User:Boris Tsirelson/Sandbox1: Difference between revisions
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Then the statement fails badly: every triangle ''X'' evidently is similar to some triangle inside ''Y'', and the other way round; however, ''X'' and ''Y'' need no be similar. | Then the statement fails badly: every triangle ''X'' evidently is similar to some triangle inside ''Y'', and the other way round; however, ''X'' and ''Y'' need no be similar. | ||
==Notes | ==Notes== | ||
{{reflist}} | {{reflist}} | ||
==References== | |||
{{Citation | |||
| last = Borel | |||
| first = Émile | |||
| title = Probabilities and life | |||
| year = 1962 | |||
| publisher = Dover publ. (translation) | |||
}}. | |||
{{Citation | |||
| last = Bourbaki | |||
| first = Nicolas | |||
| title = Elements of mathematics: Theory of sets | |||
| year = 1968 | |||
| publisher = Hermann (original), Addison-Wesley (translation) | |||
}}. | |||
{{Citation | |||
| year = 1992 | |||
| editor-last = Casacuberta | |||
| editor-first = C | |||
| editor2-last = Castellet | |||
| editor2-first = M | |||
| title = Mathematical research today and tomorrow: Viewpoints of seven Fields medalists | |||
| series = Lecture Notes in Mathematics | |||
| volume = 1525 | |||
| publisher = Springer-Verlag | |||
| isbn = 3-540-56011-4 | |||
}}. | |||
{{Citation | |||
| last = Feynman | |||
| first = Richard | |||
| author-link = Richard Feynman | |||
| title = The character of physical law | |||
| edition = twenty second printing | |||
| year = 1995 | |||
| publisher = the MIT press | |||
| isbn = 0 262 56003 8 | |||
}}. | |||
{{Citation | |||
| year = 2008 | |||
| editor-last = Gowers | |||
| editor-first = Timothy | |||
| title = The Princeton companion to mathematics | |||
| publisher = Princeton University Press | |||
| isbn = 978-0-691-11880-2 | |||
}}. | |||
{{Citation | |||
| last1 = Lawvere | |||
| first1 = F. William | |||
| last2 = Rosebrugh | |||
| first2 = Robert | |||
| title = Sets for mathematics | |||
| year = 2003 | |||
| publisher = Cambridge University Press | |||
| isbn = 0-521-80444-2 | |||
}}. | |||
{{Citation | |||
| last = Mathias | |||
| first = Adrian | |||
| year = 2002 | |||
| title = A term of length 4,523,659,424,929 | |||
| journal = Synthese | |||
| volume = 133 | |||
| issue = 1/2 | |||
| pages = 75–86 | |||
| url = http://www.springerlink.com/content/x28504221108023t/ | |||
}}. | |||
(Also [http://personnel.univ-reunion.fr/ardm/inefff.pdf here].) |
Revision as of 13:35, 1 September 2010
The general idea of the Cantor–Bernstein–Schroeder theorem and related results may be formulated as follows. If X is similar to a part of Y and at the same time Y is similar to a part of X then X and Y are similar. In order to be specific one should decide
- what kind of mathematical objects are X and Y,
- what is meant by "a part",
- what is meant by "similar".
In the classical Cantor–Bernstein–Schroeder theorem
- X and Y are sets (maybe infinite),
- "a part" is interpreted as a subset,
- "similar" is interpreted as equinumerous.
Not all statements of this form are true. For example, let
- X and Y are triangles,
- "a part" means a triangle inside the given triangle,
- "similar" is interpreted as usual in elementary geometry: triangles related by a dilation (in other words, "triangles with the same shape up to a scale factor", or equivalently "triangles with the same angles").
Then the statement fails badly: every triangle X evidently is similar to some triangle inside Y, and the other way round; however, X and Y need no be similar.
Notes
References
Borel, Émile (1962), Probabilities and life, Dover publ. (translation).
Bourbaki, Nicolas (1968), Elements of mathematics: Theory of sets, Hermann (original), Addison-Wesley (translation).
Casacuberta, C & M Castellet, eds. (1992), Mathematical research today and tomorrow: Viewpoints of seven Fields medalists, Lecture Notes in Mathematics, vol. 1525, Springer-Verlag, ISBN 3-540-56011-4.
Feynman, Richard (1995), The character of physical law (twenty second printing ed.), the MIT press, ISBN 0 262 56003 8.
Gowers, Timothy, ed. (2008), The Princeton companion to mathematics, Princeton University Press, ISBN 978-0-691-11880-2.
Lawvere, F. William & Robert Rosebrugh (2003), Sets for mathematics, Cambridge University Press, ISBN 0-521-80444-2.
Mathias, Adrian (2002), "A term of length 4,523,659,424,929", Synthese 133 (1/2): 75–86. (Also here.)