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| In [[set theory]], a branch of mathematics, the '''Milner – Rado paradox''', found by {{harvs|txt|first1=Eric Charles|last1=Milner|author1-link=Eric Charles Milner|first2=Richard|last2=Rado|author2-link=Richard Rado|year=1965}}, states that every [[ordinal number]] α less than the [[Successor cardinal|successor]] ''κ''<sup>+</sup> of some [[cardinal number]] κ can be written as the union of sets ''X''<sub>1</sub>,''X''<sub>2</sub>,... where ''X''<sub>''n''</sub> is of [[order type]] at most ''κ''<sup>''n''</sup> for ''n'' a positive integer.
| | I'm Hayden and I live in a seaside city in northern Italy, Castrignano Del Capo. I'm 40 and I'm will soon finish my study at Computing and Information Science.<br><br>Feel free to visit my blog post: [https://www.youtube.com/watch?v=-BD2sJnaQEg address here] |
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| ==Proof==
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| Let ''<math>\alpha</math>'' be a limit ordinal, and for each ''<math>\beta<\alpha</math>'', let ''<math>\{X_\beta^n\}_n</math>'' be the obvious thing.
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| Fix an increasing sequence <math>\{\beta_\gamma\}_{\gamma<\mathrm{cf}\,(\alpha)}</math> [[Cofinality|cofinal]] in <math>\alpha</math> with <math>\beta_0=1</math>.
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| Note <math>\mathrm{cf}\,(\alpha)\le\kappa</math>.
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| Define:
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| :<math>X^\alpha _0 = \{0\};\ \ X^\alpha_{n+1} = \bigcup_\gamma X^{\beta_{\gamma+1}}_n\setminus \beta_\gamma</math>
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| Observe that:
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| :<math>\bigcup_{n>0}X^\alpha_n = \bigcup _n \bigcup _\gamma X^{\beta_{\gamma+1}}_n\setminus \beta_\gamma = \bigcup_\gamma \bigcup_n X^{\beta_{\gamma+1}}_n\setminus \beta_\gamma = \bigcup_\gamma \beta_{\gamma+1}\setminus \beta_\gamma = \alpha \setminus \beta_0</math>
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| and so ''<math>\bigcup_nX^\alpha_n = \alpha</math>''.
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| Let <math>\mathrm{ot}\,(A)</math> be the [[order type]] of ''<math>A</math>''. As for the order types, clearly <math>\mathrm{ot}(X^\alpha_0) = 1 = \kappa^0</math>.
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| Noting that the sets <math>\beta_{\gamma+1}\setminus\beta_\gamma</math> form a consecutive sequence of ordinal intervals, and that each <math>X^{\beta_{\gamma+1}}_n\setminus\beta_\gamma</math> is a tail segment of <math>X^{\beta_{\gamma+1}}_n</math> we get that:
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| :<math>\mathrm{ot}(X^\alpha_{n+1}) = \sum_\gamma \mathrm{ot}(X^{\beta_{\gamma+1}}_n\setminus\beta_\gamma) \leq \sum_\gamma \kappa^n = \kappa^n \cdot \mathrm{cf}(\alpha) \leq \kappa^n\cdot\kappa = \kappa^{n+1}</math>
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| ==References==
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| *{{Citation | last1=Milner | first1=E. C. | last2=Rado | first2=R. | title=The pigeon-hole principle for ordinal numbers | doi=10.1112/plms/s3-15.1.750 | mr=0190003 | year=1965 | journal= Proc. London Math. Soc. (3) | volume=15 | pages=750–768}}
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| *[http://math.stackexchange.com/questions/440184/how-to-prove-milner-rado-paradox How to prove Milner-Rado Paradox? - Mathematics Stack Exchange]
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| {{DEFAULTSORT:Milner-Rado paradox}}
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| [[Category:Set theory]]
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| [[Category:Paradoxes]]
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| {{mathlogic-stub}}
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I'm Hayden and I live in a seaside city in northern Italy, Castrignano Del Capo. I'm 40 and I'm will soon finish my study at Computing and Information Science.
Feel free to visit my blog post: address here