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Thanks to all who improved my English! (See also my articles about Julis König and Jules Richard.)
But is self-describing and non-selfdiscribing correct? Would non-self-describing be better? If so, please correct the headline. Regards, WM
Hi David:
Cantor's diagonal proof would not deserve to be included in this article unless the following was true:
The diagonal number, as a finitely definable number belongs to a countable set of real numbers. (Compare König's paradox.) Therefore, Cantor's proof proves the existence of uncountably many real numbers by producing a real number belonging to a countable set.
The problem of doubly-hyphenated words is a nasty one. I don't like constructions like "non-self-describing". I'd prefer to use "not self-describing", or maybe even "autological" and "heterological" as the authors did in Grelling-Nelson paradox. I'm not sure that this is a hard and fast rule in English (no double hyphens), but it's what my English teachers drilled into me at an early age. I'll give that some more thought.
I'm still confused by the paragraph about Cantor's diagonal proof of the uncountability of the set of real numbers (and by your comment above). The proof I remember is a straightforward proof by contradiction – assume that the set of real numbers is countable, and then construct a real number via the diagonal method that's not in the enumerated list of real numbers.
Are we thinking of the same thing here? I see that you've changed the paragraph back to the way it was first written. I must confess that it still doesn't quite make sense to me. Maybe it just needs to be reworded a little bit. What I think you're trying to say is that in any enumeration of the real numbers one can use the diagonal argument to construct a real number that's not in the list; and if that one is added to the list, we can construct another one that's not in the list, and so forth. Comments?
Thanks for a nice article. I realize that these little contradictions are addressed elsewhere in Wikipedia, but it's instructive to see them all interconnected in one article arranged in chronological order. Good work! DavidCBryant17:56, 7 April 2007 (UTC)[reply]
Hi David, I will join your anti-doubly-hyphenation-attitude. But I do not like so much words like "autological", because the reader has to learn a new word, while with "self-describing" it is obvious what is meant and he can concentrate on the meaning. (In German it is easier to construct such a word.)
Cantor's proof is straightforward and, as I said, would not be considered a paradox. It simply shows that there are more reals than naturals. The paradox is, in my opinion, that it is impossible to construct a real number which is not finitely defined. The construction of Cantor's list and of its diagonal number is finitely defined (as is Richards number). We know that there are only countably many finitely defined real numbers. One of them is Cantor's diagonal number. So Cantor proves the existence of uncountably many real numbers by pointing to another number of a countable set of real numbers. Isn't that a paradox?
Does someone care to expound on / fix that section? What does the following sentence mean?
The set of all positive even numbers is the union of all finite sets.
And the following sentences don't help at all either.
Thank you for the hint. It must read " The set of all positive even numbers is the union of all finite sets of positive even numbers." I have fixed it. 141.82.28.2208:07, 19 April 2007 (UTC)[reply]
"By building on Cantor's idea in a natural way we see that the whole universe contains no more points than a grain of dust."
What exactly is a "point" in this context? Is this accurate? I admit to having a very limited understanding of physics, but I thought that, according to the quantization of matter, a grain of dust would be composed of finitely many particles and so would have fewer "points", in this sense, than the entire universe. —The preceding unsigned comment was added by 24.57.214.150 (talk) 17:40, 27 April 2007 (UTC).[reply]
While it is nice to see a page that collects various paradoxes from historical paradoxes of (naive) set theory, this article contains information which is unclear, misleading, and at times flat out wrong.
To start with, it conflates "size" and "number" in their colloquial sense with the modern exact mathematical meanings as defined by cardinality and ordinal numbers.
It uses "enumerate" in a non-standard way: one cannot "enumerate" the reals. One can biject R with R^2, and this is indeed surprising to many, including Cantor.
The comment that
Georg Cantor explained this by the difference between "reality" and "number". Both sets have the same cardinality, but one has more "reality" than the other.
is misleading at best. Cantor "explains the difference" (resolves the paradox) by noting that there is a bijection between the two sets. And which set is supposed to have more "reality" than the other?
The section entitled: "The set of all sets" contains the (unrelated) Burali-Forti paradox. Also, the "non-self-describing" section is a paradox of description / defining ("old" may be an old word, but such a description is not a mathematical one), and is not related to the "set of all sets" description. It should be classed with Berry's paradox.
What is the the author's meaning of "Paradox by change of language"? Does he mean the fallacy of equivocation, as it appears?
The author of this article makes no reference to the modern resolution of his description of Konig's paradox and the oddly named "Uncountability of the reals" by the definition of computable number. Yes, the set of computable real numbers is countable; no, we cannot apply the diagonal argument here because we cannot compute (construct) a list of all computable numbers.
Under Konig's paradox, "The assumption that the real numbers can be well-ordered leads to a contradiction" is simply false.
Under Skolem, the fact that there exists a countable model does not mean that there does /not/ exist an uncountable set /in that model/.
Under Banach-Tarski, the statement "The actual construction, however, is not possible" is misleading. The construction is indeed "possible", i.e., provable, given the axiom of choice. Of course such a construction is not possible in the real world, but this is true of most mathematical constructions.
Under "Tristam Shandy", the author oddly places his own ruminations regarding his false conclusions about finite sets and infinite sets. The fact that every finite set of naturals contains a largest member, and therefore (some conclusion) does not apply to infinite sets of naturals, which have no largest member.
The Ross-Littlewood paradox is an example of a supertask; this should be referenced.
"Paradoxes of order" talks about "gaps" in the rational numbers without defining what those gaps are in a mathematical sense. There simply is no paradox here, except one of poor understanding and definition. And the axiom of choice is irrelevant to this question.
The "paradox of the binary tree" is a paradox of the author's own making; see the sci.math thread "Review of Muckenheim's Book". Wikipedia is not for original research.
In summary: the article is poorly organized, lacks historical context, mixes together paradoxes of naive set theory with assertions of axiomatic set theory, makes incorrect or misleading statements, and needs some serious work.
It uses "enumerate" in a non-standard way: one cannot "enumerate" the reals. One can biject R with R^2, and this is indeed surprising to many, including Cantor.
I used the term "to enumerate in an extended sense". This expression was coined by Cantor (in German, of course). It helps to avoid the technical expression "bijection" in this article.
The comment that
Georg Cantor explained this by the difference between "reality" and "number". Both sets have the same cardinality, but one has more "reality" than the other.
is misleading at best. Cantor "explains the difference" (resolves the paradox) by noting that there is a bijection between the two sets. And which set is supposed to have more "reality" than the other?
Cantor asserted that the set of all natural numbers has more reality than the set of all even natural numbers.
The section entitled: "The set of all sets" contains the (unrelated) Burali-Forti paradox.
The set of all ordinals is closely related to the set of all cardinals and to the set of all sets and thus to the context.
Also, the "non-self-describing" section is a paradox of description / defining ("old" may be an old word, but such a description is not a mathematical one)
There are some readers, which are not mathematicians, I suppose.
What is the the author's meaning of "Paradox by change of language"? Does he mean the fallacy of equivocation, as it appears?
This is explained in detail in the article Jules Richard. Too detailed explanations would overload the present article.
The author of this article makes no reference to the modern resolution of his description of Konig's paradox and the oddly named "Uncountability of the reals" by the definition of computable number. Yes, the set of computable real numbers is countable; no, we cannot apply the diagonal argument here because we cannot compute (construct) a list of all computable numbers.
So it is. But countability is defined as the possibility to set up a bijection with the natural numbers. Such a bijection is a list.
Under Konig's paradox, "The assumption that the real numbers can be well-ordered leads to a contradiction" is simply false.
Under Skolem, the fact that there exists a countable model does not mean that there does /not/ exist an uncountable set /in that model/.
These are just biased statements. Please do not mix up paradox and antinomy.
Under Banach-Tarski, the statement "The actual construction, however, is not possible" is misleading. The construction is indeed "possible", i.e., provable, given the axiom of choice. Of course such a construction is not possible in the real world,
It is provable but it is not possible to do, neither in the real world nor in anyone's imaginaton. I think the average reader will understand that as it stands.
but this is true of most mathematical constructions.
That depends on what is understood by mathematics.
The Ross-Littlewood paradox is an example of a supertask; this should be referenced.
I would not oppose.
"Paradoxes of order" talks about "gaps" in the rational numbers without defining what those gaps are in a mathematical sense. There simply is no paradox here, except one of poor understanding and definition.
The talk about gaps stems from Fraenkel, Abraham A., Levy, Azriel: "Abstract Set Theory" (1976), p. 160: "The irrational numbers may just be defined as the set of gaps" in Q. To understand this does not require more than secondary school knowledge. And again, please do not mix up "paradox" with "antinomy". Regards, WM11:32, 23 May 2007 (UTC)[reply]
Note - it is more standard to reply in-line on Talk pages, rather than copying and pasting. This saves space, and ensures that the original material is not mangled. I have replaced your "*"'s with colons; which indent your replies; and combined my original statements with your responses below to the best of my ability. The history for the talk page contains the previous version, so I apologize in advance if I have lost some sense of your statements in this process. 71.198.111.24522:29, 10 June 2007 (UTC)[reply]
Okay, let's take this one section at a time. First, initial line:
This article contains a discussion of paradoxes of set theory.
should be amplified; there is a perfectly good article on Wikipedia which clarfies the intended meaning of "paradox" (as opposed to antimony) intended here.
By "Paradoxes of set theory", I assume we mean something more than simply relating basic confusions about the statement of some problem in mathematical terms. Otherwise, we might as well include on this page the "Paradox of the bellboy" where three men rent each rent a room at a hotel and end up tipping the bellboy such that we wonder "where is the extra dollar"?
Next, the statement:
The smallest infinite set is the set of all natural numbers N.
is false: there simply is no "the smallest infinite set" under Cantor's definition of cardinality. Not all sets are Von Neumann ordinals! This is just sloppy wording; the set of all even numbers has equal reason to be called "the smallest infinite set" in Cantor's theory.
No. There is no smaller infinite set. We have to observe didactics. It would be inappropriate to start with too much complications. That there are many smallest infinite sets becomes clear from the following sentence. Regards, WM09:48, 10 June 2007 (UTC)[reply]
On the contrary: In the context where we will soon be asking "which is smaller, the set of all naturals or the subset of the naturals which are squares?" it is important that we do not first muddy the waters by first claiming the the latter is the smallest infinite set, and then claim that they are the same size. In English usage, "the smallest" carries the connotation: "smaller than any other". 71.198.111.24522:29, 10 June 2007 (UTC)[reply]
Similarly:
Besides the cardinality, which describes the number of elements in a set, the order of elements within the set is also a subject of set theory.
Yes, order theory can be considered as a subset of set theory. But again, in mathematical articles, "the" means "the one and only". There is no such "the" order for an arbitrary set; there are many such orders. So this statement needs rewording.
The axiom of choice guarantees that every set can be well-ordered: Every subset which contains at least one element has a first element with respect to the order.
While essentially correct (except for a similar complaint regarding "the" well-order: the reals have uncountably many non-isomorphic well-orderings),...
the first element is related to the well-order in which it is the first element. But I wll use "that" instead of "the" in order to stress that there may be more than one well-orderings. Regards, WM09:48, 10 June 2007 (UTC)[reply]
I would word this more carefully: "The axiom... : There is an ordering of the elements such that every subset with at least one element has a first element, with respect to that order." 71.198.111.24522:29, 10 June 2007 (UTC)[reply]
... the most recent revert claimed that this article is "more understandable to the layman" in this form. Why does the layman need know about the application of the axiom of choice to an uncountable set in order to understand Berry's paradox (for example)?
That does not answer my question: The paradoxes of Konig, Richard and Berry are not particularly paradoxes of well-order; they are similar paradoxes of definability. The "paradox of the gaps" does not rely on a well-order, but a total order. Etc. 71.198.111.24522:29, 10 June 2007 (UTC)[reply]
The order of a set is described by an ordinal number which does not belong to the set.
Again, this is false. "The" order, as in "the one and only ordering" of a set doesn't exist;
The order given to a set is "the" order given to a set.
instead there are in general /many/ orders for a set. And if we refer to "the ordinality" as /the/ unique ordinal that can be bijected with an arbitrary set, then for infinite sets this doesn't exist in general.
That need not be described in this article, in my opinion. But if you disagree, I invite you to add some brief description. Regards, WM09:48, 10 June 2007 (UTC)[reply]
Secondly, the ordinality of (in the sense of "the unique ordinal which can be bijected with") the set {{}, 3, {{}, {3}}} is 3, and yet 3 is a member of that set. This needs to be reworded to be correct; again, it is sloppy wording. In any case, what is the relevance of this statement as regards any of the valid paradoxes listed in the article?
For instance, 3 is the ordinal number of the set {0, 1, 2} = 3;
Yes, no ordinal is a member of itself. This is not an unusual state of affairs in set theory. In fact, in most theories with foundation/regularity, no set is a member of itself. What paradox in this article requires that we know this bit of information?
Then it should be amplified at the reference for the Burali-Forti paradox, instead of being presumed as a general requirement for understanding the article. 71.198.111.24522:29, 10 June 2007 (UTC)[reply]
Please don't simply revert the article. There are many issues which have been raised in the Talk page which need to be addressed. It serves Wikipedia better if you would engage and address these issues. 71.198.111.24519:48, 11 June 2007 (UTC)[reply]
... and ω is the ordinal number of the set of all natural numbers. Neglecting the order, we are left with the cardinal number |N| = |ω| = \aleph_0.
What does "Neglecting the order, we are left with..." mean here? ω is still ω, whether we consider it as a set /with/ an ordering (e.g., by inclusion), or we instead simply consider it as a set with no ordering. How does it illuminate any of the paradoxes mentioned in the article?
I don't see how it points to that paradox, because I'm unable to make heads or tails of the sentence. I might suggest instead: "In set theory, every set has a cardinality, which is itself a particular set. For example the cardinality of the set of all natural numbers is the set \aleph_0". 71.198.111.24522:29, 10 June 2007 (UTC)[reply]
Furthermore, why is there any mention of order and ordinals here in the paragraph regarding basics at all? "Ordinality" is not key in the paradoxes that follow, with the possible exception of Konig's paradox, whose resolution in any case is not based on the well-ordering of the reals, but on other issues (is the set of all definables itself definable?). Konig's paradox still exists without the axiom of choice, and is often reworded as "the least undefinable ordinal has just been defined by this sentence".
Ordinality is the key to the first paradox of this kind ever discovered, namely the paradox of all ordinal nunmbers by Burali-Forti. Regards, WM09:48, 10 June 2007 (UTC)[reply]
Either I am reading this article from a relatively naive understanding of set theory, or from a relatively experienced viewpoint. In the former case, there is not enough space to adequately describe the concept of ordinal numbers here (there is space at the article ordinal); and I will get much more out of the essentially similar paradox of the Barber. That is, after all, the reason that Russel described it: it is easy to understand for a novice. On the other hand, if I am relatively experienced with the ideas of set theory, then I already know what an ordinal is, and I don't need it explained here. I don't see how the article is well-served by a vague and incomplete definition of ordinals when an equivalent description of the paradox is available. 71.198.111.24522:29, 10 June 2007 (UTC)[reply]
Lastly, the other issues brought up in the section "Basics" are only really of importance for a handful of these "paradoxes": in particular, those that are concerned with surprising but perfectly valid results regarding cardinality. Berry's paradox, for example, has nothing to do with the existence of uncountable, or even "actual" infinite sets. The paradoxes which have to do with understanding the set theory definition of cardinality as a measure of set sizes ("larger", "more than", etc.) should be grouped together with these basics (as I did in the previous edit), rather than simply plopped up top of the article as if they somehow are required before understanding any of the "paradoxes" which follow.
For this reason, I am re-reverting this section to eliminate incorrect, unclear, or unnecessary statements. After further review, I would recommend refactoring to place those "paradoxes" which actually depend on the definition of "size of a set" as its cardinality into one section (as I did previously). 71.198.111.24501:20, 5 June 2007 (UTC)[reply]
I would not like to renounce the introductory "Basics", so I reintroduce it, but invite you to correct what you think is false. But please keep in mind that this is not a mathematical encyclopaedia.
I think it really does the article a disservice, for reasons I gave above and will summarize here:
* There is not sufficient space in this article to give useful definitions of the concepts of set theory adequate to /properly/ understand, for example, the Skolem paradox. Where this level of depth is /required/ we should simply refer people to articles whose task /is/ to provide that understanding.
* The paradoxes such as the Barber, Berry's, Aristotle's Wheel, and so on were devised to translate the more abstract paradoxes into language which can be understood by a more general readership. They do not require an extended knowledge of the terms of set theory.
Either the readers of this article have little knowledge of set theory, in which case they should be directed to an article which is dedicated to explaining "ordinals"; or else they already know what an ordinal is, in which case there is no need to explain it to them here. 71.198.111.24522:29, 10 June 2007 (UTC)[reply]
Also Burali-Forti's paradox as the first one ever discovered, must remain there.
And I don't know why you disagree with the binary tree (in the discussion below). It is no "original research". It is but a simple observation which meanwhile has become popular in a wide range (see Google and my book). Regards, WM09:48, 10 June 2007 (UTC)[reply]
[As noted before, please address your comments on the Talk page in-line.] The reason why I removed it is simple: see WP:NOT. Referencing your own works is considered self-promotion; and the fact that there is discussion with you on sci.math regarding this notion is not evidence that this notion is widely held (in fact, it is evident that it is widely disputed in that forum that your argument makes any mathematical sense at all). 71.198.111.24522:29, 10 June 2007 (UTC)[reply]
Sorry the book has been written by a professor of mathematics at a German university of applied sciences. Further this book is not of vanity but hard cover and available at several libraries in Germany and Austria. May I ask for your qualification?217.94.210.21510:42, 25 July 2007 (UTC)[reply]
First, to claim that Cantor "resolved" Galileo's paradox by saying that the set of all natural numbers has more "reality" than the set of squares of naturals is very uninformative. In modern mathematical terms, what does "The set X has more reality than the set Y" mean? If you do not answer this question either directly in the article or by easily available reference, then the statement hardly seems to "resolve this paradox" or illuminate Cantor's thought processes, any more than if Cantor had said (translated directly from the German of 1900) "the set of naturals has more pregnancy than the set of squares of naturals". To my mind, this type of statement creates more confusion than it dispels, and should be removed.
Cantor did not "solve" Galileo's paradox. But he saw that there is a problem when applying his bijection. Only in order to show that Cantor, the foremost discoverer and defender of the actual infinite, was also of the opinion that there are problems to be explained, this phrase, reporting his explanation, is important. Regards, WM11:00, 10 June 2007 (UTC)[reply]
Cantor's definition of cardinality certainly does resolve the paradox; it clearly defines "size of a set" in such a way that the two sets provably have the same "size". If there were an article on Wikipedia called "Cantor's reality of sets", then the statement that one set has more "reality" than another might make sense. But as I noted, I (nor I suspect any other reader) cannot deduce what Cantor might or might not have meant by this statement: so it is pointless to include it unless your intention is to say "Cantor was still a very confused person when it came to infinite sets". 71.198.111.24522:29, 10 June 2007 (UTC)[reply]
Second, look at the wikipedia entry for enumeration. It explicitly states that an enumeration of a set X is a surjective or bijective mapping from N -> X. While Cantor may have used a German word which can be translated into English as "enumeration"
In algebra, the French term "corps" is the English term "field", even though the literal translation of "corps" is "body". English usage is simply the standard in the English Wikipedia, and English usage is not always a direct translation from German. 71.198.111.24522:29, 10 June 2007 (UTC)[reply]
when he wrote his theories a hundred years ago, this use of the English term to refer to a mapping from R -> X is outdated and incorrect in modern usage.
The goal of this article should not be to increase confusion, but to be clear. This is the English wikipedia; and modern English usages (not nearly century old German usages translated directly into English) should be its standard. 71.198.111.24522:29, 10 June 2007 (UTC)[reply]
How do you justify its use in this article, when in many other articles on wikipedia, we have the statement "there is no enumeration of the real numbers"? Why do you object to using the modern usage "has the same size as, as defined by cardinality" instead of the arguably incorrect "is enumerated by" or the unclear and peculiar "is enumerated by, as defined by extended numeration"?
Why should Cantor's remark of "enumerating in an extended sense" (which is a direct translation of his collected works, p. 447: Alle Mengen sind daher in einem erweiterten Sinne "abzählbar", im besonderen alle "Kontinua".) be removed in an article which is mostly about history? Regards, WM11:00, 10 June 2007 (UTC)[reply]
For the reasons I mentioned above; and more importantly, because it does not serve to explain to a modern reader why the result is paradoxical. The paradox doesn't revolve around the use of the direct translation "extended enumeration", but instead because it showed that a line has "as many" points as an area (compare with the Aristotle Wheel paradox).
As regards the purpose of the article, I disagree that it is primarily a history article. It is not titled, for example, "a history of set theory through its paradoxes". In addition, it seems to largely stop its historical overview at around the early 1920s, whereas many other paradoxes have arisen in the meantime, and a whole bunch of research has been developed classifying most of these paradoxes as variants of the Liar's paradox.
On the other hand, many people encounter the same sense of paradox today when they first encounter the idea that there are "more" reals than naturals, although both sets are infinite; and in that sense the paradox is equally "modern" as historical. Our goal should be presenting such types of paradoxes, simply, and in modern terms. We should introduce or reference terms just to give a reader the information they need to know to understand the paradox.
A general article on the historical development of set theory would be an interesting addition to Wikipedia; but set theory did not progress solely (or even primarily) by the development and resolution of paradoxes. 71.198.111.24522:29, 10 June 2007 (UTC)[reply]
I moved the subsection "Uncountably many real numbers" from the article to discuss it. It seems out of place; the other articles in the section were all variants of the Barry Paradox.
Further, the diagonal proof can easily be modified to produce an uncountable number of reals not on the list.
Removed section:
Cantor's diagonal proof is the most famous proof of the existence of uncountably many real numbers. The diagonal number, as a finitely definable number belongs to a countable set of real numbers. Therefore, Cantor's proof proves the existence of uncountably many real numbers by producing a real number belonging to a countable set.
I made some signficant edits this morning. I cut some things that seemed unrelated to the topic - paradoxes of set theory - and expanded some of the sections. Please forgive me if my edits were too bold. Here are some notes for myself and others:
Sentences that say things like "John defined a set that cannot be defined" are incomprehensible. They can be rewritten so that they make sense, for example, "John proposed a definition of a set and showed that the existence of a set satisfying the definition leads to a contradiction".
The lede is woefully inadequate. It should discuss the difference between naive and axiomatic set theory (naive is in natural language, while axiomatic keeps a distinction between the formal language and the metalanguage). It should also summarize the rest of the article in a paragraph.
I'm not convinced the Tristam-Shandy stuff is completely relevant. At best, it needs to be rewritten to explain how it is related to set theory.
The basics section is too long; we can wikilink to the respective articles. But I agree that a short review here is appropriate.
The section on paradoxes of enumeration needs more work, and should be merged with the section on AC, or the section paradoxes of order, or with the Tristam-Shandy stuff; there is some redundancy there. paradoxes o
It would be nice to find references to the historical papers where these paradoxes were first published. Some of them are in certain books of historical logic papers such as From Frege to Godel. There is a long list of references at the bottom, so I may have missed some there.
Great start! I ended up breaking the overview into three sections, which I think should be mirrored by the order and organization of the content of the article. These seem to be:
a section relating directly to cardinality/size/measure/ordering. I would include "je voix mais..." and Banach-Tarski as examples where cardinality and other forms of "size" such as measure are simply shown to be different animals. While B-T relies on well-ordering, I don't know that I would call it a paradox of order (more a paradox of AoC implies Vitali sets; I'd call it a paradox of measure); and in fact I wonder whether there are any paradoxes of order to be found here.
a section relating to supertasks (such as Tristam Shandy); where there is an assumption of an infinite process which "completes". This is the least worthy of being a "paradox of set theory"; but it does provide an opportunity to address the common misconception that, for example, infinite sums are supertasks (as opoosed to limits); or that infinite unions are supertasks.
a section relating to the formal language issues of set theory (set of all sets, paradoxes of definition, Skolem and its philosophical implications). Although various statements regarding cardinality and well-ordering arise here, the resolution of these paradoxes have to do with their interpretation in some language. Skolem is a nice capper in terms of the limits of these languages to fully capture what we had hoped (I'm thinking of the non-categoricty described at http://www.earlham.edu/~peters/courses/logsys/low-skol.htm).
I give them the above order since the first two refer most to problems with intuitions regarding infinity, while the latter is more an issue of formal language and logic.
I'd like to try to place the content of the "Basics" section more as part of the development of the exposition of the sections: for example, we can define cardinality as part of the difference between the two options given in Galileo's paradox - it's a one-to-one match, which Cantor used to define "same size as", and so on; followed by the inability of of providing such a bijection between N and P(N); and so on. Ordinals could be introduced under Burali-F (although I still wonder whether ordinals need to be defined in detail here at all).
Of course, all of these "paradoxes" are frequently presented by certain persons to "prove" that infinite sets do not exist; but we can still make lemonade out of lemons! Chas zzz brown22:09, 31 August 2007 (UTC)[reply]
The current lead of this article is strange both in its contents and the placement. I struggled quite a bit trying to understand what is it trying to express. At best, it appears to be an out of place section with the analysis of the possible nature of (misconceptions about) paradoxes of set theory, not an introduction to the topic or the summary of the article. Can we not tuck it to the end, and replace with a proper lead? Arcfrk23:39, 2 September 2007 (UTC)[reply]
I appreciate your comments. They reveal the fact that, at this point, I'm just struggling to figure out what the actual content and topic of this article should be to start with; the lead was an attempt to at least organize the existing content into some kind of coherent whole.
The problem is that this article was originally a collection of various (basically unrelated) arguments against "completed infinity"; and since this concept is currently formalized in set theory, more general arguments against set theory were also included to that same end. Several of the least relevant arguments have already been removed; but we are still left with a bit of a mish-mash.
For example: Is "Galileo's paradox" actually a paradox of /set theory/ (naive or otherwise) or a paradox of poor definition? Does Cantor's surprise regarding the bijection R -> R^2 constitute a /paradox/ of set theory? Do supertasks represent a /set theory/ paradox, or some other class of paradox? Is B-T a paradox of set theory, or measure theory? Is Berry's paradox a paradox of set theory, or of informal logical systems?
Yes; if we accept that (almost) all of modern mathematics can be translated into the lingua franca of ZFC, all paradoxes are in some sense "paradoxes of set theory". But how useful is that to the reader? Without some sort of organizing principle, it seems like this article could simply be replaced by a list of links to existing WP articles regarding these "paradoxes" as more general "paradoxes of mathematics".
A key question is: who is the intended audience of this article?
"Most" people find it quite bizarre that there should be a bijection between an infinite set and a proper subset; and from that point of view, a bijection from R to R^2 or the lack of one between Z and R is indeed paradoxical; and we would like to serve this audience, which includes many with a fair amount of mathematical experience such as engineers and physicists. But those who find such things difficult to grapple with are not going to be able to make heads or tails of Konig's or Skolem's paradox without a good deal of exposition.
Conversely, for someone who can understand Skolem's paradox, the idea of a bijection between Z and nZ or R and R^2 isn't paradoxical at all - instead, it's quite natural and obvious. But the resolution of Konig's argument is not at all obvious, and it's interesting to see the links between the related "meta-mathematical" paradoxes; and so we want to serve this audience as well.
So, if it is not to be just a list of "various surprising mathematical results", what is the "story line" for this article? Some ideas:
Break the article into two distinct articles: paradoxes of infinite cardinality, and paradoxes of formal systems and proof. The former covers "surprising" facts about the formalization of infinite sets that contradict our intuitions regarding number, size, order, measure, etc. The latter covers "surprising" facts regarding proof, mathematical truth, etc. which arise from the more general program of axiomatization. This allows us to serve the two distinct audiences indicated above.
Paradoxes arise as a result of formalizing our intuitions. We start with unformalized mathematics; standards of proof (say, in the 17th or 18th century) were much more lax than is currently accepted (as are the current standards of proof typically used by engineers and physicists). Naive set theory began to formalize certain intuitive notions: infinity, size, order, measure. These formalizations contradict the natural language version of these terms and our intuitions based on physical experience; resulting in various paradoxes. Axiomatic set theories in turn formalize our intuitive notions of proof, mathematical truth, and definability. These formalizations contradict the natural language version of these terms and our intuitions of what should be provable, definable, etc.; resulting in various paradoxes.
Or some other thing... any ideas? For the time being, I am going to re-organize the content along the lines I discussed above; without removing content. Chas zzz brown23:21, 3 September 2007 (UTC)[reply]
Since we have articles on the individual paradoxes, I see nothing wrong with bringing this list down to paradoxes involving, in some way, completed infinity – upon which the article should really be renamed "Paradoxes of infinity". As a start, I'm taking out Berry's paradox. The lead could state that our intuition is inconsistent and not enough in dealing with the infinite, but that initial attempts at formalizing our intuition did not managed to exclude these inconsistencies. --Lambiam06:34, 4 September 2007 (UTC)[reply]
Re Chaz Brown: I think if we limit this article to things that are generally labeled "paradox" that will give us a good selection of material. I wonder... could it just be titled "mathematical paraadoxes" instead of paradoxes in set theory? I like the way the material is rearranged. — Carl (CBM · talk) 12:35, 4 September 2007 (UTC)[reply]
The infinite binary tree contains all infinite sequences of binary numerals and, therefore, represents all real numbers of the interval between 0 and 1. The set of nodes of any level is countable. This implies that any possible cross section of the infinite tree contains at most countably many paths while, according to Cantor's theorem, the infinite binary tree contains more paths than nodes.
I am not sure what is meant by "cross section" here; if you pick any collection of nodes of the tree, the set of paths that include at least one of those nodes is uncountable. I can guess what was intended, but I'm not sure. I am also not certain this is often called a "paradox", rather than just a fact about trees. If we do put it in the article, a link to Aronszajn tree would be appropriate. — Carl (CBM · talk) 12:40, 22 January 2008 (UTC)[reply]
Start with one path (like 0.000...). Add another path-extension in the first node. Add another path-extension in the next node (level 1, left hand side). Add another path-extension in the next node (level 1, right hand side). Continue to do so over all countably many nodes. Then you have the complete tree. No path-extension is missing. Hence there cannot be more than countably many paths in the tree.
"Cross section" is simply an abbreviation of the number of distinct paths at some level.
The "Hence" argument is wrong (of course) because it ignores the fact that the paths through the tree form a closed set. So if you start with 0000..., and add 10000.., 110000, ... , 111000, ... , ... then you end up with the path 1111... even though you never specifically added it. My actual concern is whether this is often classified as a paradox. I do see the point that if you take the set of finite paths through the tree, this is a countable set, but the set of infinite paths is uncountable. — Carl (CBM · talk) 16:41, 22 January 2008 (UTC)[reply]
The set of nodes in the infinite tree is countably infinite. So you can add a path-extension to each node. Do you agree that this procedure will yield all paths of the complete infinite tree? If you agree to this statement, then you get representations of all real numbers between 0 and 1. If not, tell me which path extension is missing. Of course you are right that the set 0000..., 10000.., 110000, ... , 111000, ... ends up with the path 1111... . But I do not see why this invalidates the argument. The paradox has become famous during the last years, particularly in mathematical news groups.91.7.222.140 (talk) 17:27, 22 January 2008 (UTC)[reply]
"The set of nodes in the infinite tree is countably infinite. So you can add a path-extension to each node. Do you agree that this procedure will yield all paths of the complete infinite tree? " I don't agree; my example shows how you can get "extra" paths besides the ones you add. Is there a published reference that considers this paradoxical? — Carl (CBM · talk) 17:39, 22 January 2008 (UTC)[reply]
"my example shows how you can get "extra" paths besides the ones you add." So you mean that there may be paths which do not differ by a node? Are you sure that such paths are really existing, i.e. there are different numbers which do not differ by digits however? And if so, how could such numbers be treated in Cantor's diagonal argument where diffing by digits is essentially presupposed for different numbers?91.7.222.140 (talk) 18:46, 22 January 2008 (UTC)[reply]
Here are two English references: The first has been printed, the second is in press. Proceedings of the First International Symposium of Mathematics and its Connections to the Arts and Sciences, A. Beckmann, C. Michelsen, B. Sriraman (eds.), Franzbecker, Berlin 2005, p. 134 - 141, available as http://arxiv.org/pdf/math.GM/0505649. SudDansk Universitet Press 2008, preprint available as http://arxiv.org/abs/0709.410291.7.203.183 (talk) 21:37, 22 January 2008 (UTC)[reply]
Those two Arxiv references are, unfortunately, not quite what I'm looking for. I'm looking for papers in established journals or texts by well respected publishers. Is the book published by Shaker-Verlag available for purchase in print, or just online? Was it refereed? — Carl (CBM · talk) 22:01, 22 January 2008 (UTC)[reply]
The finite trees represent numbers in [0, 1) which can be represented as n/2^k for positive integers n and k. The whole infinite tree includes much more: rationals with denominators other than powers of 2, irrational algebraic numbers, and transcendentals. "1/3" cannot be represented as a finite binary expansion, but it is a real number. Ditto with and .CRGreathouse (t | c) 16:52, 22 January 2008 (UTC)[reply]
Here only the infinite tree is under consideration. (But it can be constructed by the union of all finite trees.) 1/3 is represented by a path in the infinite tree, namely 0.010101..., like every transcendental number. 91.7.222.140 (talk) 17:27, 22 January 2008 (UTC)[reply]
But you've only shown that the each cross-section is countable, not that the infinite tree is countable. Yes, I can show you the first binary place where pi/4 and 25/32 differ, but pi/4 doesn't appear as a 'cross-section' ever. Since its binary representation is infinite you'll never count it when you count the finite trees. That's why your count is wrong. A true well-ordering of real numbers would be able to place any real number at a particular location, which your method doesn't do (except for the numbers with denominators which are powers of 2, as noted above). CRGreathouse (t | c) 17:50, 22 January 2008 (UTC)[reply]
"But you've only shown that the each cross-section is countable". Sorry, the set of all nodes is countable. "pi/4 doesn't appear as a 'cross-section' ever." What I called the cross section (at some level) is the number of distinct paths at some level. It is good to have this discussion in order to see where I have to be clearer. Please forget the cross section. "Since its binary representation is infinite you'll never count it when you count the finite trees. That's why your count is wrong." I count all nodes. Every extension of a path begins at a node. Therefore there cannot be more extensions than nodes (well at most there can be twice as many, if we forget the nodes above). "A true well-ordering of real numbers ..." I have no well-ordering of the reals. I have only an upper threshold of their cardinal number. —Preceding unsigned comment added by 91.7.222.140 (talk) 18:33, 22 January 2008 (UTC)[reply]
The reason you don't have a well-ordering of the reals is that none exist in ZF. You say you have an upper bound, but you don't -- just because you have the prefixes doesn't mean you have the numbers. You do have a dense subset of the reals, but dense subsets need not have the same cardinality.
Point-by-point: "I count all nodes.": no, only the ones with powers of 2 as the denominators. "Every extension of a path begins at a node.": yes, assuming that by "node" you mean "binary prefix" and "extension of a path" you mean "real number". "Therefore there cannot be more extensions than nodes (well at most there can be twice as many, if we forget the nodes above).": no, each 'node' has an infinite number of 'paths'.
Let's call the first n binary digits the beginning of a real number and the remaining digits the extensionof this real number. Can there be extensions without beginnings? No. "I count all nodes" means I count all binary digits. Each node has an infinite number of paths. Of course. What we consider is an infinite tree. But no extension is possible which does not start at a node. And at each node start two extendions. Therefore there cannot be more extensions than nodes.91.7.219.134 (talk) 09:19, 23 January 2008 (UTC)[reply]
Notice. Above you stated earlier: "Further, the diagonal proof can easily be modified to produce an uncountable number of reals not on the list." This is wrong. The set of producible or constructible numbers is countable, because the numbers of constructions is countable (finite expressions over a finite alphabet). This is why one can say that Cantor's diagonal procedure produces only numbers of a countable set. Therefore it is a paradox that this procedure is a proof of uncountable sets. 141.82.28.22 (talk) 11:12, 23 January 2008 (UTC)[reply]
You don't have to construct each element, simply show that it's there. Instead of changing a digit at each step (producing 1 number at the end) you can easily choose 8 digits at each step -- nicely avoiding the issue with 9s and 0s. Then you have an uncountable number of members not on the list, . CRGreathouse (t | c) 14:13, 23 January 2008 (UTC)[reply]
The same material is presented in this text and apparently also in a book I haven't seen by the same author: W. Mückenheim, Die Mathematik des Unendlichen, Shaker-Verlag, Aachen 2006. It appears that the author thinks the paths (always starting from the root) in the union of a set of trees are the union of the paths in the trees in that set (see page 6); this false assumption leads to a contradiction. This appears to me to be at about the same level as the false reasoning that, as all finite ordinals have a finite number of elements (using the Von Neumann definition of ordinals), the same must apply to the limit ordinal ω, being the union of all finite ordinals. The anonymous editor (COI?) seems to be suffering from the same delusion: when I stated (in the edit summary) "Of course there can be more infinite bit sequences than finite prefixes of such sequences, which is the same",[2] he replied "Wrong. Every pair of different infinite sequences must differ at a finite place. Each infinite bit sequence consists of bits at finite places only."[3] --Lambiam17:02, 22 January 2008 (UTC)[reply]
"...the false reasoning that, as all finite ordinals have a finite number of elements ...the same must apply to the limit ordinal ω, being the union of all finite ordinals. "Let me put a question: Do you think that a set consisting of aleph_0 identical elements 1 has more than 1 element? Do you think that a set consisting of aleph_0 elements 1 and aleph_0 elements 2 has more than two elements? Certainly you do not. Why then do you think that a set of infinitely many elements of size less than omega has omega elements? 91.7.203.183 (talk) 21:50, 22 January 2008 (UTC)[reply]
Are you serious? I see no relationship between the questions about identical elements and the last question. I believe that 1 is the successor of 0 because it is so by definition. Likewise, I believe ω has ω elements (using the Von Neumann cardinal assignment to define cardinality) because this is so by definition. --Lambiam00:17, 23 January 2008 (UTC)[reply]
There is a relationship between size of elements and cardinal number of sets for natural numbers. To make it easy: There are not 10 natural numbers less than 5. And there are not infinitely many numbers less than 100. In order to have a set with 100 natural numbers you need at least one number larger than 99 in ths set. By the same argument there cannot be an infinite number (aleph_0) of finite numbers, if aleph_0 is to be understood as a number larger than every natural number. See a recent discussion: http://groups.google.com/group/sci.math.research/browse_frm/thread/e251d1529dc6df9e/ab40b7c8e8f619ae?hl=de#ab40b7c8e8f619ae —Preceding unsigned comment added by 141.82.28.22 (talk) 10:57, 23 January 2008 (UTC)[reply]
Consider only the path-extensions which start at some node. As there are countably many nodes there cannot be more than countably many path-extensions, even if you apply two of them to each node. Every pair of different infinite sequences must differ at a finite place. Each infinite bit sequence consists of bits at finite places only. Do you disagree? 91.7.222.140 (talk) 17:27, 22 January 2008 (UTC)[reply]
Repeating the statement that there cannot be more than countably many path-extensions infinitely often does not make it any less false. --Lambiam00:17, 23 January 2008 (UTC)[reply]
I'm not particularly worried about whether this "is" or "isn't" paradoxical (although the wording could be improved). Most of the "paradoxes" listed here are not paradoxes at all, but misunderstandings about infinite sets. What I am concerned about is whether this is generally regarded as a paradox, or even generally described as a paradox. It would be possible to make this page arbitrarily long if we added every possibly counterintuitive statement about set theory. So the standard for inclusion is the reliable published sources describe the phenomenon as a paradox. I don't treat Arxiv as a reliable source in general, since anyone can publish anything there. I don't know about the book Die Mathematik des Unendlichen. Was it refereed? Are there any other authors, besides the author of this book, who have described this tree phenomenon as a paradox? — Carl (CBM · talk) 14:50, 23 January 2008 (UTC)[reply]
First it is wrong that everybody can publish in the arxiv. Second I gave printed sources. Third it should be generally regarded as a paradox if there are more tails than heads. If there are any paradoxes listed here, than the binary tree is a paradox.141.82.28.22 (talk) 07:51, 24 January 2008 (UTC)[reply]
It is true that not everyone can publish in the Arxiv, but at the same time papers in the arxiv are not subject to peer review or refereeing in the same way as published papers. I don't want to argue about whether it should be regarded as a paradox. I do see that it could be considered a paradox, but I don't see that it is at all common to do so. The general standard for inclusion on Wikipedia is not whether an argument is correct, but whether it is (correct and) published in reliable sources. This standard is important because otherwise our articles would be, like the arxiv, a forum for the dissemination of new theories, which we do not want to be. — Carl (CBM · talk) 12:40, 24 January 2008 (UTC)[reply]
"I do see that it could be considered a paradox," that is nice to hear, "but I don't see that it is at all common to do so" because most people don't know it. "otherwise our articles would be, like the arxiv, a forum for the dissemination of new theories, which we do not want to be." There is a slight difference. Set theory and the binary tree are not new theories. In bringing both together we do not create a new theory but mention only a simple observation. It is the same as if one describes historical errors in some movie. It is enough to have observed them. Everybody can look at the movie and observe them too. I think I have seen such articles in Wikipedia. (40 errors or so in "Back to the Future I", if I remember correctly.) And there is no problem for a mathematician to see that the binary tree must contain more extensions than beginnings, if set theory is correct. So what do you require additionaly ?141.82.28.22 (talk) 13:14, 24 January 2008 (UTC)[reply]
'40 mistakes in Back to the Future' does not belong in Wikipedia unless it's referenced. As for your theory, here or elsewhere, I think you should concentrate on explaining why you think that there can be no more 'beginnings' than 'extensions'. CRGreathouse (t | c) 14:28, 24 January 2008 (UTC)[reply]
You are right. I saw it elsewhere, (and meanwhile there are 112 known errors). Nevertheless I believe that simple obeservations in Wiki are not always referenced. Regarding your second sentence: Every path can be divided into a beginning and an extension at a node at level n with 0 < n < oo . As a node is required to define a beginning there cannot be more beginnigs than nodes. A path without a beginning simply isn't path. Hence every extension must be the continuation of a beginning. This restricts the set of extensions to cardinality aleph_0. Sorry if I do not see another outcome. Do you?141.82.28.22 (talk) 14:48, 24 January 2008 (UTC)[reply]
I agree with what you wrote, but that proves that every binary expansion has a prefix, not that the number of prefixes is at least as great as the number of binary expansions. I readily cede that every binary expansion has a prefix -- that's true -- but you haven't even attempted a proof of what you claim to have showed. CRGreathouse (t | c) 15:05, 24 January 2008 (UTC)[reply]
Assume that two different paths have the same prefix (= beginning = the first n nodes). Then they must differ by their extensions. That means they must differ by some node with m > n. Include this node into their prefixes. Then they have different prefixes. You can also see this by construction: Construct one infinie path, e.g. 0-0-0-.... Attach at some node another path-extension. Attach at some other node another path-extension. Continue until you have treated all nodes. If you have done so for every node, you have the complete infinite binary tree. There is no node disconnected from the root. This construction is possible because there are countaby many nodes (and it shows that there are as many paths-extensions).141.82.28.22 (talk) 15:49, 24 January 2008 (UTC)[reply]
Even easier: Attach two (different) path-extensions to every node. Then there does not remain any space for further paths and path-extensions. 141.82.28.22 (talk) 15:53, 24 January 2008 (UTC)[reply]
Your first reply ("Assume that two different paths...") shows that every two unequal real numbers differ at some finite place in their binary expansion. I agree with this, but it still doesn't prove your point. Your second reply shows (well, claims, really) that there exist an infinite number of distinct real numbers with power-of-two denominators. You haven't justified the claim that "there does not remain any space", though. In fact the claim is quite false, as justified by this example: for each prefix , choose if n is odd and otherwise. It is easy to check that the are distinct, and yet 1/3 does not appear on the list. Thus there does remain "space"; in fact most rationals do not even appear. CRGreathouse (t | c) 03:19, 25 January 2008 (UTC)[reply]
Do you claim that not every real number can be written as a binary number? Then you should claim the same for decimal representation. It would be quite unusual. Please let me know. If you agree however, that every real can be represented in binary, then there is no chance to find a real of [0, 1]which is not represented in the binaray tree. Then there does not remain space for anything else in the tree.91.7.217.84 (talk) 08:56, 25 January 2008 (UTC)[reply]
Edit: Oh, you wanted two distinct reals for each prefix, not one. Just add to get the second distinct solution -- this choice (one of many, many possibilities) makes it easyto check that all remain distinct. CRGreathouse (t | c) 03:21, 25 January 2008 (UTC)[reply]
No, I do not want two distinct reals for each prefix, but I have given an upper estimation for the number of paths in the binary tree. Even if two paths (going left and right) started at every node, their number was countable. --- And what about your producing of uncountably many reals?91.7.217.84 (talk) 08:56, 25 January 2008 (UTC)[reply]
You haven't given an upper estimate, because I just constructed a list larger than yours which isn't nearly complete. The step where you say that you have an upper bound is unjustified and where the argument goes wrong. As for "Do you claim that not every...", certainly I agree that every real number has a binary expansion and a decimal expansion -- in fact each has either 1 or 2 such expansions. (There are three possibilities for the two: one decimal expansion and one binary expansion [e.g. sqrt(2)]; two decimal expansions and one binary expansion [e.g. 1/5], and two decimal expansions and two binary expansions [e.g. 1.5].) But I don't claim that each one has a finite binary expansion, which is essentially what you're claiming. CRGreathouse (t | c) 13:44, 25 January 2008 (UTC)[reply]
The binary which I talk about is infinite. So there is no arguing about finite expansions. If you claim to have constructed one more number than is given by the complete infinite binary tree, then you should be able to say either why it is not in the infinite binary tree or why it is not covered by all possible paths. Because all possible infinite paths are already occupied by my upper estimation.91.7.251.116 (talk) 15:37, 25 January 2008 (UTC)[reply]
You said: (1) certainly I agree that every real number has a binary expansion, (2)there is a number which is not on your binary tree. By definition the binary tree contains all decimal expansions of real numbers between 0 and 1. Please take this definition, if you have problems to understand that your statements (1) and (2) are contradicting each other. 91.7.231.167 (talk) 08:50, 26 January 2008 (UTC)[reply]
the definition of a cardinal number given in the Cardinal numbers section is irrational. it's circular. it somehow defines a number as a set, and not the obvious set of one number equal to that cardinal number. it should be fixed or expanded to make at least a little sense. 68.96.49.125 (talk) 06:55, 25 October 2012 (UTC)[reply]
We have a subsection entitled, "Je le vois, mais je ne crois pas." While it's undeniable that Cantor did write that in a letter to Dedekind, nevertheless Gouvêa argues convincingly in his article "Was Cantor Surprised?" in the March 2011 American Mathematical Monthly that Cantor was not reporting astonishment at the result (of which he was quite convinced), but rather a "worry about the correctness of his proof."
Under the circumstances, although the result (that |R| = |Rn| for any fixed n) is a fine example of a paradox of set theory, using that quote to introduce the result seems unfair, or at least misleading.
A. Using plain 'English' arguments these people notice that matching members of a finite size set to members of another or the same finite size set will prove the size of the first is only one of >, =, or < the size of the second. So they will also work with ∞ly large sets. However any that are 'proved' = in size by a sub-proof are also proved > and < in size by two more sub proofs using the same method! Same with a set and itself, this is one way of two they 'prove' ∞ (+, ×, or ↑) a +number = the same ∞ and a +number (+, ×, or ↑) ∞ = the same ∞, except they contradict exponentiation of infinity? Since these results are different than for finite sets their reason for it also working for ∞ sets is without foundation. Further they claim they are only = not > or < in 'size' (cardinality) not size, presumably because they are hanging onto what they want to believe, so when contradictory things are proved they use one to contradict the others.
WHAT IS GOING ON is they use ∞te increase (therefore ∞te increase without limit), and also translation rules (out of time translating members of a set to those of another such as changing naturals to reals and back), in their plain language arguments, apparently they can use either in any argument. Since both work for any one argument they both have the same characteristic, 'without limit', therefore their ∞ is unbounded thus the ∞te sets are unbounded thus have no last element. The size of any set is the position of its last element. Therefore these sets have no size, causing no relative size as evidenced by >, =, < all for the same 2 sets and for a set and itself. Since these sets have no size presumably the real numbers etc have no size thus same cardinality.
B. As above the same two methods that can prove for example 2 × ∞a = ∞a can also prove 2 ↑ ∞b = ∞b, but since they can prove 2 ↑ ∞b = 2 ↑ ∞b they accept the 2nd which thus disproves the first so again when contradictory things are proved they use one to contradict the others. Also considering all the above of this section presumably both are actually true. The same-∞ results come from one process, the different-∞ result from another. Presumably the 2nd procedure would also work with × or +, against the continuum hypotheses not using D. below. To save space I won't go into these......
C. AS THEY ARE STATED the diagonal arguments are false, proving sets are uncountable, but because of size rather than non-countability. The 'proof' must prove a universal negative, to do so it must but dosn't leave enough room for the count which is obviously size 2 ↑ the width of the diagram, but the height is the same as the width. For example if one did that for naturals starting with non-0 digit they would not fit. I even have a diagonal 'proof', which even uses multiple 'diagonals' ( one to one matching of each colomn to one row for each row GENERALLY ). But the naturals are known to be countable!
D. The supposedly uncountable sets are countable. If someone attempts to count by flipping the bits of the naturals to the other side of the decimal point they don't accept it, indicating they are adding something unstated to the axiomatic definition of Counting. And in fact it does not make sense for π or e to be exponential of ∞ ÷ approximately 2 thru a count to be acceptable which is insinuated. Correcting, count instead using a fully developed or universal generating function of bit strings, GENERATING FUNCTION. from the reals → input to the GENERATING FUNCTION, said having true random bit generator function like pseudorandom functions, that is a number in gets a random bit out for reproducibility. The diagonal arguments change a lot so a large Wikipedia page would be needed for both... Victor Kosko (talk) 13:58, 30 May 2017 (UTC)[reply]
