Talk:Schröder-Bernstein theorem/Draft

"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":

$S_{1}\subset S\subset A\Rightarrow f_{\ast }(S_{1})\subset f_{\ast }(S)$

probably should be

$S_{1}\subset S\subset A\Rightarrow \sigma _{\ast }(S_{1})\subset \sigma _{\ast }(S)$

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": "$A_{2}:=A\setminus A_{2}$" — rather, $A\setminus A_{1}$.

"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 a≤b and b≤a 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)

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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)