View source for Talk:Schröder-Bernstein theorem/Draft

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"Details": "the induced induced image"? [[User:Boris Tsirelson|Boris Tsirelson]] 06:29, 26 September 2010 (UTC)
: Thanks. Corrected. --[[User:Peter Schmitt|Peter Schmitt]] 12:26, 28 September 2010 (UTC)

"Proof:Proof":
: <math>  S_1 \subset S \subset A
          \Rightarrow f_\ast (S_1) \subset f_\ast (S)
  </math>
probably should be
: <math>  S_1 \subset S \subset A
          \Rightarrow \sigma_\ast (S_1) \subset \sigma_\ast (S)
  </math>
 
:: Oops -- correct but not what is needed. --[[User:Peter Schmitt|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. --[[User:Peter Schmitt|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. --[[User:Peter Schmitt|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. --[[User:Peter Schmitt|Peter Schmitt]] 23:37, 5 October 2010 (UTC)

"Details": probably also (4) is needed, explaining what are ''A''<sub>2</sub>, ''B''<sub>1</sub> and ''B''<sub>2</sub> (which is easy) and why ''B''<sub>1</sub> is the image of ''A''<sub>1</sub> under ''f'' and ''A''<sub>2</sub> is the image of ''B''<sub>2</sub> under ''g'' (which is less easy).

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

[[User:Boris Tsirelson|Boris Tsirelson]] 12:19, 5 October 2010 (UTC) 

:: Done. Thanks. --[[User:Peter Schmitt|Peter Schmitt]] 23:37, 5 October 2010 (UTC)

:::It seems, some new "decreasing" should rather be "increasing". [[User:Boris Tsirelson|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 &sigma; starting at ''A''. --[[User:Peter Schmitt|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": "<math>A_2 := A \setminus A_2</math>" — rather, <math>A \setminus A_1</math>.

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

:: Done. --[[User:Peter Schmitt|Peter Schmitt]] 23:52, 13 October 2010 (UTC)

In addition, some commas and fullstops after displays are missing. [[User:Boris Tsirelson|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 ...  --[[User:Peter Schmitt|Peter Schmitt]] 23:52, 13 October 2010 (UTC)

== Toward approval ==

Nominated. [[User:Boris Tsirelson|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. [[User:David Finn|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. --[[User:Peter Schmitt|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? [[User:Milton Beychok|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'',<sup>47</sup> ...
: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).
:[[User:Boris Tsirelson|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 [http://planetmath.org/?op=getobj&from=objects&name=ProofOfSchroederBernsteinTheoremUsingTarskiKnasterTheorem PlanetMath]. [[User:Boris Tsirelson|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 [http://books.google.com/books?id=cT8mLaDpjPQC&pg=PA200&dq=Schr%C3%B6der-Bernstein+theorem&hl=en&ei=7Fy3TNjGL8TNswbd7cCSCw&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDIQ6AEwAg#v=onepage&q&f=false Google book]. [[User:Boris Tsirelson|Boris Tsirelson]] 19:48, 14 October 2010 (UTC)

:An ocean of sources about the history (pointed by WP): [http://www.hinkis.org/HTML_pages/CBT_papers.html Papers on the history of the Cantor-Bernstein theorem]. [[User:Boris Tsirelson|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. 
:: --[[User:Peter Schmitt|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. --[[User:Peter Schmitt|Peter Schmitt]] 23:34, 14 October 2010 (UTC)
:::Nice, but please also let me know where can I find these facts (for verification). [[User:Boris Tsirelson|Boris Tsirelson]] 08:00, 15 October 2010 (UTC)