Talk:Cantor's diagonal argument/Archive 3

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Archive 2

Archive 3

Constructivist interpretations

The late User:Futonchild added a problematic section which I tried to shape into something that was at least meaningful, but frankly it's still problematic. It goes like this at present:

The interpretation of Cantor's result will depend upon one's view of mathematics, and more specifically on how one thinks of mathematical functions. In the context of classical mathematics, functions need not be computable, and hence the diagonal argument establishes that, there are more infinite sequences of ones and zeros than there are natural numbers. To those constructivists who countenance only computable functions, Cantor's proof (merely) shows that there is no recursively enumerable set of indices (for example, Gödel numbers) for the programs computing them.

Now, I think it is true that constructivists, or at least some of them, do not accept the "quantity" interpretation of the argument. It seems to me, though, that to deny the quantity interpretation, they're pretty much obliged by the argument to deny that the sequences of zeroes and ones can be collected into a completed totality. They are not really saying, that is, that there are only as many sequences of zeroes and ones as there are natural numbers, even if the only sequences they accept are the computable ones, because for coherency's sake the only enumeration of sequences they could accept is a computable one (say, a computable enumeration of Turing machines that produce total sequences, or a global computable function giving the nth bit of the mth sequence), and Cantor's argument (which by the way is intuitionistically valid) excludes that possibility. (I don't know any intuitionistic proof, on the other hand, that there's no injection from the sequences into the naturals; it's conceivable that some intuitionists believe that the existence of such an injection is "not false".)

Anyway, I think the current text is unsatisfactory, but I don't want to just delete it out of hand; there probably are constructivist interpretations that deny the proof is about "quantity", and they should be fairly represented. Can anyone help out here, especially with versions that might be attributed to specific constructivist/intuitionist thinkers? --Trovatore 07:00, 14 March 2007 (UTC)

Your parenthetical remark about injections is close. I'm not sure but I think your proposed injection is still inconsistent, on the other hand it is quite consistent to assert a partial surjection from the naturals to the infinite bit sequences, i.e. that the infinite bit sequences are subcountable. Clearly this contradicts the "quantity interpretation" because although the naturals and infinite bit sequences are not in bijection, each one can be seen as a partial image of the other. Now pardon me while I fix this redlink. --99.234.59.230 (talk) 03:34, 4 January 2008 (UTC)

Could someone explain how $f:s\rightarrow T$ can have a partial surjection (i.e. be subcountable)? If the proof shows that there exists $T_{x}$ such that there does not exist a corresponding $s_{x},$ doesn't that mean that f is not surjective? If I'm understanding partial surjection correctly, it just means that the function is surjective and it is also a partial function. If f is not surjective, how can it have a partial surjection? StardustInTheDark (talk) 15:10, 25 November 2017 (UTC)

Confusion

In the section An uncountable set, it says

This sequence s_i is countable, as to every natural number n we associate one and only one element of the sequence.

And then below the example sequence, it goes on to state

For each m and n let s_m,n be the n^th element of the m^th sequence on the list. So, for instance, s_2,1 is the first element of the second sequence.

Is the n above and below the same? The same goes for the words "sequence" and "element".

Above the example sequence, the text appears to use the terms "sequence" and "element" in terms of the higher hierarchy, i.e. "sequence" referring to the list of numbers, and "elements" referring to individual sequences (i.e. lines), and n appears to refer to the numbering of the elements of the list of sequences.

Below the example sequence, individual sequences from the list (i.e. the "meta-sequence") are discussed as sequences, and now "element" refers to the position after decimal point within each individual sequence.

Could this be clarified? Maybe I'm just not understanding it properly, but I believe it's not just superficially confusing but actually nonsensical because the same terms (and the variable n) are used first in higher-order context, then in lower-order context. Or am I missing something here? --84.44.158.71 (talk) 18:38, 6 October 2012 (UTC)

No, there's no confusion as both n are preceded by words 'every' or 'each'. These two are separate statements, each of them describes its domain and each of them uses its own n to denote a variable from the domain. Of course we might force using different character to denote variables for each sentence in the text, but there is no need to do so. Anyway our alphabet is too short to do that even in medium-size mathematical text, not to mention integral tables... --CiaPan (talk) 18:51, 7 October 2012 (UTC)

And there's no 'decimal point' in any of the sequences discussed in that section. --CiaPan (talk) 18:38, 9 October 2012 (UTC)

You don't understand the mathematics involved. --213.196.218.202 (talk) 11:48, 6 November 2012 (UTC)

Does this create only a single sequence that falls outside of the enumeration rules?

I'm not a mathematician, but the way the article is currently worded, a very cursory glance implies that this argument results in the construction of one and only one sequence that is present in the collection but not given in the enumeration.

If that is the case, why couldn't the enumeration rules be redefined to simply give that one single special case as S1 as the rest of the sequences starting at S2, S3, S4, etc. ? Seems like a rather obvious solution. If functions can have piecewise definitions; why not enumerations? Element S1 equals == Foo, Elements S2-S[infinity] == Bar.

The article on countability says "Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a unique natural number." Right. I think I get that. So, uhhhhhh...just adjust your enumeration ordering rules to be slightly messier. Problem solved? At least in the context of Cantor's Argument.

But if Cantor's argument implies the existence of more than one sequence excluded from the enumeration and these exclusions cannot be ordered, perhaps the article could make that more clear?

I'm not trying to claim I've revolutionized mathematics here with my 5 minutes of article skimming and thought; I'm just saying the description of the proof as-is fails to clearly explain why one could not write a set of piecewise-defined enumeration rules to put the elements of T in a one to one relationship with the natural numbers. Blue Rock (talk) 19:52, 25 October 2017 (UTC)

No, the proof is fine as it is. What you're missing is that the proof works for any proposed enumeration. If there is one, you reach a contradiction; therefore, there isn't one. This is a fundamental point of logic.

If you want to discuss how the point can be made clearer, we can do that here, but if you want more explanation on a personal level, please ask a question at WP:RD/Math. --Trovatore (talk) 20:01, 25 October 2017 (UTC)

I understand how proof by contradiction works, thank you. Please read my post more carefully. The article does not explain at all how this would apply to any possible enumeration scheme; in fact it completely glosses over Cantor's chosen scheme, which I have not dissected or thought about yet in any great detail. If Cantor's diagonalization argument does not construct more than this one single contradiction, then one could simply construct a new enumeration scheme simply by specifying that one known, definable sequence as my S1, then use Cantor's rules--new S2 is Cantor's S1, new S3 is Cantor's S2, etc. I'm still reading but the article on this argument, as it is currently worded, does not appear to address how Cantor's diagonalization could be used to rule out piecewise defined enumeration rules. Blue Rock (talk) 20:15, 25 October 2017 (UTC)

The article does in fact explain how it would apply to any possible enumeration scheme. --Trovatore (talk) 20:18, 25 October 2017 (UTC)

Mmmm... sort of. Not terribly well but yes after a few more minutes of thought I think I see what the main idea is that I've completely overlooked here. The construction of the contradiction is permitted to be even more "piecewise" as you are basically are allowed to "look" at the entire sequence and contradict each term individually. Uh, well ok then. Nevermind. There is actually still an interesting argument (not for this talk page, sure sure) to be had here in that what happens if you allow the rules for enumeration to explicitly contradict the rules for generating the contradicting sequence, which results in an undecidable stalemate (between the enumeration ruleset would be trying to defeat the ruleset for generating the contradiction). I grant that said stalemate would result in no enumeration being possible for an infinite set of infinite sequences, so in that respect I guess Cantor wins. But what do you call a self-referential set of rules that cannot in fact end up generating any enumeration because they (impossibly) seek to deny the possibility of this particular hand-crafted contradiction? The rule set I am describing exists. It has no logically consistent interpretation, but it surely exists in the same way that undecidable statements in formal logic exist.

So...on reflection, I must insist I do still have a point, but it's convoluted enough to fall outside of the scope of the article. Carry on, then. Blue Rock (talk) 21:02, 25 October 2017 (UTC)

Summary: my own brief comprehension issue was that in the "Uncountable set" section, the article did not seem to provide either the function to generate the ordered sequences or the function to generate the contradiction. If it had, it might have been clear at a glance that a generalization of the formalization to generate the contradiction was, well, just a wrapper around the function to generate the sequences. In that case, of course you can't ever beat it. But you CAN "tie" it. If the function to generate the ordered sequences were able to refer to the function that generates the proof-by-contradiction sequence--if it were a FAIR fight, so to speak, with the contradiction function and the enumeration function both allowed to refer to each other--you could at least create an unsolvable contradiction such that no stable enumeration was possible. Which I maintain is not the same thing as merely generating an incomplete enumeration. The significance of that observation, I don't know. But I do believe I have a small point here about the clarity of the article, and there is thus an argument for a more formal treatment in the Uncountable Set section, though I'd understand if we're just trying to keep the example as simple to grasp as possible. But these misunderstanding will tend to crop up when infinity is involved. Blue Rock (talk) 21:51, 25 October 2017 (UTC)

"Cantor" as agent in the argument

As the article currently explains the argument, there's a lot of Cantor does this, Cantor does that, as though he were personally constructing the counterexample to a proposed enumeration.

I know we didn't actually invent this style; I've seen it before, and it probably does exist in some RS. But that doesn't mean we have to use it. It kind of grates on me. Does anyone object to changing the exposition so that we can give Cantor's spirit some rest, instead of constantly requiring him to be involved in the argument? --Trovatore (talk) 22:18, 25 October 2017 (UTC)

Cantor killed my father. He deserves no rest. Blue Rock (talk) 22:19, 25 October 2017 (UTC)

(I have no opinion on it as a matter of style but I did feel unable to resist adding that, sorry) Blue Rock (talk) 22:21, 25 October 2017 (UTC)

Trovatore makes a good point that Cantor is mentioned too often in the diagonal argument proof. I eliminated some of the mentions of Cantor and only kept those relevant to the strategic decisions that he made. Namely, he separated his proof into two parts: a constructive part and a proof by contradiction to prove uncountability. This mention of Cantor is important because it's historically accurate and because Cantor used this approach whenever possible. Too many accounts of Cantor's work leave out the constructive part of his proofs. This has led to Cantor being credited for non-constructive proofs that he did not publish and then these proofs are critiqued as just pure existence proofs. Another example of Cantor giving constructive arguments can be found in Cantor's first set theory article where his constructive argument leads to both the construction of transcendental numbers and the uncountability of the real numbers. By the way, I also made a small change to the non-constructive proof of "T is uncountable" that clarifies the contradiction. --RJGray (talk) 21:50, 14 January 2018 (UTC)

Interpretations section

This is currently very problematic, as it claims that the reals can be subcountable in constructivist logics, which as far as I know is not true. In constructivism, Cantor's diagonal argument simply shows that the natural numbers (any countable set) cannot "exhaust" the reals, but indeed there is a trivial injection from the naturals into the real, so the reals can hardly be subcountable!

2A02:C7F:C617:6600:8426:D508:2908:C46F (talk) 21:46, 28 October 2017 (UTC)

Subcountability is about sujections (as opposed to injections, which you mention), so it's not quite clear how you draw that conclusion. 62.218.65.235 (talk) 13:33, 24 May 2020 (UTC)

The article does not include any proof that "there are actually more infinite sequences of ones and zeros than there are natural numbers."

The article asserts that:

the diagonal argument establishes that, although both sets are infinite, there are actually more infinite sequences of ones and zeros than there are natural numbers.

No, it doesn't. The diagonal argument only establishes that, given an enumeration of real numbers, which is not an enumeration of all real numbers, then another real number can be defined in terms of that enumeration. Form this we can further conclude that there cannot be an enumeration of all real numbers, since then we would have a number that was both in and not in the enumeration.

What the article omits is that the notion that the reals constitute some sort of "bigger" infinity arise from one specific argument, which involves different levels of language, for example:

If every real number could be expressed in some language as some finite combination of symbols, then we could easily have a method of matching the natural numbers to these combinations of symbols. All that is required is an alphabetical style ordering of all the symbols used to define real numbers (in the same way as a, b, c … is the order for the English alphabet), and then you simply list them in the same way as you would order the words in a dictionary. And if you could list them, you can obviously attach natural numbers to every item in the list. But this would contradict the Diagonal argument, so there must be real numbers that cannot have any finite representation.

However, that argument forces the enumeration of the reals to be in a different language system (a meta-language) from the real numbers of the list, where every combination of symbols that defines a real number in that system is simply an object in that meta-language, and which has no inherent numerical value in that meta-language. Hence that meta-language has no method of generating a diagonal number from that enumeration.

In short, that argument introduces different levels of language while at the same time completely ignoring whether that introduction of different levels of language might affect the overall proof. — Preceding unsigned comment added by Jamesrmeyer (talk • contribs) 14:56, 12 January 2018 (UTC)

Different languages are irrelevant to the proof. A particular real number stays the same real number however it is represented and they can all be represented as a sequence of 0's and 1's. The construction would produce a number not in the sequence . If it was in the weird symbols sequence then the number in 0's and 1's would be in the 0's and 1's sequence. Dmcq (talk) 08:54, 15 January 2018 (UTC)

The assertion that all real numbers "can all be represented as a sequence of 0's and 1's" is patently false. On the other hand it can be asserted that every real number has an expansion in whatever base one chooses. And that irrationals have infinite expansions in every base - hence irrationals cannot have any representation as a sequence of 0's and 1's.

The point made was that the diagonal argument, of itself, does not establish that there are "more infinite sequences of ones and zeros than there are natural numbers." And as I pointed out above, any proof that there are "more infinite sequences of ones and zeros than there are natural numbers" requires an argument that specifically necessitates an introduction of different levels of language.

Hence the claim that "Different languages are irrelevant to the proof" is inherently absurd, since different levels of language are necessarily used to obtain the result that there are more real numbers than natural numbers, which is a result that is additional to the result that there cannot be an enumeration of real numbers.

I would note that the definition of a diagonal number is a definition that is stated in terms of the enumeration function, and hence that definition has to be in the same language system as that enumeration function. If one wants to imagine a method of circumventing this inconvenient fact, then one can plead a reliance on Platonist assertions that real numbers somehow "exist" independently of any definition, and by this pleading any considerations of language are ignored. When that is done, it is hardly surprising that one ends up with the conclusion that there "exist" real numbers that have no finite definition. Jamesrmeyer (talk) 13:07, 15 January 2018 (UTC)

Using base two all real numbers can be represented as a sequence of 0's and 1's, exactly the same as in base ten they can be represented as a sequence of the digits from 0 to 9. There are no real numbers which can be represented in one but not the other. Hyper languages do not magically generate more real numbers. For instance π in binary starts 11.001001000011111101101010100010001000010110100011... Dmcq (talk) 14:19, 15 January 2018 (UTC)

You don't seem to understand the difference between a representation and a definition that defines an infinite expansion (which can never be represented). The base is irrelevant - irrationals have infinite expansions in every base.

But tell you want, just write down a representation of an irrational's infinite expansion for me and then I will concede. Jamesrmeyer (talk) 19:32, 15 January 2018 (UTC)

I am sorry but it is not my job to convince you of anything. The purpose of this page is to propose improvements using WP:reliable sources. If you can find some reliable source for what you are saying then I'll look at that. Dmcq (talk) 20:42, 15 January 2018 (UTC)

It is the job of Wikipedia to convince people of what it asserts. I pointed out that while there is a claim under the Interpretation heading that claims

"the diagonal argument establishes that, although both sets are infinite, there are actually more infinite sequences of ones and zeros than there are natural numbers."

the previous section only referred to a proof regarding the presence or absence of a total enumeration. Hence there should be some evidence provided for the additional assertion that there are more elements of one set than the other. Jamesrmeyer (talk) 22:13, 15 January 2018 (UTC)

It isn't, see WP:NOT. "Wikipedia is not a soapbox or means of promotion", "Wikipedia is not a manual, guidebook, textbook, or scientific journal". It is an encyclopedia and is just supposed to report what is in reliable sources with due weight and a neutral point of view. See Cardinality for the standard definition of the size of infinite sets. Definition 3 is the basis for saying the number of reals is greater than the number of natural numbers. Dmcq (talk) 22:37, 15 January 2018 (UTC)

I'm not saying that it has to be as a manual, guidebook, textbook, or scientific journal, I'm saying that it should provide sufficient information - which means that it should direct the reader to where he can find information that supports what its articles state. I pointed out that that in a particular place that information is lacking. Your reference to the definition of cardinality is merely a definition that asserts the cardinality of A is less than B if there is an injective function, but no bijective function, from A to B. That is neither proof nor evidence that there are more elements in B, it's simply a definition. Jamesrmeyer (talk) 23:23, 15 January 2018 (UTC)

While it can be said "it's simply a definition", it should be noted that definitions are carefully chosen in mathematics to make intuitive concepts precise. I suggest reading Controversy over Cantor's theory#Cantor's argument which delves into the reasons behind the definitions of "having the same number" or "having the same cardinality" as well as the definition of "having greater cardinality." For example, this article states [note that understanding clause (2) requires that you read what comes earlier in the article]:

The concept of "having greater cardinality" can be captured by Cantor's 1895 definition: B has greater cardinality than A if (1) A is equinumerous with a subset of B, and (2) B is not equinumerous with a subset of A. Clause (1) says B is at least as large as A, which is consistent with our definition of "having the same cardinality". Clause (2) implies that the case where A and B are equinumerous with a subset of the other set is false. Since clause (2) says that A is not at least as large as B, the two clauses together say that B is larger (has greater cardinality) than A.

Using this definition on this article: T has greater cardinality than N (the set of natural numbers) since (1) N is equinumerous with the subset of T consisting of the strings that are all 0's except for a 1 in the n-th place, and (2) T is not equinumerous with a subset of N because if you assume that it is equinumerous with a subset of N, you get a contradiction using the diagonal argument. So if you accept proof by contradiction, it is not equinumerous with a subset of N. --RJGray (talk) 01:43, 16 January 2018 (UTC)

It was König who in 1905 first gave the argument that I indicated at the start of this topic - that if all elements had finite definitions, then there would be an enumeration of those elements, and concluded that that indicates that there must be elements that cannot be finitely defined. However, a more up to date reference would be preferable to referencing a German text couched in rather arcane language, and for which, as far as I am aware, there is no suitable English translation.

On the other hand, if it actually is the case that the mathematical community do not consider that method of proof acceptable, and prefers to rely instead on the acceptance of definitions that encapsulate someone's intuition and which produce the assertion that there are more real numbers than rational numbers, then surely the details of that should be incorporated into the Wiki article. If there are no objections and no-one suggests a suitable proof as an alternative, it would be worthwhile for that to be done. Jamesrmeyer (talk) 09:14, 16 January 2018 (UTC)

I don't know what contradiction you see between the view expressed in the article and König's view. They are both true. Only ℵ₀ zero-one strings have finite definitions, and therefore there must be zero-one strings that do not have such definitions, because the number of zero-one strings that exist is greater than ℵ₀. This fits together with König's view, as you've expressed it, perfectly well. --Trovatore (talk) 10:02, 16 January 2018 (UTC)

What I am saying is that if König's argument is relevant to the Wiki article, then that should be acknowledged within the Wiki article. Then people have that information and they can decide if they accept the Wiki assertions at face value as they stand, or they can access König's argument directly and make up their own minds without relying completely on Wiki. Isn't that the whole ethos underlying Wiki? Jamesrmeyer (talk) 12:40, 17 January 2018 (UTC)

It is standard in mathematics to talk like a Platonist whether you are one or not. It saves a lot of time and avoids dragging in details that are often not relevant.

I don't think König is directly relevant here. He may have been the first to make this particular argument directly (this might be an interesting thing to mention at the definable real number article). But the "more" issue is not about definability; it's just the standard description of the phenomenon at issue. --Trovatore (talk) 22:03, 17 January 2018 (UTC)

...to talk like a Platonist whether you are one or not...saves a lot of time and avoids dragging in details that are often not relevant. That must be one of the most asinine and unfounded comments I have read in a long time. Jamesrmeyer (talk) 13:23, 18 January 2018 (UTC)

Well I also agree with the statement so that is two asinine people who disagree with you. Perhaps you should reconsider. Dmcq (talk) 16:19, 18 January 2018 (UTC)

Here's a translation of that paper [1]. The intro explains why it is not often referenced. Of course there are only a countable number of computable numbers. There's even the Löwenheim–Skolem theorem about first order theories always having a countable model. That is all quite irrelevant here. Definitions are what mathematics works from. This is not about the computable reals. It is not about what you get if you start with a different set of axioms like in Constructivism (mathematics). It is not about physical reality or people's conceptions of it. It is about the usual standard mathematics which includes for instance things like the axiom of choice even if many mathematicians do work which does not assume it or assume other variants. Dmcq (talk) 10:42, 16 January 2018 (UTC)

I'm not sure what your point is here. You say ""Definitions are what mathematics works from", but I'm not sure what definition you think applies to using the word "more" in relation to infinite sets. The definitions in the Wiki article on Cardinality make no mention of "more", so the assertion in the Wiki Cantor's diagonal argument article "there are actually more infinite sequences of ones and zeros than there are natural numbers" is not substantiated by those definitions of cardinality. Jamesrmeyer (talk) 12:22, 17 January 2018 (UTC)

In the article on Cardinality I pointed to in definition 3 it says 'A has cardinality strictly less than the cardinality of B if there is an injective function, but no bijective function, from A to B.' 'More' is just a way of referring to this relation swapped around, |A| < |B| is the same as |B| > |A|. Dmcq (talk) 18:59, 17 January 2018 (UTC)

Perhaps that is being "less than" logically precise. The terms "less than" and "greater than" can be used to indicate relationships that are in some way ordered but not necessarily by specific numerical quantities, whereas "more than" does indicate such a relationship. Jamesrmeyer (talk) 13:11, 18 January 2018 (UTC)

I don't know where you get that from or why you think it is relevant to anything here. Dmcq (talk) 16:19, 18 January 2018 (UTC)

Sets of differing quantities of elements imply an ordered relationship. However it is an elementary logical error to assume that the converse is true (A implies B does not imply that B implies A).

As an example, consider vectors in 3D orthogonal space defined by 3 axes. One can have an ordered relationship based on whether a vector is defined by 1, 2 or 3 axes. But that ordered relationship is completely independent of the scalar length of the vectors; the fact that the vectors have different lengths does not imply that ordered relationship. And such an ordered relationship can apply for any number of dimensions.

For finite sets it holds that the term for cardinality concurs with the term for number of elements in the set - the terms are exactly equivalent. But there is no logical inference from that to the assertion that the terms are precisely equivalent for infinite sets.

One could equally well describe the cardinality of infinite sets by natural numbers 1, 2, 3, ... where 1 is the cardinality of the set of natural numbers, and the complete information of whether there can be an injection or bijection between any two sets is determined by cardinality described by this terminology. Furthermore using such terms does not in any way diminish the amount of information regarding the quantity of elements in such sets. Jamesrmeyer (talk) 10:42, 20 January 2018 (UTC)

See Bijection and Injective function about those words. The definition of cardinality and less than etc is a formalization of the intuitive notion extended to infinite sets. Nobody has found any problem with the formal theory. One can always set up other formal theories but making them useful and reflecting intuitive notions is a bit more difficult. I guess you are trying to say something about the intuitive use or how it was formalized but I am unable to make out anything about what it is. Dmcq (talk) 11:17, 20 January 2018 (UTC)

References

du Bois-Raymond and Cantor's diagonal argument

An edit saying du Bois-Raymond invented the argument earlier was reverted, correctly I believe. However there probably should be something here about the attribution and why it is wrong and there is the chance that Cantor was inspired by d Bois-Raymond's proof as he had an interest in infinity and infinitesimals. See note 1 page 187 in Simmons, Keith (1993). Universality and the Liar: An Essay on Truth and the Diagonal Argument. CUP. p. 187.. Dmcq (talk) 18:02, 9 January 2018 (UTC)

The line removed was

Historically, the diagonal argument first appeared in the work of Paul du Bois-Reymond in 1875.

Du Bois-Reymond, Paul (1875), "Über asymptotische Werte, infinitäre Approximationen und infinitäre Auflösungen von Gleichungen", Mathematische Annalen, 8 (3): 363–414, doi:10.1007/bf01443187

Dmcq (talk) 18:09, 9 January 2018 (UTC)

I will restore the reference to du Bois-Reymond together with a book on his work by G. H. Hardy, where the diagonal argument is explained.nikita (talk) 17:37, 25 July 2020 (UTC)

What's the problem with this disproof?

Hi, I'm not a mathematician, but I've always thought this theorem was false, and I thought somebody might have an intelligent opinion. Here is an explanation of why:

1. Define an enumeration A of all expressions in language (e.g. by enumerating sequences of letters and symbols).

2. Define an enumeration B of all infinite sequences of binary digits, as those items in A which define infinite sequences of binary digits. This will be the enumeration used for Cantor's Theorem.

3. Cantor's additional sequence must be within A, because it is written in language. For example, A must contain "The additional sequence that will be defined in terms of B for Cantor's Theorem."

4. If this additional sequence defines an infinite sequence of binary digits, then it will be an element of B with an index n. However, because of the nature of its definition, its nth digit would be its own complement.

5. Therefore, the additional sequence does not define an infinite sequence of binary digits. The sequence Cantor proposes to generate does not exist in this case.

Furthermore, there are multiple ways of writing a description of a sequence in language, implying a 1-to-1 mapping of describable sequences to natural numbers, but not of natural numbers to describable sequences. There are more natural numbers than describable sequences.

I mentioned this to my high school calculus teacher in class around 2001, and this point is where the argument stalled. Xloem (talk) 21:03, 10 October 2018 (UTC)

Hi Xloem. Per the talk page guidelines, the article talk pages are for discussing improvements to the article. If you have general questions about the subject matter, you can ask at the math reference desk. Also, this talk page has an arguments subpage. --Trovatore (talk) 21:11, 10 October 2018 (UTC)

Pinging OP: Xloem. --CiaPan (talk) 07:29, 11 October 2018 (UTC)

A quickie though, any more should go to the arguments page, in step one you are at best describing the computable numbers. And the argument can be used to show one can't list out the computable numbers even though they are countable.This is the Halting problem. Dmcq (talk) 10:58, 11 October 2018 (UTC)

Cantor's diagonal argument, float to integer 1-to-1 correspondence, proving the Continuum Hypothesis

Moved to talk:Cantor's diagonal argument/Arguments#Cantor's diagonal argument, float to integer 1-to-1 correspondence, proving the Continuum Hypothesis. --Trovatore (talk) 19:53, 1 December 2018 (UTC)

Automatic archiving

Apparently, there's some confusion with automatic archiving of this talk page. As far as I can see, there are the following archive pages:

• Talk:Cantor's diagonal argument/Archive 1
• Talk:Cantor's diagonal argument/Archive 2 (/Archives/ 1)
• Talk:Cantor's diagonal argument/Archive 3 (/Archives/ 2, /Archives/Archive 2)

Only the first archive page in that list is referenced by the archive index. Can someone more knowledgable than me please fix this? In the mean time I have managed to only make the mess worse by moving Talk:Cantor's diagonal argument/Archives/ 2 to Talk:Cantor's diagonal argument/Archives/Archive 2. Sorry about that. – Tea2min (talk) 08:28, 2 December 2018 (UTC)

Hello, Tea2min! Archive pages' history shows they are created and managed by User:ClueBot III. As the bot's userpage informs, it is operated by User:Cobi. So maybe Cobi could explain naming discrepancy, and help in fixing it? --CiaPan (talk) 15:47, 6 April 2019 (UTC)

Hi, Tea2min, Cobi is not very active recently, as can be seen from the contributions list – just six edits this year on five distinct days, most recently on 21 March 2019. So I think we should not expect a prompt reply. --CiaPan (talk) 07:38, 20 May 2019 (UTC)

no reals

moved to talk:Cantor's diagonal argument/Arguments#no reals --Trovatore (talk) 19:04, 6 April 2019 (UTC)

The proof is (EDIT: NOT) incomplete

Moved to arguments page. --Trovatore (talk) 06:34, 16 July 2021 (UTC)