Talk:Paradoxes of set theory

message: 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. 141.82.28.22 08:07, 19 April 2007 (UTC)
thread_title: Tristam Shandy

message: "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)
thread_title:  Dust vs. the universe 

message: 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.
71.198.111.245 20:44, 29 April 2007 (UTC)
thread_title:  General Review 

message: 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, WM 11:32, 23 May 2007 (UTC)
thread_title:  Reply to the General Review 

message: The section on "The Paradox of the Binary Tree" violates wikipedia policy regarding no original research (see WP:NOT). As such, I have removed it. 71.198.111.245 01:29, 5 June 2007 (UTC)
replies: text: The book written by a physicist wich has been published by vanitiy publisher is no reliable source, so i'll remove this section. --Mathemaduenn 08:41, 25 July 2007 (UTC)
replies: text:  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?217.94.210.215 10:42, 25 July 2007 (UTC)
replies: text: I have made only a short look at his lit of publications and I have not found any publication in a known mathematical journal, so I think he is a physicist.( look also  Tätigkeitsbereich: Physik). --Mathemaduenn 17:49, 25 July 2007 (UTC)
thread_title:  Wikipedia: No Original Research 

message: 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.
CRGreathouse (t | c) 05:08, 22 July 2007 (UTC)
thread_title:  Diagonal proof 

message: :::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)
thread_title:  Move to [[Mathematical paradoxes]] ? 

message: ::::: '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)
replies: text: 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)
replies: text:  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)
replies: text:  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)
replies: text:  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 n/2^k, choose r_{n,k}=n/2^k if n is odd and r_{n,k}=(4n+\pi/n)/2^{k+2} otherwise.  It is easy to check that the r_{n,k} 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)
replies: text:  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)

text:  Edit: Oh, you wanted two distinct reals for each prefix, not one.  Just add 5/2^{k+9} 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)
replies: text: 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)
replies: text:  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)
replies: text:  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)
replies: text:  I am saying that there is a number which is not on your binary tree -- I constructed an explicit example.  CRGreathouse (t | c) 21:30, 25 January 2008 (UTC)
replies: text:  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)

text: Yes, that's right. That's the purpose behind the anon's edits: to show that there are only a countable number of real numbers. CRGreathouse (t | c) 21:30, 25 January 2008 (UTC)
replies: text: Then there is no point in continuing the discussion. This page is for improvements to the article, and the "paradox" is original research.  --Lambiam 22:20, 25 January 2008 (UTC)
thread_title:  paradox of binary trees, part 2 

message: 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)
thread_title:  cardinal number 

message: 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.
Sorry, the quote was too good to be applicable. Any good ideas how to resolve this?—PaulTanenbaum (talk) 22:42, 19 February 2013 (UTC)
replies: text: Never mind. I've come up with a fix and shall apply it.—PaulTanenbaum (talk) 00:48, 20 February 2013 (UTC)
thread_title:  Misquote of Cantor? 

message: 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)
thread_title:  4 Disproofs