{"id":14288,"date":"2024-02-14T07:03:08","date_gmt":"2024-02-14T07:03:08","guid":{"rendered":"https:\/\/science-hub.click\/%E3%82%A6%E3%83%AB%E3%83%88%E3%83%A9%E3%83%97%E3%83%AD%E3%83%80%E3%82%AF%E3%83%88-%E5%AE%9A%E7%BE%A9\/"},"modified":"2024-02-14T07:03:08","modified_gmt":"2024-02-14T07:03:08","slug":"%E3%82%A6%E3%83%AB%E3%83%88%E3%83%A9%E3%83%97%E3%83%AD%E3%83%80%E3%82%AF%E3%83%88-%E5%AE%9A%E7%BE%A9","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=14288","title":{"rendered":"\u30a6\u30eb\u30c8\u30e9\u30d7\u30ed\u30c0\u30af\u30c8 – \u5b9a\u7fa9"},"content":{"rendered":"

\u5c0e\u5165<\/h2>

\u6570\u5b66\u3067\u306f\u3001\u30a6\u30eb\u30c8\u30e9\u7a4d\u306f<\/b>\u4e3b\u306b\u62bd\u8c61\u4ee3\u6570\u3068\u30e2\u30c7\u30eb\u7406\u8ad6 (\u6570\u7406\u8ad6\u7406\u5b66\u306e\u5206\u91ce) \u3067\u4f7f\u7528\u3055\u308c\u308b\u69cb\u9020\u3067\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u3053\u308c\u3089\u3068\u540c\u3058\u7279\u6027\u3092\u6301\u3064\u5b9f\u6570\u306e\u62e1\u5f35\u3001\u8d85\u5b9f\u6570\u3092\u53d6\u5f97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>

\n\"\u30a6\u30eb\u30c8\u30e9\u30d7\u30ed\u30c0\u30af\u30c8<\/figure>

\u610f\u5473<\/h2>

\u30a6\u30eb\u30c8\u30e9\u7a4d\u3092\u69cb\u7bc9\u3059\u308b\u4e00\u822c\u7684\u306a\u65b9\u6cd5\u306f\u3001\u30a4\u30f3\u30c7\u30c3\u30af\u30b9I<\/i>\u306e\u30bb\u30c3\u30c8<\/a><\/span>\u3001 I<\/i>\u306e<\/i>\u5404\u8981\u7d20i<\/i><\/sub>\u306e\u69cb\u9020\u4f53Mi<\/i> (\u3059\u3079\u3066\u540c\u3058\u7f72\u540d\u3092\u6301\u3064)\u3001\u304a\u3088\u3073I<\/i>\u4e0a\u306e\u30a6\u30eb\u30c8\u30e9\u30d5\u30a3\u30eb\u30bf\u30fc<\/a><\/span>U<\/i>\u304b\u3089\u59cb\u307e\u308a\u307e\u3059\u3002\u901a\u5e38\u306e\u9078\u629e\u306f\u3001 I \u3092<\/i>\u7121\u9650\u3068\u3057<\/a><\/span>\u3001 U \u3092<\/i>\u975e\u81ea\u660e\u306b\u3059\u308b\u3053\u3068\u3001\u3064\u307e\u308aI<\/i>\u306e\u6709\u9650\u90e8\u5206\u3092\u542b\u307e\u306a\u3044\u3053\u3068\u3067\u3059 (\u305d\u3046\u3067\u306a\u3044\u5834\u5408\u3001\u8d85\u7a4d\u306f<\/a><\/span>\u305d\u306e\u56e0\u6570\u306e 1 \u3064\u3068\u540c\u578b\u306b\u306a\u308a\u307e\u3059)\u3002<\/p>

\u30c7\u30ab\u30eb\u30c8\u7a4d<\/a><\/span>\u306e\u4ee3\u6570\u6f14\u7b97<\/p>

$$ {\\prod_{i \\in I} M_i } $$<\/div><\/dd><\/dl>

\u306f\u901a\u5e38\u306e\u65b9\u6cd5\u3067\u5b9a\u7fa9\u3055\u308c (\u305f\u3068\u3048\u3070\u3001(2 \u5024) \u6f14\u7b97 +\u3001( a<\/i> + b<\/i> ) i<\/i><\/sub> = a<\/i> i<\/i><\/sub> + b<\/i> i<\/i><\/sub> )\u3001\u6b21\u306e\u5834\u5408\u306b\u9650\u308a\u3001 a<\/i> ~ b<\/i>\u306b\u3088\u308b\u6f14\u7b97\u3068\u4e92\u63db\u6027\u306e\u3042\u308b\u540c\u5024\u95a2\u4fc2\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002 <\/p>

$$ {\\left\\{ i \\in I: a_i = b_i \\right\\}\\in U.} $$<\/div><\/dd><\/dl>

\u6b21\u306b\u3001\u3053\u306e\u65cf\u306e\u8d85\u7a4d<\/b>( U<\/i>\u306b\u95a2\u3059\u308b) \u306f\u3001\u5546\u69cb\u9020\u3092\u5099\u3048\u305f\u3001\u3053\u306e\u540c\u5024\u95a2\u4fc2\u306e\u30c7\u30ab\u30eb\u30c8\u7a4d\u306e\u5546\u96c6\u5408\u3067\u3059 (\u3064\u307e\u308a\u3001\u305d\u306e\u8981\u7d20\u306f\u7a4d\u306e\u540c\u5024\u30af\u30e9\u30b9\u3067\u3059)\u3002\u3060\u304b\u3089\u3053\u305d\u6ce8\u76ee\u3055\u308c\u308b\u3053\u3068\u3082\u3042\u308b<\/p>

$$ {\\prod_{i\\in I}M_i \/ U . } $$<\/div><\/dd><\/dl>

A<\/i> \u2208 U<\/i>\u306e\u5834\u5408\u306fm<\/i> ( A<\/i> ) = 1\u3001\u305d\u308c\u4ee5\u5916\u306e\u5834\u5408\u306fm<\/i> ( A<\/i> ) = 0 \u3068\u8a2d\u5b9a\u3059\u308b\u3053\u3068\u3067\u3001\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u306e\u30bb\u30c3\u30c8\u306b\u5bfe\u3057\u3066\u5c3a\u5ea6m<\/i> (\u6709\u9650\u52a0\u6cd5) \u3092\u5b9a\u7fa9\u3067\u304d\u307e\u3059\u3002\u30c7\u30ab\u30eb\u30c8\u7a4d\u306e 2 \u3064\u306e\u8981\u7d20\u306f\u3001\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u306e\u30bb\u30c3\u30c8\u306e\u307b\u307c\u3069\u3053\u3067\u3082\u7b49\u3057\u3044\u5834\u5408\u3001\u7b49\u4fa1\u3067\u3059\u3002<\/p>

\u4ee3\u6570\u6f14\u7b97\u306e\u307b\u304b\u306b\u3001\u95a2\u4fc2\u3082\u540c\u3058\u65b9\u6cd5\u3067\u62e1\u5f35\u3067\u304d\u307e\u3059: R<\/i> ([ a<\/i> 1<\/sup> ],\u2026,[ a n<\/sup><\/i> ]) if and Only if <\/p>

$$ {\\left\\{ i \\in I: R^{M_i}(a^1_i,\\dots,a^n_i) \\right\\}\\in U,} $$<\/div><\/dd><\/dl>

\u3053\u3053\u3067\u3001[ a<\/i> ] \u306f\u95a2\u4fc2 ~ \u306b\u5bfe\u3059\u308ba<\/i>\u306e\u540c\u5024\u30af\u30e9\u30b9\u3092\u6307\u5b9a\u3057\u307e\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u3059\u3079\u3066\u306eMi<\/i>\u304c\u9806\u5e8f\u4ed8\u3051\u3055\u308c\u305f\u30dc\u30c7\u30a3\u3067\u3042\u308b\u5834\u5408<\/i><\/sub>\u3001ultraproduct \u306b\u3064\u3044\u3066\u3082\u540c\u3058\u3053\u3068\u304c\u5f53\u3066\u306f\u307e\u308a\u307e\u3059\u3002<\/p>

\u30a6\u30eb\u30c8\u30e9\u30d1\u30ef\u30fc<\/b>\u306f\u3001\u3059\u3079\u3066\u306e\u4fc2\u6570M<\/i> i<\/i><\/sub>\u304c\u7b49\u3057\u3044\u30a6\u30eb\u30c8\u30e9\u7a4d\u3067\u3059\u3002 <\/p>

$$ {M^\\kappa\/U=\\prod_{\\alpha<\\kappa}M\/U.\\,} $$<\/div><\/dd><\/dl>

\u3088\u308a\u4e00\u822c\u7684\u306b\u306f\u3001 U \u304c<\/i>X<\/i>\u306b\u5bfe\u3059\u308b\u30d5\u30a3\u30eb\u30bf\u30fc<\/span>\u306e\u307f\u3067\u3042\u308b\u5834\u5408\u3001\u524d\u306e\u69cb\u7bc9\u306f\u5f15\u304d\u7d9a\u304d\u5b9f\u884c\u3067\u304d\u307e\u3059\u3002\u7d50\u679c\u3068\u3057\u3066\u5f97\u3089\u308c\u308b\u69cb\u9020

$$ {\\prod_{i\\in I}M_i \/ U} $$<\/div>\u3092\u9084\u5143\u54c1<\/b>\u3068\u3044\u3044\u307e\u3059\u3002<\/p>
\n\"\u30a6\u30eb\u30c8\u30e9\u30d7\u30ed\u30c0\u30af\u30c8<\/figure>

\u30a6\u30a9\u30b7\u30e5\u306e\u5b9a\u7406<\/h2>

\u0141o\u015b \u306e\u5b9a\u7406\u306f\u3001\u57fa\u672c\u8d85\u7a4d\u5b9a\u7406<\/i>\u3068\u547c\u3070\u308c\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u304c\u3001 Jerzy \u0141o\u015b<\/span> (\u767a\u97f3: \u02c8w\u0254\u0255<\/span> \u3001\u304a\u3088\u305douosh<\/i> ) \u306b\u3088\u308b\u3082\u306e\u3067\u3059\u3002\u3053\u308c\u306f\u3001\u5f0f\u304cM<\/i> i<\/i><\/sub>\u3067\u771f\u3068\u306a\u308b\u3088\u3046\u306a\u30a4\u30f3\u30c7\u30c3\u30af\u30b9i<\/i>\u306e\u30bb\u30c3\u30c8\u304cU<\/i>\u306e\u8981\u7d20\u3067\u3042\u308b\u5834\u5408\u306b\u9650\u308a\u3001ultraproduct \u3067\u3042\u3089\u3086\u308b 1 \u6b21\u5f0f\u304c\u771f\u3067\u3042\u308b\u3068\u4e3b\u5f35\u3057\u307e\u3059\u3002\u3088\u308a\u6b63\u78ba\u306b\u306f:<\/p>

\u03c3 \u3092\u7f72\u540d\u3001 U \u3092<\/i><\/span>\u30bb\u30c3\u30c8I<\/i><\/span>\u4e0a\u306e\u9650\u5916\u30d5\u30a3\u30eb\u30bf\u30fc\u3001\u304a\u3088\u3073\u305d\u308c\u305e\u308c\u306b\u3064\u3044\u3066

$$ { i \\in I } $$<\/div> M<\/i> i \u3092<\/i><\/sub><\/span>\u03c3\u69cb\u9020\u3068\u3059\u308b\u3002 M \u3092<\/i><\/span>U<\/i><\/span>\u306b\u95a2\u3059\u308bMi<\/i><\/sub>\u306e<\/i><\/span>\u8d85\u7a4d\u3068\u3059\u308b\u3001\u3064\u307e\u308a\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002
$$ { M = \\prod_{ i\\in I }M_i\/U. } $$<\/div><\/p>

\u3060\u304b\u3089\u3001\u3059\u3079\u3066<\/a><\/span>\u306b\u3064\u3044\u3066

$$ { a^{1}, \\ldots, a^{n} \\in \\prod M_{i} } $$<\/div> \u3001 \u307e\u305f\u306f
$$ { a^{k} = (a^{k}_{i})_{i \\in I} } $$<\/div> \u3001\u304a\u3088\u3073\u5404 \u03c3 \u5f0f\u03c6<\/span>\u306b\u3064\u3044\u3066<\/p>

$$ { M \\models \\phi[[a^1], \\ldots, [a^n]] } $$<\/div>\u3082\u3057\u3001\u305d\u3057\u3066\u305d\u306e\u5834\u5408\u306b\u9650\u308a<\/p>
$$ { \\{ i \\in I\u00a0: M_{i} \\models \\phi[a^1_{i}, \\ldots, a^n_{i} ] \\} \\in U. } $$<\/div><\/dd><\/dl>

\u3053\u306e\u5b9a\u7406\u306f\u3001\u5f0f\u03c6<\/span>\u306e\u8907\u96d1\u3055<\/a><\/span>\u306b\u95a2\u3059\u308b\u5e30\u7d0d\u6cd5\u306b\u3088\u3063\u3066\u8a3c\u660e\u3055\u308c\u307e\u3059\u3002 U<\/i>\u304c\u9650\u5916\u30d5\u30a3\u30eb\u30bf\u30fc (\u5358\u306a\u308b\u30d5\u30a3\u30eb\u30bf\u30fc\u3067\u306f\u306a\u3044) \u3067\u3042\u308b\u3068\u3044\u3046\u4e8b\u5b9f\u306f\u3001\u5426\u5b9a\u306e\u5834\u5408\u3092\u6271\u3046\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u3001\u9078\u629e\u516c\u7406<\/a><\/span>\u306f\u5b58\u5728\u91cf\u6307\u5b9a\u5b50\u306e\u5c0e\u5165\u306e\u516c\u7406\u306b\u5fc5\u8981\u3067\u3059\u3002<\/p>

\n\"\u30a6\u30eb\u30c8\u30e9\u30d7\u30ed\u30c0\u30af\u30c8<\/figure>

\u4f8b<\/span><\/h3>

R \u3092<\/i>\u69cb\u9020M<\/i>\u306e\u5358\u9805\u95a2\u4fc2\u3068\u3057\u3001 *<\/sup> M \u3092<\/i>M<\/i>\u306e\u8d85\u7a4d\u3068\u3057\u307e\u3059\u3002\u305d\u308c\u3067\u5168\u4f53\u304c

$$ {S=\\{x \\in M|R x\\}} $$<\/div> *<\/sup> M<\/i>\u306b\u985e\u4f3c\u3057\u305f*<\/sup> S<\/i>\u304c\u3042\u308a ( *<\/sup> S \u306f<\/i>\u3001 R<\/i> x i \u304c<\/sub>\u307b\u307c\u3059\u3079\u3066\u306ei<\/i>\u306b\u5bfe\u3057\u3066\u771f\u3068\u306a\u308b\u3088\u3046\u306a\u30b7\u30fc\u30b1\u30f3\u30b9 (x i<\/sub> ) \u306e\u30bb\u30c3\u30c8 (\u540c\u5024\u30af\u30e9\u30b9) \u3067\u3059)\u3001\u30b2\u30fc\u30e0S<\/i>\u306b\u7d44\u307f\u8fbc\u307e\u308c\u305f\u4e00\u6b21\u516c\u5f0f\u306f\u4f9d\u7136\u3068\u3057\u3066\u771f\u3067\u3059\u7528\u30fb<\/sup>S\u3002<\/i>\u305f\u3068\u3048\u3070\u3001 M \u3092<\/i>\u5b9f\u6570\u3001 Rx \u3092<\/i>\u300c x<\/i>\u306f\u6709\u7406\u6570<\/a><\/span>\u3067\u3042\u308b\u300d\u3068\u3044\u3046\u95a2\u4fc2\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u5834\u5408\u3001 M<\/i>\u3067\u306f\u3001\u6709\u7406x<\/i>\u3068y<\/i>\u306e\u4efb\u610f\u306e\u30da\u30a2\u306b\u5bfe\u3057\u3066\u3001 x<\/i> < z<\/i> < y<\/i>\u3067\u3042\u308b\u975e\u6709\u7406z \u304c<\/i>\u5b58\u5728\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u3053\u308c\u306f\u4e00\u6b21\u8a00\u8a9e\u306b\u7ffb\u8a33\u3067\u304d\u307e\u3059\u3002\u0141o\u015b \u306e\u5b9a\u7406\u306f*<\/sup> S<\/i>\u304c\u540c\u3058\u6027\u8cea\u3092\u6301\u3064\u3053\u3068\u3001\u3064\u307e\u308a\u8d85\u6709\u7406\u3092\u5b9a\u7fa9\u3067\u304d\u30012 \u3064\u306e\u8d85\u6709\u7406\u306e\u9593\u306b\u306f\u975e\u8d85\u6709\u7406\u8d85\u73fe\u5b9f\u304c\u5e38\u306b\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002\u4e00\u65b9\u3001\u30a2\u30eb\u30ad\u30e1\u30c7\u30b9\u306e\u516c\u7406 (\u3053\u306e\u7121\u9650\u306e\u3059\u3079\u3066\u306e\u4e0d\u7b49\u5f0f\u306b\u5bfe\u3057\u3066\u3001 x<\/i> > 1\u3001 x<\/i> > 1 +1\u3001 x<\/i> > 1 + 1 + 1\u3001… \u3068\u306a\u308b\u3088\u3046\u306a\u5b9f\u6570x \u306f<\/i>\u5b58\u5728\u3057\u306a\u3044\u3068\u8a00\u3046\u306e\u3068\u540c\u3058\u3067\u3059) list) \u306f\u4e00\u6b21\u8a00\u8a9e\u306b\u7ffb\u8a33\u3067\u304d\u305a\u3001\u0141o\u015b \u306e\u5b9a\u7406\u306f\u9069\u7528\u3055\u308c\u307e\u305b\u3093\u3002\u307e\u305f\u3001\u8d85\u73fe\u5b9f\u03c9 \u304c<\/i>\u3053\u306e\u5f62\u5f0f\u3067\u306f\u30a2\u30eb\u30ad\u30e1\u30c7\u30b9<\/a><\/span>\u306e\u516c\u7406\u3092\u6e80\u305f\u3055\u306a\u3044\u3053\u3068\u3082\u4e0a\u3067\u898b\u307e\u3057\u305f\u3002\u4e00\u65b9\u3001\u8d85\u6574\u6570*<\/sup> N<\/b> ( \u03c9<\/i>\u306f\u305d\u306e\u4e00\u90e8) \u306e\u96c6\u5408\u3092\u5b9a\u7fa9\u3059\u308b\u3068\u3001\u0141o\u015b \u306e\u5b9a\u7406\u306b\u3088\u308a\u3001\u3059\u3079\u3066\u306e\u8d85\u6574\u6570\u3088\u308a\u5927\u304d\u3044\u8d85\u5b9f\u6570\u306f\u5b58\u5728\u3057\u306a\u3044\u3053\u3068\u304c\u4fdd\u8a3c\u3055\u308c\u307e\u3059\u3002\u6700\u5f8c\u306b\u3001\u0141o\u015b \u306e\u5b9a\u7406\u306e\u304a\u304b\u3052\u3067\u3001\u305f\u3068\u3048\u3070\u3001\u3059\u3079\u3066\u306e\u8d85\u6574\u6570\u306f\u5076\u6570\u304b\u5947\u6570\u306e\u3044\u305a\u308c\u304b\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u304c\u4f9d\u7136\u3068\u3057\u3066\u771f\u5b9f\u3067\u3042\u308b\u5834\u5408\u3001\u7279\u5b9a\u306e\u8d85\u6574\u6570\u306e\u5834\u5408\u3092\u77e5\u308b\u3053\u3068\u306f\u4e00\u822c\u306b\u4e0d\u53ef\u80fd\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u5076\u6570\u306e\u96c6\u5408\u304cU<\/i>\u306b\u5c5e\u3059\u308b\u5834\u5408\u306b\u9650\u308a\u3001 \u03c9 \u306f<\/i>\u5076\u6570\u306b\u306a\u308a\u307e\u3059\u3002<\/p><\/div>
\n\"\u30a6\u30eb\u30c8\u30e9\u30d7\u30ed\u30c0\u30af\u30c8<\/figure>

\u53c2\u8003\u8cc7\u6599<\/h2>
  1. Ultraprodukt \u2013 allemand<\/a><\/li>
  2. Ultraproduct \u2013 anglais<\/a><\/li>
  3. Ultratulo \u2013 finnois<\/a><\/li>
  4. \u05e2\u05dc-\u05de\u05db\u05e4\u05dc\u05d4 \u2013 h\u00e9breu<\/a><\/li>
  5. Ultraprodotto \u2013 italien<\/a><\/li>
  6. \u8d85\u7a4d \u2013 japonais<\/a><\/li><\/ol><\/div>\n
    \n

    \n \u30a6\u30eb\u30c8\u30e9\u30d7\u30ed\u30c0\u30af\u30c8 – \u5b9a\u7fa9\u30fb\u95a2\u9023\u52d5\u753b\n <\/h2>\n
    \n \n
    \n
    \n