Talk:Cantor's first set theory article/Archive 1: Difference between revisions

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Older discussion is archived at [[/Archive1]]

==The rationals==

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There's an explicit exercise in [[Walter Rudin]]'s ''Principles of Mathematical Analysis'' that asks the student to show for any rational number less than √2 how to find a larger rational number that is still less than √2, and similarly for those larger than √2. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 02:04, 24 January 2010 (UTC)

: For <math>a\lt<\sqrt2</math>, one possible choice would be <math>a+1/\lceil1/(\sqrt2-a)\rceil</math>. [[User:Paradoctor|Paradoctor]] ([[User talk:Paradoctor|talk]]) 13:27, 16 December 2013 (UTC)

== Proposed Changes to Article ==

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I have also added a "Notes" section, and I have added references to the current "References" section.

I highly recommend reading Cantor's original article, which is at: [http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=266194 "Über eine Eigenschaft des Ingebriffes aller reelen algebraischen Zahlen"]. A French translation (which was reviewed and corrected by Cantor) is at: [http://www.springerlink.com/content/37030699752l2573/fulltext.pdf "Sur une propriété du système de tous les nombres algébriques réels"]. Unfortunately, I have not found an English translation on-line. However, an English translation is in: Volume 2 of Ewald's ''From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics'' ({{ISBN |9780198532712}}).

Most of the material I added to this Wikipedia article comes from Cantor's article, Cantor's correspondence, Dauben's biography of Cantor ({{ISBN |0674348710}}), and the article [http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2907 "Georg Cantor and Transcendental Numbers"].

Finally, I wish to thank all the people who have worked on this Wikipedia article. Without the excellent structuring of your article and the topics you chose to cover, I suspect that I would not have written anything. (This is the first time I've written for Wikipedia.) It's much easier to add and revise rather than develop from scratch. [[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 23:30, 5 May 2009 (UTC)

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! colspan="10" | '''Height 1:'''

|-

| '''1.''' || &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; || || ||align="right" || {{color|#c0c0c0|+1·}}''x'' || {{color|#c0c0c0|+ 0}} || = 0 || &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; || ''x''₁ = 0

|-

! colspan="10" | '''Height 2:'''

|-

|-

| || || || ||align="right" || {{color|#c0c0c0|+}}2·''x'' || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | coprimereducible

|-

| '''2.''' || || || ||align="right" || {{color|#c0c0c0|+1·}}''x'' || +1 || = 0 || || -1''x''₂ = −1

|-

| '''3.''' || || || ||align="right" || {{color|#c0c0c0|+1·}}''x'' || -1−1 || = 0 || || ''x''₃ = +1

|-

| || || ||align="right" || {{color|#c0c0c0|+1·}}''x''² || {{color|#c0c0c0|+0·''x''}} || {{color|#c0c0c0|+ 0}} || = 0 || || notcolspan="2" | irreduciblereducible

|-

! colspan="10" | '''Height 3:'''

|-

| || || || ||align="right" || {{color|#c0c0c0|+}}3·''x'' || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | coprimereducible

|-

| '''5.''' || || || || align="right" || {{color|#c0c0c0|+}}2·''x'' || +1 || = 0 || || -1''x''₅ = −1/2

|-

| '''6.''' || || || ||align="right" || {{color|#c0c0c0|+}}2·''x'' || -1−1 || = 0 || || ''x''₆ = +1/2

|-

| '''4.''' || || || ||align="right" || {{color|#c0c0c0|+1·}}''x'' || +2 || = 0 || || -2''x''₄ = −2

|-

| '''7.''' || || || ||align="right" || {{color|#c0c0c0|+1·}}''x'' || -2−2 || = 0 || || ''x''₇ = +2

|-

| || || ||align="right" || {{color|#c0c0c0|+}}2·''x''² || {{color|#c0c0c0|+0·''x''}} || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | coprimereducible

|-

| || || ||align="right" || {{color|#c0c0c0|+1·}}''x''² || +{{color|#c0c0c0|1·}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | irreduciblereducible

|-

| || || ||align="right" || {{color|#c0c0c0|+1·}}''x''² || -−{{color|#c0c0c0|1·}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | irreduciblereducible

|-

| || || ||align="right" || {{color|#c0c0c0|+1·}}''x''² || {{color|#c0c0c0|+0·''x''}} || +1 || = 0 || || colspan="2" | no real root

|-

| || || ||align="right" || {{color|#c0c0c0|+1·}}''x''² || {{color|#c0c0c0|+0·''x''}} || -1−1 || = 0 || || notcolspan="2" | irreduciblereducible

|-

| || ||align="right" || {{color|#c0c0c0|+1·}}''x''³ || {{color|#c0c0c0|+0·''x''²}} || {{color|#c0c0c0|+0·''x''}} || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | irreduciblereducible

|-

! colspan="10" | '''Height 4:'''

|-

| || || || ||align="right" || {{color|#c0c0c0|+}}4·''x'' || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | coprimereducible

|-

| '''13.''' || || || ||align="right" || {{color|#c0c0c0|+}}3·''x'' || +1 || = 0 || || -1''x''₁₃ = −1/3

|-

| '''14.''' || || || ||align="right" || {{color|#c0c0c0|+}}3·''x'' || -1−1 || = 0 || || ''x''₁₄ = +1/3

|-

| || || || ||align="right" || {{color|#c0c0c0|+}}2·''x'' || +2 || = 0 || || notcolspan="2" | coprimereducible

|-

| || || || ||align="right" || {{color|#c0c0c0|+}}2·''x'' || -2−2 || = 0 || || notcolspan="2" | coprimereducible

|-

| '''8.''' || || || ||align="right" || {{color|#c0c0c0|+1·}}''x'' || +3 || = 0 || || -3''x''₈ = −3

|-

| '''19.''' || || || ||align="right" || {{color|#c0c0c0|+1·}}''x'' || -3−3 || = 0 || || ''x''₁₉ = +3

|-

| || || ||align="right" || {{color|#c0c0c0|+}}3·''x''² || {{color|#c0c0c0|+0·''x''}} || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | coprimereducible

|-

| || || ||align="right" || {{color|#c0c0c0|+}}2·''x''² || +{{color|#c0c0c0|1·}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | irreduciblereducible

|-

| || || ||align="right" || {{color|#c0c0c0|+}}2·''x''² || |-−{{color|#c0c0c0|1·}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | irreduciblereducible

|-

| || || ||align="right" || {{color|#c0c0c0|+}}2·''x''² || {{color|#c0c0c0|+0·''x''}} || +1 || = 0 || || colspan="2" | no real root

|-

| '''11.,16.''' || || ||align="right" || {{color|#c0c0c0|+}}2·''x''² || {{color|#c0c0c0|+0·''x''}} || -1−1 || = 0 || || ''x''₁₆, ''x''₁₁ = ±1/√{{overline|2}}

|-

| || || ||align="right" || {{color|#c0c0c0|+1·}}''x''² || +2·''x'' || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | irreduciblereducible

|-

| || || ||align="right" || {{color|#c0c0c0|+1·}}''x''² || -2·−2''x'' || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | irreduciblereducible

|-

| || || ||align="right" || {{color|#c0c0c0|+1·}}''x''² || +{{color|#c0c0c0|1·}}''x'' || +1 || = 0 || || colspan="2" | no real root

|-

| '''9.,15.''' || || ||align="right" || {{color|#c0c0c0|+1·}}''x''² || +{{color|#c0c0c0|1·}}''x'' || -1−1 || = 0 || || ''x''₁₅, ''x''₉ = (-1−1 ± √{{overline|5}}) / 2

|-

| || || ||align="right" || {{color|#c0c0c0|+1·}}''x''² || -−{{color|#c0c0c0|1·}}''x'' || +1 || = 0 || || colspan="2" | no real root

|-

| '''12.,18.''' || || ||align="right" || {{color|#c0c0c0|+1·}}''x''² || -−{{color|#c0c0c0|1·}}''x'' || -1−1 || = 0 || || ''x''₁₈, ''x''₁₂ = (+1 ± √{{overline|5}}) / 2

|-

| || || ||align="right" || {{color|#c0c0c0|+1·}}''x''² || {{color|#c0c0c0|+0·''x''}} || +2 || = 0 || || colspan="2" | no real root

|-

| '''10.,17.''' || || ||align="right" || {{color|#c0c0c0|+1·}}''x''² || {{color|#c0c0c0|+0·''x''}} || -2−2 || = 0 || || ''x''₁₇, ''x''₁₀ = ±√{{overline|2}}

|-

| || ||align="right" || {{color|#c0c0c0|+}}2·''x''³ || {{color|#c0c0c0|+0·''x''²}} || {{color|#c0c0c0|+0·''x''}} || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | coprimereducible

|-

| || ||align="right" || {{color|#c0c0c0|+1·}}''x''³ || +{{color|#c0c0c0|1·}}''x''² || {{color|#c0c0c0|+0·''x''}} || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | irreduciblereducible

|-

| || ||align="right" || {{color|#c0c0c0|+1·}}''x''³ || -−{{color|#c0c0c0|1·}}''x''² || {{color|#c0c0c0|+0·''x''}} || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | irreduciblereducible

|-

| || ||align="right" || {{color|#c0c0c0|+1·}}''x''³ || {{color|#c0c0c0|+0·''x''²}} || +{{color|#c0c0c0|1·}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | irreduciblereducible

|-

| || ||align="right" || {{color|#c0c0c0|+1·}}''x''³ || {{color|#c0c0c0|+0·''x''²}} || -−{{color|#c0c0c0|1·}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | irreduciblereducible

|-

| || ||align="right" || {{color|#c0c0c0|+1·}}''x''³ || {{color|#c0c0c0|+0·''x''²}} || {{color|#c0c0c0|+0·''x''}} || +1 || = 0 || || notcolspan="2" | irreduciblereducible

|-

| || ||align="right" || {{color|#c0c0c0|+1·}}''x''³ || {{color|#c0c0c0|+0·''x''²}} || {{color|#c0c0c0|+0·''x''}} || -1−1 || = 0 || || notcolspan="2" | irreduciblereducible

|-

| ||align="right" || {{color|#c0c0c0|+1·}}''x''⁴ || {{color|#c0c0c0|+0·''x''³}} || {{color|#c0c0c0|+0·''x''²}} || {{color|#c0c0c0|+0·''x''}} || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | irreduciblereducible

|-

! colspan="10" | '''Height 5:'''

|-

| || || || ||align="right" || {{color|#c0c0c0|+}}5·''x'' || {{color|#c0c0c0|+0}} || = 0 || || notcolspan="2" | coprimereducible

|-

| || || || || || ''':'''

! colspan="10" | ...

|}

Line 231 ⟶ 223:

::I hope you don't mind all my suggestions (I can be a bit of a perfectionist when it comes to tables). I think your table is an excellent addition to the article and will definitely help readers understand the ordering. In fact, it motivated me to reread Cantor's article and I noticed a detail that I had forgotten: Cantor gives the number of algebraic reals of heights 1, 2, and 3, which (of course) agree with your table. --[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 18:20, 17 December 2013 (UTC)

:::I changed the table according to your suggestions (perfectionism in writing optimizes the overall workload, since the table is written only once, but read -hopefully- a lot of times). Maybe the indices like in ''x''_'''3''' should not be in boldface? And: are you sure that no algebraic number may occur as root of two different irreducible polynomials? I've forgotten almost all my algebra knowledge... - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 20:40, 17 December 2013 (UTC)

== Contrast 2nd theorem with sequence of rational numbers? ==

::I like your attitude about perfectionism—I agree, we should think about the reader's workload. I also like the way you nicely simplified the table to have just 2 columns, by putting using "''x_n'' =" with the roots. I think that ''x''₃ is preferable to ''x''_'''3''' because the text doesn't use boldface and it looks better. Some other suggestions: I found double indexing "''x''_11,16" confusing. Try "''x''₁₁, ''x''₁₆" or, perhaps better, "''x''₁₆, ''x''₁₁" to match the way that the + of the ± goes with ''x''₁₆, and the – goes with ''x''₁₁ (or maybe there's a minus-plus symbol with minus on top of the plus). Also, I see no need for the large space between the "''x_n'' =" and the roots at the top of the table. I can see you're lining up with the roots at the bottom of the table, but on a first reading, many users may not go to the bottom of the table and may wonder about the space. Finally, try moving the "…" over a bit at the end of the table.

::Your question about the possibility of an algebraic number occurring as the root of two different irreducible polynomials is very relevant. At the site: [http://www.encyclopediaofmath.org/index.php/Algebraic_number Algebraic Number (Encyclopedia of Math)], you can read about the minimal polynomial of an algebraic number. This minimal polynomial is the polynomial of least degree that has α as a root, has rational coefficients, and first coefficient 1. It is irreducible. By multiplying by the least common denominator of all its coefficients, you obtain α's irreducible polynomial with integer coefficients that Cantor uses. The minimal polynomial ''Φ(x)'' of the algebraic number α can be easily shown to be a factor of any polynomial ''p(x)'' with rational coefficients that has root α. You start by dividing ''p(x)'' by ''Φ(x)'' using long division. This gives: ''p(x)'' = ''q(x)'' ''Φ(x)'' + ''r(x)'' where deg(''r(x)'') < deg(''Φ(x)''). Assume ''r(x)'' ≠ 0. Since ''p''(α) = ''Φ''(α) = 0, we then have ''r''(α) = 0 which contradicts the fact that the minimal polynomial ''Φ(x)'' is the polynomial of least degree with root α. So ''r(x)'' must be 0. Therefore: ''p(x)'' = ''q(x)'' ''Φ(x)'' so the minimal polynomial is a factor of ''p(x)''. --[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 20:32, 18 December 2013 (UTC)

:::I didn't have web access during xmas holidays, but now I updated the table according to your recent suggestions. There is a "∓" symbol, but I think it looks unusual in an expression, so I instead changed the order of the lhs variables. I moved the final dots into the "=" column and simulated vertical dots by a colon, as I couldn't find an appropriate symbol or template.

:::I like your suggestion for a footnote containing our table. As you are currently editing the article anyway, would you insert your footnote and move the table? Maybe it is best to remove it from the talk page, to avoid confusion about where to do possible later table edits.

:::Last not least: Thank you for your explanation why there is only one minimal irreducible polynomial for an algebraic number; it helped me to bring back my memories about algebra. - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 14:09, 27 December 2013 (UTC)

::Sorry to be so slow in getting back to you. I've been busy and haven't watching my Watchlist. I see that you've already made the necessary changes, which is great--you deserve the credit. I think that the way you improved your table is much better than my suggestion. Keep up your excellent Wikipedia work! --[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 18:36, 8 April 2014 (UTC)

== Contrast 2nd theorem with sequence of rational numbers? ==

Cantor's 2nd theorem seems obvious at first glance to many people, as we usually are unable to imagine a sequence that could completely fill a whole interval. However, there are sequences (like that of all positive rational numbers) whose set of [[accumulation point]]s equals a whole interval (or even whole ℝ₊; cf. the picture [[accumulation point|there]]). Mentioning this in the article might prevent novice readers from thinking "''Mathematicians make a big fuzz proving things that are obvious, anyway''", and might generally help to sharpen one's intuition about what a sequence ''can'' do in relation to an interval and what it ''cannot''. It would require, however, to explain the notion of an ''accumulation point'' (which is poorly represented in English Wikipedia in general). - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 11:52, 17 December 2013 (UTC)

{{Talk:Georg Cantor's first set theory article/GA1}}