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# Talk:Schröder-Bernstein theorem/Draft

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 Definition:  A classic theorem of set theory asserting that sets can be ordered by size. [d] [e]

"Details": "the induced induced image"? Boris Tsirelson 06:29, 26 September 2010 (UTC)

Thanks. Corrected. --Peter Schmitt 12:26, 28 September 2010 (UTC)

"Proof:Proof":



probably should be


Oops -- correct but not what is needed. --Peter Schmitt 23:37, 5 October 2010 (UTC)

"Monotone" in general may be understood as "either increasing or decreasing"; it is meant "(momotonely) increasing" or "isotone".

Yes, that was negligent. --Peter Schmitt 23:37, 5 October 2010 (UTC)

"Proof:Proof":

"By assumption, there are injective functions (...) that induce..."

I'd say

"By assumption, there are injective functions (...); they induce..."

because the second part of the phrase is not a part of the assumption (but its consequence).

True -- that is better. --Peter Schmitt 23:37, 5 October 2010 (UTC)

"Outline": the reader can guess what is denoted by f and g (or see the details), but we'd better let him know.

I forgot that I did not introduce it before. --Peter Schmitt 23:37, 5 October 2010 (UTC)

"Details": probably also (4) is needed, explaining what are A2, B1 and B2 (which is easy) and why B1 is the image of A1 under f and A2 is the image of B2 under g (which is less easy).

I was lazy -- I thought this is "obvious". --Peter Schmitt 23:37, 5 October 2010 (UTC)

Boris Tsirelson 12:19, 5 October 2010 (UTC)

Done. Thanks. --Peter Schmitt 23:37, 5 October 2010 (UTC)
It seems, some new "decreasing" should rather be "increasing". Boris Tsirelson 13:18, 6 October 2010 (UTC)
Fixed. My only excuse is that it was very late and the mental image was the decreasing sequence produced by σ starting at A. --Peter Schmitt 09:14, 7 October 2010 (UTC)

## More remarks

"Outline": "This defines a mapping of subsets of A to subsets of A that is monotone" — rather, increasing.

"Details": "(2) σ is a monotone function" — rather, increasing.

"Details": "" — rather, .

"Details": "that has the desired properties:" — either a continuation will follow, or the colon should be a fullstop.

Done. --Peter Schmitt 23:52, 13 October 2010 (UTC)

In addition, some commas and fullstops after displays are missing. Boris Tsirelson 17:33, 12 October 2010 (UTC)

This is purpose. I know that it typographically it is not correct. But I find both versions (punctuation inside or outside the display) irritating. I tried to avoid this situation but haven't always. I am not sure what to do ... --Peter Schmitt 23:52, 13 October 2010 (UTC)

## Toward approval

Nominated. Boris Tsirelson 08:49, 14 October 2010 (UTC)

This article is about the Schröder-Bernstein theorem, yet I can't help noticing the lack of either Schröder or Bernstein in the text. Presumably that refers to two people who "discovered" this or however you put it, so there must be some element of history in there that is lacking in the article itself. The Schröder-Bernstein property article is the same - lots about what it is, none about where it came from or how it got named. David Finn 09:46, 14 October 2010 (UTC)

You are right, David, and I am currently researching what is known about the history. I agree that, in general, it is nice to have a historical section but, on the other hand, missing it is not a crucial gap if the mathematical treatment is adequate. In any case, I think it is better to have nothing on the history than something superficial and incomplete. --Peter Schmitt 13:06, 14 October 2010 (UTC)
Peter, don't you think that an article to be approved should have at least one reference? Milton Beychok 17:53, 14 October 2010 (UTC)
About the history, here is a quote from
Bourbaki, Nicolas (1984 (original), 1994 (translation)), Elements of the history of mathematics, Masson (original), Springer (translation). Page 28:
... Cantor had been unable to establish the existence of a well ordering between arbitrary cardinals. This gap was going to be filled, on the one hand by the theorem of F. Bernstein (1897) showing that the relations ab and ba imply a=b,47 ...
and a footnote 47:
This theorem had already been obtained by Dedekind in 1887, but its proof was not published ([79], v. III, p. 447).
Boris Tsirelson 19:14, 14 October 2010 (UTC)
About a reference: probably the given proof can be found in
Thomas Forster, Logic, induction and sets, Cambridge University Press, Cambridge, 2003.
I did not see the book, but it is mentioned in PlanetMath. Boris Tsirelson 19:40, 14 October 2010 (UTC)
A better ref: "Schaum's outline of theory and problems of Boolean algebra and switching circuits" by Elliott Mendelson, page 200, see Google book. Boris Tsirelson 19:48, 14 October 2010 (UTC)
An ocean of sources about the history (pointed by WP): Papers on the history of the Cantor-Bernstein theorem. Boris Tsirelson 19:51, 14 October 2010 (UTC)
Boris, thank you for the references. I did not intend to say that I don't know where to look, but I do want to check and cite as many of the original sources as possible.
Milt, I did not nominate the article -- you will have to ask Boris :-) Seriously, I do not think that (inline) references are needed, but there are already two papers on the theorem and its history in the bibliography, and I'll add the original sources.
--Peter Schmitt 21:09, 14 October 2010 (UTC)
Because of the interest :-) I have added what I have ready so far -- needs some more work. --Peter Schmitt 23:34, 14 October 2010 (UTC)
Nice, but please also let me know where can I find these facts (for verification). Boris Tsirelson 08:00, 15 October 2010 (UTC)
The sources are in the Bibliography. The original sources are, however, not yet complete. --Peter Schmitt 20:38, 15 October 2010 (UTC)

I see.

"Cantor is often added because he first stated the theorem in 1895" — really, 1895? I has absolutely no opinion about this, but looking at Deiser's book I see on page 71: "Cantor hatte diesen Satz bereits 1883 formuliert". Another source says: "in Cantor's letter to Dedekind of November 5, 1882 Cantor stated the Cantor-Bernstein theorem". Maybe, 1895 is rather the year of the first publication of Cantor's "prediction"? Also, "stated the theorem" is not very clear; it should be clear that he formulated it without proof (and therefore it was not yet a theorem but rather a conjecture).

Dates like "(11 July 1887)" seem to me too detailed for this article; isn't the year enough?

"while Schröder's name is often omitted because his proof was not correct" — after seeing some sources I think maybe the phrase could be continued as "..., and Dedekind's name is omitted because his proof was published much later". And "1887 Richard Dedekind proved it in unpublished notes" could continue "to be published only in 1932". Boris Tsirelson 16:13, 16 October 2010 (UTC)

I know that the text needs "polishing".
As to Cantor 1882/1883: I have read this, too, and I try to find out more about it. (Cantor may have considered it proven because it follows from his theory of ordinal numbers (needing the Axiom of Choice).
Perhaps I should continue to include König, Zermelo, Peano.
--Peter Schmitt 00:12, 20 October 2010 (UTC)
I have now added some material and partially rewritten the text. While some details may be added later, it is now reasonably satisfactory, I believe. --Peter Schmitt 13:49, 20 October 2010 (UTC)
I agree. Boris Tsirelson 14:28, 20 October 2010 (UTC)

This version of this article is set to be locked on October 20. All seems to be in line for this to happen, unless someone moves the date or removes the ToApproval notice. D. Matt Innis 17:00, 18 October 2010 (UTC)

Right. But I am watching, see User talk:Peter Schmitt#The approval, and shall act according to the situation. Boris Tsirelson 18:31, 18 October 2010 (UTC)
Good to see you know what you are doing. I don't have to help you guys anymore! D. Matt Innis 18:40, 18 October 2010 (UTC)
The newer version (of 13:45, 20 October 2010) is set to be locked on October 21. We should be thankful to Peter for the "History" section. Boris Tsirelson 14:28, 20 October 2010 (UTC)
Excellent. I will return tomorrow to lock this version. D. Matt Innis 15:44, 20 October 2010 (UTC)
Tomorrow, you'll have more urgent things to do -- counting ballots ;-) --Peter Schmitt 17:43, 20 October 2010 (UTC)

Congratulations All, another good collaborative effort! Thanks, a good way to start election day! D. Matt Innis 13:56, 21 October 2010 (UTC)

## Reaching a larger audience

As an encyclopaedia, this would be more relevant to a greater number of people if the symbols were spelled out in plain English. Many non-math people do not know what they symbols mean and a simple English version would be invaluable. This could be done by posting footnotes with each mathematical statement and thereby avoid complicating the text in the main article itself. Thomas Simmons 23:18, 18 November 2010 (UTC)

(1) Have you any example of such approach, either here or in Wikipedia?
(2) How many such footnotes would you like to insert? About 5? About 20?
(3) Symbols can be spelled out by words, but these words are quite special terms; should their meaning be also explained here?
(4) It is reasonable to expect that a man wishing to read such an article either has already the needed background, or is ready to prepare himself by reading some other articles or, if needed, some texbooks. We all know that, depending on the case, it may take years of time and tons of textbooks....
Boris Tsirelson 05:59, 19 November 2010 (UTC)
No Boris, it is not reasonable. This is an encyclopaedia. If it takes large, complex, advanced math skills to read the article, why read the article? There are better places to get the information as it stands and those who understand the subject matter already would not care to waste their time. This is a place to inform and teach, not pose and obfuscate. I am rereading Weinberg's The First Three Minutes. HIs whole purpose was to take a complex subject and make it accessible. He did so quite well, because, as Feynman would have opined, he understood the subject matter. FYI, we are here to do it better than Wikipedia. Thomas Simmons 22:07, 19 November 2010 (UTC)
If you suggest to explain symbols, then please give an example from CZ or WP showing that it is feasible.
But if you rather suggest us to restrict ourselves to such masterpieces of popular science, then at least (a) drive out all authors less brilliant than Weinberg, and (b) delete all articles on topics less exciting than the birth of the Universe. Boris Tsirelson 15:44, 20 November 2010 (UTC)
Not at all sure why this has taken this particular turn. Suggesting a read out of the sentences written in one language in a more accessible language - in this case English - is not unreasonable or unusual.
• 1+1=2 is read "one plus one equals two";
• : "The intersection of set A with set B equals".
etc.
The suggestion and the request actually, is that these articles be made accessible by rendering them less opaque. I can not see why some engineer from the JPL or a theoretical physicist from MIT's CTP would be reading these articles except to check their accuracy. We are attempting to reach the secondary and post-secondary level and possibly - here is a wild thought - those who have an interest but never made it past elementary school (of which there are many). Ever wonder why such books as the "XYZ for Dummies" series sell so well? No one explains this stuff and after a certain level - which just whizzes by for most - it is lost to us. What is the purpose of an encyclopaedia but to inform. There are many levels with which this information may be provided. It is feasible. Maybe the cultural divide is too wide to explain it here but people of note such as Richard Feynmann and Steven Weinberg thought it worth while.
As for the crack about driving out those less brilliant than Weinberg, and avoiding topics less exciting, that is simply not what this is about and does not serve to advance this discussion. If it is not what someone can do, then they need not bother.
The suggestion has been made, the examples given and the rationale articulated. Thomas Simmons 18:49, 20 November 2010 (UTC)
Maybe... I did not write this article, I only approved it; Peter wrote it. If he is enthusiastic to rewrote it, I'll be glad. But I cannot agree that "the suggestion has been made, the examples given". Once again: the example does not meet the suggestion. Worse, the example differs from the suggestion by many orders of magnitude. "First three minutes" differs from this article not by notation explained. Not at all.
Another question to you. The article Oxytocin was approved recently. Do you believe it is accessible for me? Yes, I can try to find out what are "magnocellular neurosecretory cells", "supraoptic nucleus", "paraventricular nucleus of the hypothalamus" and "the posterior lobe of the pituitary gland", but really, it will give me very little, unless I'll learn these matters during years. So, why do you expect this article to be more accessible than that one? And also more accessible than most other math articles in CZ? Boris Tsirelson 19:05, 20 November 2010 (UTC)
Well, A) Rewriting is not necessary and was never suggested so this is inferring what was never implied. B) Examples are given and they are appropriate to explain what is being suggested, commonly used and certainly feasible. "The First Three Minutes" and other works, and say Feynmann's Physics lectures, do an excellent job of trying to reach a simpler level of understanding and are certainly a models for our use here even if those model can be further extended. C) I have no idea what any reader may understand about Oxytocin. That is an underlying aspect of learning to inform on many levels. D) And last of all, I have not targeted a specific article - clearly this is a suggestion that can be utilised in many articles as time and talent permits. It is important to be able to see how a principle can be applied. If that is not apparent then the suggestion can not be fully understood. The principle is this, make things simpler when possible to reach a broad base of readers. It has been suggested before that articles be graded but it would not be at all necessary to rewrite an article. Given the convenience of hypertext, these explanations can be imbedded and the reader can simply click on them to go to a more in-depth explanation. Thomas Simmons 20:01, 20 November 2010 (UTC)
This sounds like a good idea, Thomas. If you're feeling motivated and a little bored, why not jump in and do that? This is the draft and probably a good place to experiment with the idea. D. Matt Innis 20:29, 20 November 2010 (UTC)
Yes, I feel it is about math, not about this article. In fact, sometimes I do my best; see Space (mathematics) and Theory (mathematics). But those are relatively lightweight surveys; they do not prove, just inform. The "Schröder-Bernstein theorem" is a narrow topic; there is nothing to survey here. It is possible to comment much more on the formulation of the theorem, giving examples. The proof is necessarily technical. Frankly, I do not know, whether proofs should be included at all (rather than delegated to the "proof" subpage).
I frankly believe that in order to make it essentially better one must rewrite it. But if anyone wants to try local improvement, I wish him every success. There is a link to Set theory; most notations are explained there. Also WP should be helpful; it contains several proofs; usually each proof is preferred by some readers, tastes differ. Boris Tsirelson 21:04, 20 November 2010 (UTC)
I have a few ideas. I have been looking for similar attempts and not surprisingly found a number of sites that do provide this for the blind. It may be possible to redirect hypertext links to another page specifically for that article or to build a subpage. Thomas Simmons 02:13, 21 November 2010 (UTC)
Happy editing! Boris Tsirelson 09:13, 21 November 2010 (UTC)

## Used by Wikipedia

"Peter Schmitt contributed the History section to Citizendium which TakuyaMurata copied into this article" - from [1]. --Boris Tsirelson 12:30, 18 January 2011 (UTC)

Wow -- a reference by name instead of "from CZ" :-) --Peter Schmitt 00:24, 20 January 2011 (UTC)
Peter, a question -- where should we be discussing continuing developments with WP-CZ? Unfortunately from my perspective, discussion areas other than motion-specific ones have been discouraged on the EC Wiki -- except sometimes. Nevertheless, I saw this just as I was considering how to continue discussion of these important topics in a constructive environment. Howard C. Berkowitz 00:36, 20 January 2011 (UTC)