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

Revision as of 18:20, 17 December 2013

Revision as of 20:40, 17 December 2013

Line 107:

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

|-

| '''1.''' ||           || || ||align="right" || {{color|#c0c0c0|+1·}}''x'' || {{color|#c0c0c0|+ 0}} || = 0 ||           || ''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''_{'''11''','''16'''} || = ±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''_{'''9''','''15'''} || = (-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''_{'''12''','''18'''} || = (+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''_{'''10''','''17'''} || = ±√{{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" | ...

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::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? ==