More saturated ideals
More saturated ideals
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更饱和的理想
DOI：
10.1007/bfb0071691
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
M. Foreman
中科院分区：
文献类型：
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作者：
M. Foreman
我们现在做一些定义：设~是可数语言，有一元谓词U。称ε-结构~具有（~，k）型当且仅当I~ I= K且I~ Ik.若K>~' X> k'，则称（K，X）--（K '，k'）当且仅当每一个（K，X）型结构都有一个（K '，k'）型初等子结构~<~。理想~ c ~称为（~-c~ zplete）当且仅当[X 7：7<~} c ~且< G，UX（~~）.一个集合Ac_ P（~）称为正的，如果A#~。J是7< 6正规的当且仅当对任意正集Acf和任意定义在A上的回归函数f，存在K使得[（~：F（~）=~}是正的.称3是k饱和的当且仅当它是正规的且~（K）/J具有k链条件。(We永远不会考虑“非正常”饱和理想。）关于饱和理想有大量的文献。(See[6]，[7].）设j：V* M是从V到可迁类M的元素嵌入.设~ 0是j（即被j移动的第一个序数）的临界点，设Ki+ 1= j（~ i）.我们称j为n-巨大嵌入，称~ 0为n-巨大基数当且仅当M在~ n-序列下是闭的。(This意味着如果（x~：~<Kn）c M，则
We now make some definitions: Let~ be a countable language with a unary predicate U. A£-structure~ is said to have type (~, k) iff I~ I= K and I~ I k. If K>~' X> k'we say that (K, X)--(~', k') iff every structure of type (K, X) has an elementary substructure~<~ of type (~', k'). An ideal~ c_~ is said to be (~-c~ zplete iff whenever [X7: 7<~} c_~ and< G, UX (~~~. A set Ac_ P (~) is said to be positive if A#~. J is 7< 6 normal iff for every positive set A c~ and every regressive function f defined on A there is a~~ K such that [(~: F ((~)=~} is positive. 3 is said to be k-saturated iff it is normal and~(K)/J has the k-chain condition.(We will never consider" non-normal" saturated ideals.) There is an extensive literature on saturated ideals.(See [6],[7].)Let j: V* M be an el~ nentary embedding frcm V into a transitive class M. Let~ 0 be the critical point of j (ie the first ordinal moved by j), Let Ki+ 1= j (~ i). We will call j an n-huge embedding and~ 0 an n-huge cardinal iff M is closed under~ n-Sequences.(This means that if (x~:~< K n) c M then