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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Milner–Rado paradox ： ウィキペディア英語版 Milner–Rado paradox In set theory, a branch of mathematics, the Milner – Rado paradox, found by , states that every ordinal number α less than the successor ''κ''+ of some cardinal number κ can be written as the union of sets ''X''1,''X''2,... where ''X''''n'' is of order type at most ''κ''''n'' for ''n'' a positive integer. ==Proof== The proof is by transfinite induction. Let ''$\backslash alpha$'' be a limit ordinal (the induction is trivial for successor ordinals), and for each '''', let ''$\backslash \_n$'' be a partition of ''$\backslash beta$'' satisfying the requirements of the theorem. Fix an increasing sequence $\backslash \_$ cofinal in $\backslash alpha$ with $\backslash beta\_0=0$. Note $\backslash mathrm\backslash ,(\backslash alpha)\backslash le\backslash kappa$. Define: :$X^\backslash alpha\; \_0\; =\; \backslash ;\backslash \; \backslash \; X^\backslash alpha\_\; =\; \backslash bigcup\_\backslash gamma\; X^X^\backslash alpha\_n\; =\; \backslash bigcup\; \_n\; \backslash bigcup\; \_\backslash gamma\; X^\}\_n\backslash setminus\; \backslash beta\_\backslash gamma\; =\; \backslash bigcup\_\backslash gamma\; \backslash beta\_\backslash setminus\; \backslash beta\_\backslash gamma\; =\; \backslash alpha\; \backslash setminus\; \backslash beta\_0$ and so ''$\backslash bigcup\_nX^\backslash alpha\_n\; =\; \backslash alpha$''. Let $\backslash mathrm\backslash ,(A)$ be the order type of ''$A$''. As for the order types, clearly $\backslash mathrm(X^\backslash alpha\_0)\; =\; 1\; =\; \backslash kappa^0$. Noting that the sets $\backslash beta\_\backslash setminus\backslash beta\_\backslash gamma$ form a consecutive sequence of ordinal intervals, and that each $X^\}\_n$ we get that: :$\backslash mathrm(X^\backslash alpha\_)\; =\; \backslash sum\_\backslash gamma\; \backslash mathrm(X^(\backslash alpha)\; \backslash leq\; \backslash kappa^n\backslash cdot\backslash kappa\; =\; \backslash kappa^$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Milner–Rado paradox」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.