand? Am I missing something or should I just scrap this?Explanation of Cantor's diagonal argument.This topic has great significance in the field of Engineering & Mathematics field.In this guide, I'd like to talk about a formal proof of Cantor's theorem, the diagonalization argument we saw in our very first lecture.Now in order for Cantor's diagonal argument to carry any weight, we must establish that the set it creates actually exists. However, I'm not convinced we can always to this: For if my sense of set derivations is correct, we can assign them Godel numbers just as with formal proofs.I studied Cantor's Diagonal Argument in school years ago and it's always bothered me (as I'm sure it does many others). In my head I have two counter-arguments to Cantor's Diagonal Argument. I'm not a mathy person, so obviously, these must have explanations that I have not yet grasped.

The underlying function is the Cantor pairing function. Yesterday I was writing codes to hash two integers and using the Cantor pairing function turns out to be a neat way. Formally, the Cantor pairing function π is defined as: π: N × N → N π ( k 1, k 2) := 1 2 ( k 1 + k 2) ( k 1 + k 2 + 1) + k 2. It can also be easily extended to .... Savickas career construction theory

cantors diagonal

A diagonally incrementing "snaking" function, from same principles as Cantor's pairing function, is often used to demonstrate the countability of the rational numbers. The graphical shape of Cantor's pairing function, a diagonal progression, is a standard trick in working with infinite sequences and countability.Contrary to what most people have been taught, the following is Cantor's Diagonal Argument. (Well, actually, it isn't. Cantor didn't use it on real numbers. But I don't want to explain what he did use it on, and this works.): Part 1: Assume you have a set S of of real numbers between 0 and 1 that can be put into a list.Cantor’s diagonal argument, the rational open interv al (0, 1) would be non-denumerable, and we would ha ve a contradiction in set theory , because Cantor also prov ed the set of the rational ...Read Grog Cantor's "Diagonal Argument" from the story Banach - Tarski Paradox By: DJ - Pon 3 by DJPon3ation (Portal Shot) with 244 reads. If you don't unde.11. I cited the diagonal proof of the uncountability of the reals as an example of a `common false belief' in mathematics, not because there is anything wrong with the proof but because it is commonly believed to be Cantor's second proof. The stated purpose of the paper where Cantor published the diagonal argument is to prove the existence of ...Cantor's 1891 Diagonal proof: A complete logical analysis that demonstrates how several untenable assumptions have been made concerning the proof. Non-Diagonal Proofs and Enumerations: Why an enumeration can be possible outside of a mathematical system even though it is not possible within the system.Cantor diagonal argument. Antonio Leon. This paper proves a result on the decimal expansion of the rational numbers in the open rational interval (0, 1), which is subsequently used to discuss a reordering of the rows of a table T that is assumed to contain all rational numbers within (0, 1), in such a way that the diagonal of the reordered ...2. Cantor's diagonal argument is one of contradiction. You start with the assumption that your set is countable and then show that the assumption isn't consistent with the conclusion you draw from it, where the conclusion is that you produce a number from your set but isn't on your countable list. Then you show that for any.I'm trying to grasp Cantor's diagonal argument to understand the proof that the power set of the natural numbers is uncountable. On Wikipedia, there is the following illustration: The explanation of the proof says the following: By construction, s differs from each sn, since their nth digits differ (highlighted in the example).4. The essence of Cantor's diagonal argument is quite simple, namely: Given any square matrix F, F, one may construct a row-vector different from all rows of F F by simply taking the diagonal of F F and changing each element. In detail: suppose matrix F(i, j) F ( i, j) has entries from a set B B with two or more elements (so there exists a ...$\begingroup$ This seems to be more of a quibble about what should be properly called "Cantor's argument". Certainly the diagonal argument is often presented as one big proof by contradiction, though it is also possible to separate the meat of it out in a direct proof that every function $\mathbb N\to\mathbb R$ is non-surjective, as you do, and ...I have found that Cantor’s diagonalization argument doesn’t sit well with some people. It feels like sleight of hand, some kind of trick. Let me try to outline some of the ways it could be a trick. You can’t list all integers One argument against Cantor is that you can never finish writing z because you can never list all of the integers ...Maybe the real numbers truly are uncountable. But Cantor's diagonalization "proof" most certainly doesn't prove that this is the case. It is necessarily a flawed proof based on the erroneous assumption that his diagonal line could have a steep enough slope to actually make it to the bottom of such a list of numerals.Cantor's diagonal argument provides a convenient proof that the set of subsets of the natural numbers (also known as its power set) is not countable.More generally, it is a recurring theme in computability theory, where perhaps its most well known application is the negative solution to the halting problem.. Informal description. The original Cantor's idea was to show that the family of 0-1 ...Use Cantor's diagonal argument to show that the set of all infinite sequences of the letters a, b, c, and d are uncountably infinite. Engineering & Technology Computer Science COMPUTER CS323. Comments (0) Answer & Explanation. Solved by verified expert. Rated Helpful1. The Cantor's diagonal argument works only to prove that N and R are not equinumerous, and that X and P ( X) are not equinumerous for every set X. There are variants of the same idea that will help you prove other things, but "the same idea" is a pretty informal measure. The best one can really say is that the idea works when it works, and if ...Cantor's theorem shows that that is (perhaps surprisingly) false, and so it's not that the expression "$\infty>\infty$" is true or false in the context of set theory but rather that the symbol "$\infty$" isn't even well-defined in this context so the expression isn't even well-posed..

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