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<\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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:A description of the recommendations for math article assessments is at [[Wikipedia:WikiProject_Mathematics/Wikipedia_1.0/Grading_scheme]]. However, the "A" class is in limbo right now: there was a system set up to try to review articles before they were rated A class, but that system never caught on, and now it is defunct. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 00:23, 25 January 2010 (UTC)

On Feb. 20, I followed your suggestion of having an example of generating an irrational number by using Cantor's 1874 method. This follows the example of generating an irrational number by using Cantor's diagonal method. &mdash;&nbsp;[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 01:19, 3 March 2010 (UTC)

== Restrict polynomials to irreducible ones in proof of countability of algebraic numbers? ==

As far as I understood Cantor's 1874 article, he considers in his proof of countability of algebraic numbers only [[irreducible polynomial]]s (p.258: "''und die Gleichung (1.) irreducibel denken''" = "''and consider equation (1.) to be irreducible''"). These are sufficient to get all algebraic numbers, and each of them corresponds to at most one algebraic number, viz. its root (if in ℝ). In this setting it is more clear what it means to "''order the real roots of polynomials of the same height by numeric order''" (cited from [[Cantor's first uncountability proof#The proofs]]). Maybe the article should also restrict polynomials to irreducible ones - ?

[[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 15:10, 13 December 2013 (UTC)

:I made a list of algebraic numbers, ordered by Cantor's rank, as a collapsible table. Maybe it is illustrative to include it (in collapsed form) into the article. At least I my self learned (1) that "''irreducible''" should mean "''cannot be written as product of smaller polynomials'' '''with integer coefficients'''", and (2) an irreducible polynomial in that sense can well have several solutions; two facts that I should have remebered from my school time. - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 16:40, 13 December 2013 (UTC)

{| align="right" class="collapsible collapsed" style="text-align:left"

! colspan="10" | '''Cantor's enumeration of algebraic numbers'''

|-

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

|-

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

|-

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

|-

|-

| || || || align="right" | 2''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

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

|-

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

|-

||| || align="right" | {{color|#c0c0c0|1}}''x''² || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

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

|-

||| || || align="right" | 3''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || || align="right" | 2''x'' || +1 || = 0 || || ''x''₅ = −1/2

|-

| || || || align="right" | 2''x'' || −1 || = 0 || || ''x''₆ = +1/2

|-

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

|-

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

|-

| || || align="right" | 2''x''² || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || +{{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || −{{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || 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 || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || {{color|#c0c0c0|+0''x''²}} || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

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

|-

| || || || align="right" | 4''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || || align="right" | 3''x'' || +1 || = 0 || || ''x''₁₃ = −1/3

|-

| || || || align="right" | 3''x'' || −1 || = 0 || || ''x''₁₄ = +1/3

|-

| || || || align="right" | 2''x'' || +2 || = 0 || || colspan="2" | reducible

|-

| || || || align="right" | 2''x'' || −2 || = 0 || || colspan="2" | reducible

|-

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

|-

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

|-

| || || align="right" | 3''x''² || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || align="right" | 2''x''² || +{{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || align="right" | 2''x''² || −{{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

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

|-

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

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || +2''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || −2''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

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

|-

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

|-

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

|-

| || || align="right" | {{color|#c0c0c0|1}}''x''² || −{{color|#c0c0c0|1}}''x'' || −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

|-

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

|-

| || align="right" | 2''x''³ || {{color|#c0c0c0|+0''x''²}} || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || +{{color|#c0c0c0|1}}''x''² || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || −{{color|#c0c0c0|1}}''x''² || {{color|#c0c0c0|+0''x''}} || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || {{color|#c0c0c0|+0''x''²}} || +{{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || {{color|#c0c0c0|+0''x''²}} || −{{color|#c0c0c0|1}}''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || {{color|#c0c0c0|+0''x''²}} || {{color|#c0c0c0|+0''x''}} || +1 || = 0 || || colspan="2" | reducible

|-

| || align="right" | {{color|#c0c0c0|1}}''x''³ || {{color|#c0c0c0|+0''x''²}} || {{color|#c0c0c0|+0''x''}} || −1 || = 0 || || colspan="2" | reducible

|-

| 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 || || colspan="2" | reducible

|-

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

|-

| || || || align="right" | 5''x'' || {{color|#c0c0c0|+0}} || = 0 || || colspan="2" | reducible

|-

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

|}

::When I wrote the section "The Proofs", my intent was to emphasize the proof of Cantor's second theorem, so I simplified his proof of the countability of algebraic numbers by leaving out "irreducible" so readers wouldn't have to know what an irreducible polynomial is. I'm sorry that you found my method less clear than Cantor's on the ordering of the algebraic numbers of a particular height. Using your enumeration table, the polynomials of height 2 give 0, -1, 1, 0 as roots, so the ordering will be -1, 0, 0, 1. In this enumeration, duplicates often appear within a height and between heights, but Cantor's proof of his second theorem does handle duplicates.

::However, you do bring up an excellent question: Shall we mention Cantor's use of irreducible polynomials? I see two ways to mention it: Add it to the text or add a footnote at the end of the paragraph that points out the text's ordering produces duplicates and that Cantor's original enumeration eliminates duplicates by using irreducible polynomials. By the way, the reason for some of the longer footnotes in this article was to explain points in more depth—readers just wanting the main points can skip the footnotes. Which is better in this case? I don't know. Maybe some readers can give us feedback.

::I like your enumeration table. A few suggestions: Label it "Cantor's enumeration of algebraic numbers". Change "not coprime" to "not irreducible". Coprime refers to a set of two or more integers so it doesn't apply to polynomials such as 2''x''. The definition of [[irreducible polynomial]] states that: "A polynomial with integer coefficients, or, more generally, with coefficients in a [[unique factorization ___domain]] ''F'' is said to be '''irreducible''' over ''F'' if it is not [[unit (ring theory)|invertible]] nor zero and cannot be factored into the product of two non-invertible polynomials with coefficients in ''F''." This definition factors: 2''x'' = (2)(''x'') and it factors: 2''x''+2 = (2)(''x''+1). Finally, the exponent 1 in your table always appears in gray and it's well understood that "''x''" means "''x''¹". Try leaving out this exponent. I think this might visually simply your table. - [[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 01:58, 15 December 2013 (UTC)

:::When I started this talk section, I had in mind the construction of rationals from integers, and I thought that algebraic numbers could be constructed from rationals in a similar way. The former is done by computing with pairs (''p'',''q'') ∈ ℤ×(ℤ\{0}) with the intended meaning ''p''/''q''; I thought the latter could be done by computing with polynomials, where one polynomial would denote one algebraic number, "viz. its root". Meanwhile I saw that even an irreducible polynomial has several roots, so that there can't be a one-to-one correspondence between polynomials and algebraic numbers, anyway. So I lost my original motivation for asking for irreducibility. Probably the proof is simplest in its current form; maybe a footenote could be added as you suggested.

:::In the enumeration table, I tried to distinguish several reasons for excluding a polynomial, a non-coprime set of coefficients being one of them, non-irreducibility being another one (admittely subsuming the former); when changing the table to produce duplicates these reasons would disappear, anyway. I used the gray parts to indicate (to myself, in the first place) the systematic way the polynomials are enumerated (nevertheless, I missed all polynomials containing ''x''³ and ''x''⁴; see the new table; I hope it is complete now ...), but you are right: at least the exponent of "''x''¹" isn't needed for that; I now deleted it. Concerning duplicates: should we have a reason "repetition" (or "duplicate"?) and not assign them a number; or should we assign them a number and mention somewhere that the enumeration is not bijective, but surjective, which suffices for countability? The former case would save some indentation space, since the ''x''⁴ column could be immediately adjacent to the leftmost (number) column, as in each row at least one of them is empty. The latter case wouldn't save much, as "(-1 ± √5) / 2" (to be kept) is about as long as "repetition". - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 12:38, 16 December 2013 (UTC)

::I think some readers may find the current text ambiguous on the question of whether duplicates appear in the sequence (of course, it doesn't matter for applying his second theorem). There are two ways to eliminate duplicates and both give the same result. Below is my first attempt at a footnote to clarify the situation and to introduce readers to Cantor's approach and your table:

::"Using this ordering and placing only the first occurrence of an real algebraic number in the sequence produces a sequence without duplicates. Cantor obtained the same sequence by using [[irreducible polynomial]]s: INSERT YOUR TABLE HERE"

::Your table is looking better, some more suggestions: remove the "·" in 2·''x'', etc. In the enumeration, you can use ''x''₁ instead of "1.", etc. (This would connect your table closer to the article where all the sequences are ''x''₁, ''x''₂, ….) Also, in front of the first coefficient, you can leave out the "+" since every polynomial starts with a positive coefficient. Finally, concerning irreducible polynomials versus coprimes, I apologize for not being clearer. I should have quoted the following from "[[Irreducible polynomial]]":

::"It is helpful to compare irreducible polynomials to [[prime number]]s: prime numbers (together with the corresponding negative numbers of equal magnitude) are the irreducible [[integer]]s. They exhibit many of the general properties of the concept of 'irreducibility' that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors:"

::This means that you factor 6''x'' = (2)(3)(''x''). Basically, the terms to use when working with factoring polynomials are "reducible" and "irreducible" (they are the counterparts to "composite" and "prime"). I think that you may be generalizing the term [[coprime]] to single integers to handle polynomials, such as 3''x'', when you call this polynomial "not coprime". I've done a Google search and I only found the term "coprime" referring two or more integers. So I think your table would be more accurate and clearer if you used the term "not irreducible". Also, I have the philosophy of placing minimal demands on the reader (whenever possible). By only using the word "irreducible", the reader is not required to understand "coprime".

::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)

::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}}