Talk:Schröder–Bernstein theorem

message: In the german article :de:Cantor-Bernstein-Schröder-Theorem, which I translated from the english version, I added a visualization of the map h.
Someone might want to translate the image displayed at that article into english and look over this trial of a translation, and then integrate it into the english article.
Here comes my translation of the section "Veranschaulichung" in the german article.
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(Before that, I added a little remark on the map h.)
Note, that this definition of h is nonconstructive, in the sense that there exists no general method to decide in finitely many steps for any given sets A, B and injections f, g, if an element x of A lies in C or not. For special sets and maps this might of course be possible.
Visualization
The definition of h can be visualized with the following diagram.
(Then the diagram)
Displayed are parts of the (disjoint) sets A and B together with parts of the mappings f and g. If you interpret the set A union B together with the two maps as a directed graph, then this bipartite graph has several connected components.
These can be divided into four types: pathes extending infinitely to both directions, finite cycles of even length, infinite paths starting in the set A, and infinite paths starting in the set B (the path passing through the element a in the diagram is infinitely long into both directions, so the diagram contains one path of every type). In general it is not possible to decide in finitely many steps, what type of path a given element of A or B belongs to.
The set C defined above contains precisely the elements of A which are part of an infinite path starting in A. The map h is then defined in such a way, that for every path it yields a bijection of the elements of A onto the elements of B direct before or after it in the path. For the both-side infinite path and for the finite cycles, we choose to map every element to its predecessor in the path.
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Hope this will be of use. --SirJective 10:54, 29 Oct 2003 (UTC)
replies: text: Still hope someone might take a look at it... :-( --SirJective 19:36, 13 Jun 2004 (UTC)

text: I now copied this section into the article (and corrected some minor mistakes). The image still needs to be translated, and I will do it as soon as someone tells me the correct english words in the context of graph theory. --SirJective 15:45, 1 Aug 2004 (UTC)
thread_title: Translation of parts from german article

message: There is a supershort proof that goes like this:
Let A and B be sets and f:A->B and g:B->A be functions. Then there is a subset A0 of B such that g(B-f[A0])=A-A0.
If f and g are injective, it follows immediatly that h(x)=f(x) for x in A0 and else h(x)=g−1(x) is a bijection.
This proof has the virtue of avoiding infinite steps. The infinite steps are needed though in proving the first assertion.
(Sry bout my mathematical syntax. I have only notepad ;) —The preceding unsigned comment was added by YohanN7 (talk • contribs) 18:59, 30 March 2007 (UTC)
thread_title:  A supershort proof hiding infinite steps 

message: I tried to follow the first proof and my head started to spin, but I found the second proof very straightforward and pleasing. Is there any reason not to lead with the simpler proof, and move the harder one into a subsection? ciphergoth (talk) 22:08, 28 April 2009 (UTC)
thread_title:  Lead with the König proof? 

message: Is it really true that most sources use Cantor's name in the title?  Most of the sources I've seen personally (and most of the first few Google hits for "Schroeder Bernstein") and most of the math people with whom I've spoken have actually just said Schroeder-Bernstein.  I don't have any source that proclaims one to be more common than the other, though.  But maybe the title should be changed to reflect what appears to me to be common practice. 207.207.126.206 (talk) 00:06, 5 August 2009 (UTC)
thread_title:  Title of theorem 

message: This is from Willard (General Topology, problem 1J, 1970).
Let A be a set, P(A) the power set of A.
If F:P(A) → P(A) is a function such that whenever C ⊆ D ⇒ F(C) ⊆ F(D), then there exists E such that E = F(E).
Proof:
Let E = ∪{ C ⊆ A : C ⊆ F(C) }.
Claim: E = F(E).
Let x ∈ E.  Then x ∈ C for some C such that C ⊆ F(C) ⊆ F(E). Hence, E ⊆ F(E).
E ⊆ F(E) ⇒ F(E) ⊆ F(F(E)) ⇒ F(E) = C such that C ⊆ F(C) for some C ⇒ F(E) ⊆ E.
Cantor-Bernstein:
If A and B are sets and f:A → B and g:B → A are injective functions, then there exists h:A → B such that h is bijective.
Proof:
Define F:P(A) → P(A) as F(C) = A - g(B - f(C)).
F preserves subsets from above, so there exists E such that E = F(E).
Define h:A → B by h(x) = f(x) for x ∈ E and h(x) = g-1(x) for x ∉ E  —Preceding unsigned comment added by Quagmirede (talk • contribs) 07:31, 30 April 2011 (UTC)
thread_title:  Set-based proof? 

message: OK, here's something that has bothered me about this article for years, and I finally decided, why not bring it up?  Not making it a formal requested move, yet; just sounding people out.
I've always known this result as the Schröder–Bernstein theorem, no Cantor.  I honestly think that's the more common usage; I don't ever recall coming across this three-part name, other than on Wikipedia.
First, let me say that I yield to no one in my admiration for Cantor, one of my great mathematical heroes.  Nor do I have any problem with the axiom of choice; I'm perfectly willing to say that AC is just plain true.
But.  The Cantor version of the proof (using the wellordering principle, from Cantor's viewpoint, or the axiom of choice as we see it post-Zermelo) is completely different in character from the piecing-together-a-bijection proof that I've always associated with the name Schröder–Bernstein theorem.  The Cantor version is really just the "law of trichotomy" for cardinal numbers.  The importance of the Schröder–Bernstein proof is that the result holds even in contexts where trichotomy does not.
So really, I would prefer to move this to Schröder–Bernstein theorem (not being too picky about the umlaut-o versus oe), both on the grounds above and on WP:COMMONNAME grounds, with a mention of Cantor and a link to law of trichotomy to explain that, arguably somewhat distinct, result. --Trovatore (talk) 08:42, 13 December 2013 (UTC)
replies: text: I have never heard it called by any other name than Schröder–Bernstein theorem, and would prefer that as the article title. —Kusma (t·c) 10:13, 13 December 2013 (UTC)
replies: text: +1 — Arthur Rubin  (talk) 14:05, 13 December 2013 (UTC)
replies: text: It will always be Schroder-Bernstein for me, but I am a bit of a traditionalist in these matters. Bill Cherowitzo  (talk) 14:34, 13 December 2013 (UTC)
replies: text: I did a few Google searches and found Cantor–Bernstein–Schroeder, Cantor–Bernstein and Schroeder–Bernstein all appearing fairly often. Schroeder–Bernstein was used in what I thought were the more authoritative sources, e.g. Halmos' Naive Set Theory (book). The root problem is that there are really two theorems: AC → (m≤n≤m iff m=n) which would be Cantor's theorem if the name weren't taken already, and (m≤n≤m iff m=n) independent of AC which (so I gather) would be the Schroeder–Bernstein theorem. But based on the common usage  criterion it appears that Schroeder–Bernstein would be the best name for the article. --RDBury (talk) 20:06, 13 December 2013 (UTC)
replies: text: By the way, some information on the history of this theorem is available here (collected by Peter Schmitt). Boris Tsirelson (talk) 18:50, 14 December 2013 (UTC)
replies: text: So it should really be the Dedekind-Cantor-Schröder-Bernstein-Dedekind-(Borel) theorem :) --RDBury (talk) 19:59, 14 December 2013 (UTC)
replies: text: Indeed! It reminds me Talk:Carathéodory's_extension_theorem#Attribution:_Carath.C3.A9odory_or_not.3F Likewise, windows (on the display) are invented not by Microsoft, nor by Apple. :-) Boris Tsirelson (talk) 20:15, 14 December 2013 (UTC)

text: I too learned it as Schröder–Bernstein. I had a suspicion this might have come about through some remixing of the theorem statement, and it looks like RDBury's research confirmed that above. Considering that most people only expect the Schröder–Bernstein version, it seems like the best outcome would be to retitle this page to Schröder–Bernstein and mention the alternative names in the lead. Rschwieb (talk) 14:59, 16 December 2013 (UTC)
replies: text: I agree with this.  I checked a few other well-known textbooks beyond Halmos (namely, Lang, Algebra; Munkres, Topology).  All call it the Schröder–Bernstein theorem, sometimes with oe instead of ö, and this is also what I learned to call it.  I would prefer the umlaut version, assuming that that is how Schröder wrote his own name. Ebony Jackson (talk) 04:47, 17 December 2013 (UTC)
replies: text: OK, let's move it, already.  We aren't interested in the name reflecting the history, but only in the name reflecting what it's really called.  (I prefer the umlaut, also, even though I may not type it.)  The only problem I see is that we have a disambiguation page at Schroeder–Bernstein theorem, not all of which are related to this theorem.  — Arthur Rubin  (talk) 01:54, 4 May 2014 (UTC)
replies: text:  - page moved. That has left a lot of double redirects, but a bot should fix them within a few hours. i will check tomorrow that it has. JohnCD (talk) 17:57, 26 October 2014 (UTC)
thread_title:  Article name 

message: I have merged the section Cantor–Bernstein–Schroeder theorem#History with the german article :de:Satz von Cantor-Bernstein-Schröder#Geschichte, including as much open-access URLs and providing as much particular page numbers as I could find out. I suggest to replace the current version of section "History" by the merge result, or by a further improved version of it. - Jochen Burghardt (talk) 16:30, 2 February 2014 (UTC)
replies: text: Looks good! Boris Tsirelson (talk) 17:20, 2 February 2014 (UTC)
replies: text:  Moved to the article's section today. - Jochen Burghardt (talk) 16:48, 12 March 2014 (UTC)
thread_title:  Merging History section with de.wikipedia 

message: About "citation needed" in Sect. "Proof": A and B may be made disjoint by a well-known simple trick: we consider A \times \{1\} = \{ (a,1) : a \in A \} instead of A; we use the evident bijective correspondence between A and A \times \{1\}; and similarly, B \times \{2\} = \{ (b,2) : b \in B \} instead of B; these new sets are disjoint.
About "citation needed" in the lead: the proof mentioned above does not use the choice axiom.
Boris Tsirelson (talk) 20:33, 22 November 2016 (UTC)
thread_title:  Doubts of 1.52.104.114 

message: There are some problems with this section (but with many interesting references and links anyway).
The proof of Schröder with is shown faulty by Korselt is the one published in 1898 (and sent in 1896). There is no other proof in 1897 : Schröder aknowledges (in a letter to Korselt, may 1902, see Korselt paper)  "Daß ich Herrn F. Bernstein die Ehre, den G. Cantorschen Satz bewiesen zu haben, allein überlasse, hatte ich einstweilen einem Freunde desselben, Herrn Dr. Max Dehn (jetzt in Münster) schon vorigen Herbst resp. Sommer -- natürlich zum Weitergeben -- gesagt" (that is Bernstein is the only one to be credited for the proof, and he says that to a friend in 1901, Schröder died in june 1902).
 
Deiser (given as reference) does not say anything about Schröder and the theorem p 501 (and in other pages too) ! Suppes (1972) is not reliable on historical matters (it was overlooked a long time that Schröder's proof was faulty). All that is known today (see for instance this review (p 164) by William W. Tait).
The "Proof sketch" of 1896 : the only given reference is Schröder 1898 (manuscript sent in 1896), nothing about this proof sketch and about Jevons, it seems to me.
Zermelo sent his proof to Poincaré in 1906 (who published a translation), no evidence that he knew Dedekind's proof at this time, and even in 1908,  Deiser p 501 says the opposite (and other historians too). Proz (talk) 20:55, 1 February 2017 (UTC)
thread_title:  History 

message: Hi,
the 2nd link on the page when I searched for this was Wolfram and it said 
"The Schröder-Bernstein theorem for numbers states that if n<=m<=n, then m=n"
To put this at the top and say "the equivalent for sets involves their sizes" (or perhaps something a bit more rigorous) would make this immediately understandable and intuitive for me. It's been 30 years since I did injective/bijective funcs, so the lead-in description left me floundering.
79.75.109.121 (talk) 18:10, 26 November 2017 (UTC)
thread_title:  Clarification from Wolfram 

message: (from the same guy who posted chunk immediately above).
I only recently came across intuitionist logic and as someone with a background in classical logic, it was a very strange thing... but anyway. 
In the article it says "However, its various proofs are non-constructive, as they depend on the law of excluded middle, and are therefore rejected by intuitionists".
If we consider only finitely-sized sets then could you not ask an intuitionist to provide a counterexample? Isn't the onus on them to support their claim in this limited domain? 
79.75.109.121 (talk) 18:15, 26 November 2017 (UTC)
replies: text: For intuitionists, some things are true for finite sets which they do not consider to be true for infinite sets. And I suspect that may be so in this case. In any event, since they reject the excluded middle as a general proposition, they would not feel that merely failing to accept the truth of Schröder–Bernstein is the same as believing it to be false. So they would not accept the onus of giving a counter-example. JRSpriggs (talk) 22:38, 26 November 2017 (UTC)
thread_title:  Intuitionistic rejection 

message: A couple of days ago, I added the small section on Statement C and a transparent proof. 
This proof is for the one-sided version of the Schröder-Bernstein Theorem, where the second injection is the identity and (hence) the second set a subset of the first one. That this trivially implies the general form is also shown.
The one-sided proof can be found by translating a proof (either Tarski-Knaster or König - which really is only Tarski-Knaster in disguise) for the general case to this situation, but that was not how I found it. Rather, it imposes itself via two well-known examples. The first one is the so-called Hilbert Hotel: the set of all natural numbers (including 0) is equipollent with the set of positive natural numbers. The second one is the fact that the closed unit interval [0,1] of the reals is equipollent with the half-open one [0,1), with the number 1 left out. Solutions for these cases straightforwardly generalise to the general case, which, I think, is more or less trivial once you understand these examples.
I now stumbled upon the french wiki page for the theorem (there called the Cantor-Bernstein Theorem). To my surprise, three proofs are presented. Next to the one using the Tarski-Knaster fixpoint-trick (to which this page only offers a link) and the König proof, there is the one-sided version that I had just added here. (I'm told now that the polish wiki has two proofs: from the three, it leaves out König.)
August 11, 2018. KeesDoe (talk) 08:43, 11 August 2018 (UTC)
replies: text: I just initiated a discussion on wikiproject math. Boris Tsirelson (talk) 10:18, 11 August 2018 (UTC)
thread_title:  On "Statement C and a transparent proof" 

message: This theorem establishes the anti-symmetry property for cardinality. Anti-symmetry is a cardinal (pun intended) notion within Set Theory, and should be explicitly acknowledged when it occurs. If I do a google search for 'anti-symmetry of cardinality' I should get a hit on this article.
 
Here's the link to a site that does what I'm suggesting:
 
https://www.whitman.edu/mathematics/higher_math_online/section04.09.html Kontribuanto (talk) 11:54, 23 February 2024 (UTC)
replies: text: I just performed the google search for "anti-symmetry of cardinality", and found only one hit (https://www.scribd.com/document/520363930/Mathematics-Mind-Map). Apparently nobody uses this term literally, not even the article at whitman.edu you cited. - Jochen Burghardt (talk) 15:29, 23 February 2024 (UTC)
thread_title:  explicit reference to standard notion needed