Cantor's first set theory article: Difference between revisions

Revision as of 21:36, 22 December 2020

Revision as of 21:56, 22 December 2020

Line 94:

! style="background: f5f5f5;" |'''Proof that for all ''n''{{space|hair|2}}: {{nowrap|{{space|thin|2}}''x''_''n'' {{space|hair|2}}∉ (''a''_''n'', ''b''_''n'')}}''' {{space|thin|2}}

|- style="text-align: left; vertical-align: top; background: white"

| style="padding-left: 1em; padding-right: 1em" | This [[lemma (mathematics)|lemma]] is used by cases 2 and 3. It is implied by the stronger lemma: For all&nbsp;''n'', (''a''_''n'',&nbsp;''b''_''n'') excludes ''x''₁,&nbsp;...,&nbsp;''x''_2''n''. This is proved by [[mathematical induction|induction]]. Basis step: Since the [[Endpoints (interval)|endpoints]] of (''a''₁,&nbsp;''b''₁) are ''x''_''1'' and ''x''_''2'' and an open interval excludes its endpoints, (''a''₁,&nbsp;''b''₁) excludes ''x''₁,&nbsp;''x''₂. Inductive step: Assume that (''a''_''n'',&nbsp;''b''_''n'') excludes ''x''₁,&nbsp;...,&nbsp;''x''_2''n''. Since (''a''_''n''+1,&nbsp;''b''_''n''+1) is a subset of (''a''_''n'',&nbsp;''b''_''n'') and its endpoints are ''x''_2''kn''+1 and ''x''_2''jn''+2 with indices ''j'',''k''&gt;2''n'' the interval (''a''_''n''+1,&nbsp;''b''_''n''+1) excludes ''x''₁,&nbsp;...,&nbsp;''x''_2''n'' and ''x''_2''n''+1,&nbsp;''x''_2''n''+2. Hence, for all&nbsp;''n'', (''a''_''n'',&nbsp;''b''_''n'') excludes ''x''₁,&nbsp;...,&nbsp;''x''_2''n''. Therefore, for all&nbsp;''n'', ''x''_''n''&nbsp;∉&nbsp;(''a''_''n'',&nbsp;''b''_''n'').{{efn-ua|Cantor does not prove this lemma. In a footnote for case 2, he states that ''x''_''n'' does ''not'' lie in the interior of the interval [''a''_''n'',&nbsp;''b''_''n''].<ref name=Ewald842 /> This proof comes from [[#Cantor's 1879 proof|his 1879 proof]], which contains a more complex inductive proof that demonstrates several properties of the intervals generated, including the property proved here.}}

|}

{{Anchor|Case3}}[[File:Cantor's first uncountability proof Case 3 svg.svg|thumb|350px|alt=Illustration of case 3. Real line containing [''a'',&nbsp;''b''] that contains nested intervals (''a''_''n'',&nbsp;''b''_''n'') for ''n''&nbsp;=&nbsp;1 to ∞. These intervals converge to the closed interval [a_∞,&nbsp;b_∞]. The number y is in this interval.|Case 3: ''a''_∞ < ''b''_∞]]