Talk:Cardinality of the continuum/Archive 2

Archive 1

Archive 2

(Counter-) intuitive argument

We temporarily stopped a discussion about the section "Intuitive argument" to discuss about the introduction. The latter discussion seems to have reached a reasonable degree of consensus, and a new introduction has been published. Feel free to continue above the discussion about the new introduction, if you like, but I think that now the "Intuitive argument" deserves more attention.

Some contributors previously agreed that the "Intuitive argument" provides an absolutely counter-intuitive proof for |R| = 2^|N|. This prooof is based on cardinal arithmetics, which is in turn based on Cantor's distinction between |N| and |R|. That distinction is used as an axiom, thus the proof is, in my opinion, of little use.

Paolo.dL (talk) 12:15, 5 June 2011 (UTC)

Proposal 1

There's probably no intuitive proof that |R| > |N|, but I propose to replace the (alleged) "intuitive argument" with an "informal introduction" such as this:

Although there are infinitely many natural numbers, it can be shown that the number of digits in any natural number is finite. 10 distinct digits are available if numbers are written using the standard decimal notation. Hence, there are 10n possible ways to compose a natural number with n digits. .... [Question: how comes then that there are infinitely many natural numbers? Can someone answer this without using cardinal arithmetic?] ... On the contrary, every real number has an infinite number of digits in its decimal expansion. For example, 1/2 = 0.50000... 1/3 = 0.33333... $\pi$ = 3.14159.... Note that this is true even when the expansion repeats as in the first two examples. In any given case, the number of digits is countable since they can be put into a one-to-one correspondence with the set of natural numbers $\mathbb {N}$. This fact makes it sensible to talk about (for example) the first, the one-hundredth, or the millionth digit of $\pi$. Since the natural numbers have cardinality $\aleph _{0},$ each real number has $\aleph _{0}$ digits in its expansion. Hence, there are $10^{\aleph _{0}}$ possible ways to compose the decimal expansion of a real number. This implies that there are $10^{\aleph _{0}}$ real numbers in the interval from 0 to 0.999... (unit interval). Any real number is composed of an integer part and a decimal expansion. Since the integer part is a natural number, and there are $\aleph _{0}$ natural numbers, then there are $\aleph _{0}\cdot 10^{\aleph _{0}}$ real numbers. Similarly, if the real numbers are represented using a binary notation, such as that used by computers, we conclude that there are $\aleph _{0}\cdot 2^{\aleph _{0}}$ real numbers. Since there is a one-to-one correspondence between the real numbers represented in the commonly used decimal numeral system, and those in other numeral systems, such as the binary one, mathematicians agree that $\aleph _{0}\cdot 10^{\aleph _{0}}=\aleph _{0}\cdot 2^{\aleph _{0}}$ Cantor showed that it is also possible to define a one-to-one correspondence between the unit interval and the whole set of real numbers. Thus $\aleph _{0}\cdot 2^{\aleph _{0}}=2^{\aleph _{0}}$ However, it is impossible to define such a correspondence between the natural numbers and the real numbers. Thus $\aleph _{0}<2^{\aleph _{0}}.$

I need your help to refine this text. Paolo.dL (talk) 12:15, 5 June 2011 (UTC)

Your bracketed question seems to be answered by the following sentence: most real numbers can't be expressed as terminating decimals. CRGreathouse (t | c) 00:23, 6 June 2011 (UTC)

This does work well as an informal introduction but only if the reader is familiar with the concept of $\aleph _{0}$ and exponentiation. It's easy to conclude that ${\mathfrak {c}}=|2^{\mathbb {N} }|$, but that shouldn't intuitively imply that $|\mathbb {N} |\neq |2^{\mathbb {N} }|$. Taking the argument at that point and saying "Cantor said so" may be very disappointing to the reader.

I put this article on my watchlist after being inspired by a conversation. I met a local math professor at a party and he described Cantor's Diagonal Argument, verbally without writing anything. To me, that was intuitive. But I already had enough background in set theory to grasp the importance of finding an element not already in the set. (Not enough, however, to have heard of aleph-naught. Intro to abstract algebra at my college stopped at the countability of rational numbers.)

Perhaps the first section can be a gentler introduction to the Diagonal Argument, or an exposition of its significance. As a bonus, the Diagonal Argument applied to scalars intuitively proves uncountability of the interval [0,1) ⊂ ℝ, which you can extend to the reals by mapping the "short" interval to the "long" one.

Or, to construct another more complete "intuitive" argument:

Consider a possibly-infinite, increasing sequence of natural numbers a_i. Such sequences clearly correspond to members of 2^ℕ since each number is either included or excluded.

Each sequence corresponds to a real number 0 ≤ r ≤ 1,

$r=\sum _{i}2^{-a_{i}}$

Therefore, |[0,1]| = |2^ℕ|. [Hmm, that equality doesn't really follow. All I can really say is |[0,1]| ≥ |2^ℕ|. Is there an "intuitive" way to deal with the asinine representations using trailing ones? Is it more appropriate to just ignore them?]

However, if we try to convert a sequence to a natural number using $n=\sum _{i}2^{a_{i}}$ (a mapping which skips no numbers), the result is infinite for infinite sequences. Such a sum has no well-defined magnitude, cannot obey the familiar rules of arithmetic and hence cannot be a natural number. It seems difficult to map elements of 2^ℕ → ℕ. However, each finite sequence does terminate, therefore does correspond to a natural number. Clearly there are "more" infinite sequences than finite sequences. Cantor's diagonal argument reveals how the endlessness of the sequences implies uncountability of the set of sequences.

Hmmmmmmmmm. I think we can each write an argument which is most intuitive to himself. Here's hoping for more advice from novice readers, and consensus… Potatoswatter (talk) 07:26, 6 June 2011 (UTC)

My text was a gentle introduction, not a proof or formal argument. It says something that the current section surprisingly omits: bijections are needed to prove equinumerosity. It states a lot of "facts" that mathematicians take for granted and most readers assume to be impossible. That's a huge step forward relative to what is written in the current "intuitive argument". How can you arouse the curiosity of a reader more than by saying that the natural numbers cannot have an infinite number of digits, while the real numbers can? That's amazing, in my opinion.

The problem is that I cannot easily explain why |N| is not a natural number. Natural numbers are countable, so if they are infinitely many, then "infinitely many" should be a natural number, which means that 2ⁿ = |N|... however, this is incompatible with the fact that the number of digits n is finite, isn't it? Explaining the reason why a natural number cannot have an infinite number of digits is crucial. I guess this is what you need to understand if you want to understand the concept of countability.

In other words, is there an infinite powerset of a finite set? If yes, how's that possible? Paolo.dL (talk) 10:26, 6 June 2011 (UTC)

No, there is no infinite powerset of a finite set. I don't really follow your reasoning as to why there should be. --Trovatore (talk) 10:59, 6 June 2011 (UTC)

It's a matter of logic symmetry between these two statements (the first of which is stated in the article):

1. There are |N| digits in the decimal expansion of a real. Hence |R| = 10^|N|
2. There are n digits in any natural number. Hence |N| = 10ⁿ

Also,

1. $\beth _{2}=2^{\beth _{1}}$
2. $\beth _{1}=2^{\beth _{0}}$
3. $\beth _{0}=|\mathbb {N} |=2^{???}$

There must be a fault in this logic, but I can't find it. Where's the mistake? Notice that I have not studied Cantor's diagonal argument yet. However, this is the reason why my contribution may be useful in this discussion. I may be able to represent the readers who need a gentle introduction.

Isn't it natural for a beginner to look for symmetry? "Intuition" is based on easy logic rules, that can be applied quickly, without effort. Symmetry is one of them. Of course, as I wrote above, intuition may lead to wrong conclusions. This is why I insisted that "the truth is counterintuitive" in this case, and "intuitive argument" is a misleading title for a section in this article. Paolo.dL (talk) 13:19, 6 June 2011 (UTC)

To Paolo.dL: "There are n digits in any natural number." is only true if it is interpreted to mean

$\forall k\in \mathbb {N} \,\exists n\in \mathbb {N} \,(k{\text{ can be written using }}n{\text{ decimal digits }})\,.$

To infer the false sentence "|N| = 10ⁿ", you would need the stronger (and false) statement

$\exists n\in \mathbb {N} \,\forall k\in \mathbb {N} \,(k{\text{ can be written using }}n{\text{ decimal digits }})\,.$

Your argument concerning $\beth _{\,-1}$ is a non sequitur (logic).

You should study Cantor's diagonal argument before you presume to write for this article! JRSpriggs (talk) 00:46, 8 June 2011 (UTC)

I wrote several parts of the new lead in this article, and you accepted most of my edits. I started this discussion because a section called "intuitive argument" is in my opinion one of the worst of its kind I have ever read on Wikipedia. I believe that the new text I proposed above is much better than the current one. I did this without knowing the details of Cantor's diagonal argument. I don't think that this section should introduce Cantor's diagonal argument. I believe it should arouse the curiosity of the reader by exploring propositions/results with simple terminology. As for the proof(s), it sufficies to explain that the existance/non-existance of a bijiection entails equinumerousity/different cardinality. A recent edit in my opinion made this section even worst, as the editor replaced the second half of the section with an argument based on counter-intuitive transfinite arithmetic, which in turn is based on what the section is supposed to introduce and readers are not supposed to know. I explained my rationale. If you disagree, just explain why. For sure, I am not here to impose my opinion, nor to gain a personal advantage, and that's what I expect from everybody else. Paolo.dL (talk) 20:58, 6 June 2011 (UTC)

Generally, your suggested sections are inaccurate and (to me) less intuitive than what is presently in the article. I'm sure there are reasonable disagreements to be had about the level of accuracy required in these sections: I would say that, apart from "(ignoring certain complications)"-type comments, the material should be strictly accurate, simply lacking details. For example, in an overview I would not feel the need to mention the minor difficulty posed by repeating decimals. CRGreathouse (t | c) 04:45, 7 June 2011 (UTC)

CRGreathouse, in the previous section, you explained the reason why one of my statements was misleading. That resulted in an accurate and easily readable short summary of an imporant section of the article, which I inserted in the lead. That summary was missing in the old version of the lead.

On the contrary, the generic comments in your latest contribution are not constructive at all. We are discussing here a section called "intuitive argument". Later, I am going to explain why in my opinion the current text is absolutely useless to beginners. The fact that that text is to you more intuitive than my version is absolutely meaningless, as that section should use a language accessible to most readers. In sum, I wish we could work together, instead of against each other. If you and JRSpriggs wish to fight, I'll quit the discussion and you'll have to find another enemy.

Paolo.dL (talk) 13:03, 7 June 2011 (UTC)

Paolo — Not knowing the diagonal argument is definitely a problem. Having a novice perspective here is nice, except that you have some foreknowledge (and preconceived notions) about the transfinite cardinals. Note the appearance of $2^{\aleph _{0}}$ in your proposed text. The diagonal argument uses (countably) infinite sequences to show that uncountability exists. It works quite simply, without transfinite symbols. Without this knowledge, how can you dictate this article's relationship to it?

Curiosity is good, but the article (and this discussion) need to stay on topic. Beth-2 has nothing to do with the cardinality of the continuum. (See beth number for that topic.) "Exploring propositions/results with simple terminology" can quickly get confusing for a reader who expects an explanation (as good readers do), not tangents. "The existance/non-existance of a bijiection entails equinumerousity/different cardinality" is misleading… the non-existence of a bijection is hard to prove. Actually, I think this gets right to why the diagonal argument should be introduced earlier: by omitting it, the current text invites the reader to invent a simpler, false argument. Or to blindly trust in "Cantor proved," which isn't very educational.

As for constructiveness vs fighting… the important thing is that we all suggest enough that some subset meets the union of our goal criteria. Without that, there's no chance of success! We simply need both criticism and suggestions to make headway. In other words, stay as organized as we were for the last round. Potatoswatter (talk) 16:54, 7 June 2011 (UTC)

The "intuitive argument" is based on a simple idea: the decimal expansion of a real number has $\aleph _{0}$ digits. Thus, $2^{\aleph _{0}}$ possible sequences of digits. In my opinion, this is the only good part of the "intuitive argument". And this is what catched my attention. Even JRSpriggs started from this idea in his attempt to show that |R| = $2^{\aleph _{0}}$ (see second part of the section). But again, I don't want to impose my opinion.

There's something however on which you, Potatoswatter, might agree with me: the current text is useless to beginners, and is supposed to be a gentle introduction for beginners. On that, there's no consensus with the other contributors to this discussion.

What's the problem in using the symbol of a cardinal number? This article is about a cardinal number. The definition of $\aleph _{0}$ is given in the lead. Paolo.dL (talk) 20:42, 7 June 2011 (UTC)

Yes, aleph-naught is introduced in the lead, but "the decimal expansion of a real number has $\aleph _{0}$ digits" is far from intuitive to someone who only just heard of it. And how are they supposed to intuitively guess that $2^{\aleph _{0}}\neq \aleph _{0}$ even though $2\cdot \aleph _{0}=\aleph _{0}$?

I am attempting to intuitively reason that |ℝ| = |2^ℕ| by counting, without transfinite symbols, which is how a derivation must be done before the introduction of transfinites — as you have admitted. Then, I go on to make a futile attempt at a bijective map between |2^ℕ| and |ℕ|, hinting at the outcome of the diagonal argument. That goes much further than the existing text or your proposal. Potatoswatter (talk) 03:52, 8 June 2011 (UTC)

Paolo, I have no wish to fight either, but I also don't see your version as progress. I look forward to your critique; hopefully it will be more useful to improving this article. CRGreathouse (t | c) 04:06, 8 June 2011 (UTC)

Thank you. May be you are right. I'll slow down a little, but as soon as I can, I'll try to explain more specifically why in my opinion the second half of the current version is useless to the readers who are supposed to need a gentle introduction. In the meantime, I'll study Potatoswatter's proposal below. Most of his ideas for the new lead were great... Paolo.dL (talk) 18:17, 8 June 2011 (UTC)

Proposal 2

I think that this does a better job at piquing the reader's interest, informing about the elementary methods used with infinite sets, and preparing to understand the significance of the diagonal argument.

Consider an increasing sequence of natural numbers ai. This sequence may be finite like { 12, 42, 987 } or infinite like { 2, 4, 6, … } . Each such sequence corresponds to a subset of the natural numbers: every number is either contained or not. There is also only one way to write each subset as a sequence. The set of sequences thus corresponds 1-to-1 to the powerset of the naturals, written $2^{\mathbb {N} }$, where 2 represents the binary decision of including or excluding each number. Each sequence can be transformed to a real number 0 ≤ r ≤ 2, $r=\sum _{i}2^{-a_{i}}$ Some pairs of sequences map to the same number,[1] therefore the cardinality of this interval is bounded: $\left|2^{\mathbb {N} }\right|\leq \left|[0,2]\right|\leq 2\cdot \left|2^{\mathbb {N} }\right|$ However, we can easily map the powerset to [0,1] or [0,4] by multiplying the summation formula by a constant factor. Thus, multiplying or dividing an infinite quantity by 2 has no effect. Applying this principle yields an exact identity: $\left|[0,1]\right|=\left|2^{\mathbb {N} }\right|$ The entire number line is infinitely long, of course. A mapping such as x → 1⁄x rearranges our finite interval to an infinite one — for this informal line of reasoning, we can ignore the resulting hole. To recap, the distinct sets of natural numbers are equinumerous with the distinct increasing series of natural numbers. The increasing series are equinumerous with the real numbers in a finite interval. The real numbers in a finite interval are equinumerous with all the real numbers. The final result is ${\mathfrak {c}}=\left|\mathbb {R} \right|=\left|2^{\mathbb {N} }\right|$ However, if we try to convert a sequence to a natural number using $n=\sum _{i}2^{a_{i}}$, the result is infinite for infinite sequences. Such a sum never stops growing and cannot equal any natural number. It seems difficult to map elements $2^{\mathbb {N} }\to \mathbb {N}$.[2] However, each finite sequence does terminate, therefore does correspond to a natural number. Clearly there are "more" infinite sequences than finite sequences. Cantor's diagonal argument reveals how the endlessness of each sequence implies uncountability for the set of sequences. 1. ^ Each finite subset pairs with an infinite subset. One example is { 0 } with { 1, 2, 3 … } ; these both map to 1. See 0.999... . 2. ^ Using Gödel numbering, much better maps can be formed. For example, we can map natural numbers to English text. The natural numbers thus map to all sets that can be described using natural language, without going on forever. Even this is too small, because infinite English text is inevitably required to describe an infinite subset with no underlying pattern. The real number line starts to seem unnaturally large!

Potatoswatter (talk) 05:29, 8 June 2011 (UTC)

I think that "cannot obey the familiar rules of arithmetic" is likely to cause greatly increased confusion in the less-mathematically-literate segment of readers. CRGreathouse (t | c) 18:51, 8 June 2011 (UTC)

Yes, I got tired of fighting with formatting and there are still a few things wrong. I've fixed that and several glaring omissions. Needs more polish and more suggestions! Potatoswatter (talk) 19:40, 8 June 2011 (UTC)

I am sorry, I know that writing this proposal was time consuming. However, I think it is too complex, and more importantly it assumes too much. For instance, you initially assume that the set you define starting from the power set of the natural numbers is equal to [0,2]. Then you assume that [0,2]*2 = [0,4]. That's the same as assuming that a segment with length 4 has the same number of points as a segment with length 2, or that |R| = |R|*2. The problem is that transfinite cardinal arithmetics is based on Cantor's results, not viceversa. This is exactly the reason why the current text is useless, in my opinion. Wouldn't it be more honest to delete the "intuitive argument" and let the readers read the rest of the article without conning them into believing that there's something intuitive in Cantor's results? Shouldn't we lead them to believe that the only way to understand that

${\mathfrak {c}}=|(a,b)|=|\mathbb {R} |=2^{\aleph _{0}}>\aleph _{0}$

is to study counterintuitive and complex arguments such as Cantor's diagonal argument?

Paolo.dL (talk) 21:43, 23 June 2011 (UTC)

About the current version of section "Intuitive argument"

The section Cardinality of the continuum#Intuitive argument as I left it last contains only arithmetic operations which are true for both finite and infinite numbers, excepting only

$\aleph _{0}+4\cdot \aleph _{0}=\aleph _{0}\,.$

That is the sense in which it is intuitive for people who are only used to finitary mathematics. The only other concession to the infinite lies in the fact that one must use less-or-equal in some cases where strictly-less would be justified for finite numbers. Thus I fail to understand how you can say that this section is counter-intuitive. JRSpriggs (talk) 06:02, 24 June 2011 (UTC)

You seem to confuse what is easy to grasp with what you are familiar with. Your two "exceptions" are more than enough to confuse a non-mathematician. Let me try to convince you about the meaning of the term intuitive. Intuition is a "gut feeling", the ability to understand "without much reflection". So, intuitive means "easy to grasp". An argument is intuitive if it can be "readily learned or understood". There's wide consensus about the fact that Cantor's idea of "different infinites" is counter-intuitive for non-mathematicians. I doubt that Cantor' results were based on intuition (i.e. obtained without much reflection). Anyway, even if his study or part of it were based on his own intuition, that would not make his results intuitive for others. There's a huge difference between the intuition of a very well-educated mind such as Cantor's or yours, and the intuition of a non-mathematician. The latter is a wild beast. You can't expect it to follow the correct path. Your mind is so used to follow that path, that you forgot what's the easiest (i.e. most intuitive) path. In this case, the easiest path is incorrect. I wish we could agree at least about this.

Here's an example. In Cardinality of the continuum#Intuitive argument, the equation $\aleph _{0}+4\cdot \aleph _{0}=\aleph _{0}$ might be easier to grasp than you think. As the title of the section is "intuitive argument", I am sure that even readers which know nothing about set theory will be encouraged to try and understand that formula. They may digest it based on the incorrect but intuitive assumption that nothing can be larger than infinite, hence there's only one infinite ($\aleph _{0}$). This is the easiest path for a non-educated mind. (I guess you can't see it because you are too familiar with the correct path, so you should trust me in this case). However, this incorrect path would lead to the incorrect conclusion that

$\aleph _{0}=2^{\aleph _{0}}$

In other words, there's a non-negligible gap in the alleged "intuitive argument": Cantor's diagonal argument is taken for granted.

Let's imagine that the reader is not clever enough to "grasp" the equation $\aleph _{0}+4\cdot \aleph _{0}=\aleph _{0}$. That would make that equation counter-intuitive. So the entire section, being based on that statement, would be perceived as counter-intuitive. I conclude that there's only one way to accept that section as intuitive: being familiar with the correct path, i.e. being a reader who does not need to read that section to understand the article.

By the way, there's also another logical gap in the current text. You did not explain the reason why the number of all possible combinations of decimal digits in a decimal expansion is ${\mathfrak {c}}\leq \aleph _{0}\cdot 10^{\aleph _{0}}$. I can understand that there are 10ⁿ ways to fill n decimal positions, but in my opinion this is not easy to guess for a beginner. Also, I can't understand why you used "≤" (I would use "="), and that is a major source of doubts.

That's why the current version of section "intuitive argument" is totally useless, and far from intuitive for those who are supposed to read it. Paolo.dL (talk) 11:10, 24 June 2011 (UTC)

I suppose that someone trying to understand this article should first understand aleph null well enough to see the truth of

$\aleph _{0}+4\cdot \aleph _{0}=\aleph _{0}\,.$

As for why one does not immediately get equality in

${\mathfrak {c}}\leq \aleph _{0}\cdot 10^{\aleph _{0}}\,,$

that is because some real numbers are represented by two decimal expansions rather than one. Of course, a little additional effort shows that this is not a problem. But why address that issue when it is irrelevant to the inequality I was trying to show in that calculation?

It would probably be no great loss to take out the section. However, what bothers me about your whole approach to this article is that you seem to be trying to use ignorance to justify degrading the article. You want to take out true and illuminating statements because they conflict with the irrational prejudices of some readers. You want to put in false or misleading statements because they are allegedly intuitive and only a little bit false (like a little bit pregnant?). This approach is counter-productive and unethical. JRSpriggs (talk) 23:30, 24 June 2011 (UTC)

How do you dare to say that I "want to put in false or misleading statements"? This is an unacceptable judgement about my intentions, which by the way reveals how biased is your opinion. I am not going to waste my time with you anymore. In the future, if you removed your blinders, you might be able to interact with someone else. Paolo.dL (talk) 11:42, 25 June 2011 (UTC)

I don't question your motives, but you do seem to add questionable or misleading material. CRGreathouse (t | c) 16:38, 25 June 2011 (UTC)

@Paolo.dL You are pretty familiar with the "good faith" aspect of Wikipedia, yes? Show your critics good faith (i.e. learn from them and have fun!) and I'm sure you will get plenty of good faith and positive feedback in return. I always remember that for every topic, there is going to be an expert that knows more than I do, and am prepared to defer. The last thing you want to do is get indignant  :) Rschwieb (talk) 20:49, 5 July 2011 (UTC)

He moved it to User talk:Rschwieb, if anyone was robbed of their chance to comment. Rschwieb (talk) 19:06, 6 July 2011 (UTC)

I am not answering personal messages here. Please use this talk page to discuss about the article. Paolo.dL (talk) 16:05, 6 July 2011 (UTC)

Moving the "intuitive argument" section

Stepping back and reading the article afresh, the "intuitive argument" section is definitely misplaced. Our first goal should be to convey the basic information about the topic at hand, and not leap directly into a quasiproof of one specific fact mentioned in the article. (It is as nonsensical as an article beginning "Cardinality of a set: 1.Intuitive argument".) The "Properties" should immediately follow the introduction, followed by the examples and "Continuum hypothesis" sections, in some order. Beth cardinals are slightly peripheral so that section goes later.

I concur with JRSpriggs that taking out the intuition section would be no loss, and I vote that it be moved, at the very least. It is both misplaced and unnecessary, but that said, I think the content in that section would make a fine external or internal link labeled "Alternative explanation for c=2^aleph_0". Rschwieb (talk) 15:35, 8 July 2011 (UTC)

It's entirely possible that no harm would be done to the article by removing the section in its present state. But I strongly support having material directed toward nonexpert readers, probably in its own section.

Most readers would not be able to follow a typical rigorous "properties" section. So to them, that content may as well not be there. In that light I can see the value of that placement of the section: all they can read is the opening and that section.

If there's a serious proposal to rework that framework, perhaps to make the other sections each open with a basic (readable by high school students, say) paragraph then I'd consider it. But if the proposal is to make the article inaccessible to lay readers, I'm not in favor.

CRGreathouse (t | c) 15:53, 8 July 2011 (UTC)

Good, because it was far from my intention to make the article inaccessible. Other introductory material motivated by the properties mentioned would be excellent, but not one person's extensive thought process about one specific fact. Highschool readers will definitely not understand carrdinal arithmetic in the section we are talking about, so I gather that you meant to say you want a basic section in that location, but you do not like the current one. Rschwieb (talk) 16:08, 8 July 2011 (UTC)

I agree that the intuitive argument section is misplaced. To immediately jump into heuristics after the introduction is akin to showing someone a single semi-random tree rather than an overview of the forest. Furthermore, to a non-expert reader an intuitive proof is just as difficult to follow as a rigorous proof.ActuariallySound (talk) 16:16, 8 July 2011 (UTC)

I'm curious to see what you have in mind, Rschwieb. CRGreathouse (t | c) 18:10, 8 July 2011 (UTC)

First we must decide: What ideas in this article are worth explaining to a layperson? My cadidates are: |R| is uncountable; and every (nondegenerate) interval has the same cardinality. I don't think that showing 2^aleph_0=c is a priority.

IMHO, an explanation of why |R| is uncountable should be based on Cantor's diagonal argument: why avoid such a beautiful piece of mathematics? I have found it rather easy to explain to laypeople, I might add. The problem with including this point is that it belongs on the CDA page. The thing about intervals definitely belong on this page, and explaining it simply is feasible. The field is open for more candidates too, but please keep in mind it is neither possible nor necessary to explain *everything* in layperson terms. Rschwieb (talk) 18:52, 8 July 2011 (UTC)

What ideas... hmm... I think that |R| ≠ |Z|, that |[a, b]| = |[c, d]| for a < b, c < d, and that |R| = |R^n|. CRGreathouse (t | c) 18:58, 8 July 2011 (UTC)

It would be pretty easy for us to explain the first two points you mentioned follow from the ones I mentioned. That |R|=|R^n| is a good one, but it kind of seems like that belongs on the cardinal arithmetic page... Rschwieb (talk) 19:13, 8 July 2011 (UTC)

I dunno, |R^n| has the cardinality of the continuum too, why exclude it? But it's not critical to include, perhaps -- though I think it's important, more so than the intervals. CRGreathouse (t | c) 02:45, 9 July 2011 (UTC)

If you have an idea for a concise proof it would be a good addition. For the interval thing, there is a very nice picture of a function from (a,b) onto R, which I think laypeople will like. Rschwieb (talk) 13:26, 9 July 2011 (UTC)

I moved section "intuitive argument" and renamed it to "Alternative explanation for c=2^aleph_0", as suggested by Rschwieb (see also the previous sections in this talk page). I appreciate your attempt to explain the contents of this article to beginners. Please consider that, in my opinion, the article Cantor's diagonal argument is not yet accessible to beginners. As I explained in the relevant talk page, at least one crucial logical step is taken for granted. So, if you are able to explain Cantor's diagonal argument to beginners, you might also want to try and fill the gaps in the main article. Paolo.dL (talk) 18:56, 9 July 2011 (UTC)

Recent reverts about cardinality of real numbers with non-unique binary representation

I have not checked this completely, I'm just arguing off the top of my head. As far as I know, the only ambiguity arising as expressing numbers as binary expansions is that 0.1=1.0 and 0.01=0.10 . At least, I think this is the case, analogously to 0.9=1.0 in decimal. If this is true that the only real numbers without unique representation are the ones terminating in 1, isn't it obvious only countably many numbers suffer this problem? It would be equivalent to counting the finite sequence of digits that come before the infinite repeating tail, and there are only countably many such finite sequences. Addendum: Rereading the reverts, maybe I have misunderstood what the problem really is. Maybe the mistake, meriting removal, is at a juncture other than this. If so, nevermind! Rschwieb (talk) 18:45, 2 February 2012 (UTC)

I quite agree with your interpretation. I don't understand the opposing view at all. — Arthur Rubin (talk) 18:52, 2 February 2012 (UTC)

I think the problem is at another juncture; Isn't it obvious that the interleaving function is well defined when only one of the inputs does not terminate in 1? Interleaving a sequence ending in 1 with one that does not would result in one that does not. Rschwieb (talk) 18:53, 2 February 2012 (UTC)

That's worse, I'm afraid. If x and y both have nonunique expansions (x₀, x₁; y₀, y₁ respectively, then all 4 possibilities are different reals:

$\left(x_{0},y_{0}\right),\left(x_{0},y_{1}\right),\left(x_{1},y_{0}\right),\left(x_{1},y_{1}\right)$

although the first and last also have different representations as reals. If only one is non-unique, then there are two possible results, both unique. — Arthur Rubin (talk) 20:09, 2 February 2012 (UTC)

Oh, it is supposed to have domain and range with real numbers. In that case, then the map works, it's just that the statement about the countability of nonuniquely expressable reals is irrelevant. That particular blurb seems to have appeared here, contributed by a user who has a history of adding not-quite-correct and questionably relevant statements to this article. Rschwieb (talk) 20:35, 2 February 2012 (UTC)

As the person that made the original revert perhaps i should clarify. The problem is that the interleaving function is not a surjection. Consider the reverse of the interleaving function with (in decimal) $z_{1}=0.25{\overline {89}}$ and $z_{2}=0.26{\overline {80}}$, both of these will map to the same $(0.2{\overline {8}},0.6)$. Since it's not a bijection it doesn't say that the cardinality of $\mathbb {R}$ and $\mathbb {R} ^{2}$ is the same. The placement in the article makes it seem as if this function was somehow an argument for their cardinalities being equal, which is why it doesn't belong there.The statement that it is only a countable number of points that should be removed is false because the function is not surjective for an uncountable number of points in the unit square. For all $(x,y)$ where y is non-unique this problem exists (for example $(0.2{\overline {8}},0.6)$ and that set is uncountable.

I'll concede the point if someone can tell me where $z_{1}=0.00{\overline {10}}\in X$ and $z_{2}=0.1{\overline {0}}\in X$ map to in $X\times X$?

Nevermind, you're right the map obviously isn't onto X! I was so distracted by the task of removing the irrelevant analysis of where the interleaving map was defined and didn't stop to question that statement. Any more surgery on the domain and range of this function is will make the interleaving example useless, so perhaps it is best to leave it out. Rschwieb (talk) 14:34, 4 February 2012 (UTC)

Proofs

(Per request on my talk page) I have added proofs for some of the examples listed because I believe that it's helpful for the reader's understanding. The list of examples itself is justified for the same reason. DavidLeighEllis (talk) 01:43, 26 April 2014 (UTC)

Well, what I really wanted to know was, how far are you planning to take this? Are you planning on adding more examples/more proofs? How many, how detailed? Do you have a grand vision, and could you let us in on it? --Trovatore (talk) 01:48, 26 April 2014 (UTC)

I'm pretty much done with this article, actually. DavidLeighEllis (talk) 01:50, 26 April 2014 (UTC)

OK, I don't want to discourage you. I think the additions are probably an improvement on balance. I was just a little concerned about where this might end up. --Trovatore (talk) 01:58, 26 April 2014 (UTC)