Talk:Schröder-Bernstein theorem/Draft: Difference between revisions

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imported>Peter Schmitt
imported>Boris Tsirelson
(→‎Toward approval: where can I find these facts)
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:: --[[User:Peter Schmitt|Peter Schmitt]] 21:09, 14 October 2010 (UTC)
:: --[[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)
:: 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)

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