On closed sets of ordinals
On closed sets of ordinals
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关于序数的闭集
DOI：
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发表时间：
1974
期刊：
影响因子：
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通讯作者：
H. Friedman
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文献类型：
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作者：
H. Friedman
我们证明了可数序数的每一个平稳集都包含任意长的可数闭子集。称序数集合A为闭的当且仅当A的每个在A中有上界的非空子集在A中有最小上界。众所周知，存在BC a 1使得B和a，1-B都不包含不可数闭子集。我们在这里证明的一个结果是，对于每个BC w1，B或a），-B包含任意长的可数闭子集。称一个序数集A为K-平稳的当且仅当Ac K和A与K的每一个幂为K的闭子集相交。我们可以把上述著名定理重述如下：存在A使得A和wo 1-A都是wl-平稳的。2我们将证明每个wl-平稳集包含任意长的可数闭子集。是否存在基数K使得对所有AC K，A或K-A包含不可数闭子集？K= aw 2时，是否成立？Karel Prikry和作者注意到，在任何情况下，关于K=W）2的陈述都不能在ZFC.3定理中被证明为真。每个wl-平稳集都包含任意长的可数闭子集。证据设A是w），-平稳的。我们通过归纳证明了在o< a）1上A有一个长度为a的闭子集。假设归纳假设是1972年11月30日编辑收到的。AMS（MOS）主题分类（1970年）。小学04 A20;中学02 K 02、02 K35。
We prove that every stationary set of countable ordinals contains arbitrarily long countable closed subsets. Call a set A of ordinals closed if and only if every nonempty subset of A which has an upper bound in A has its least upper bound in A. It is well known that there are BC a1 such that neither B nor a,1-B contains an uncountable closed subset. A consequence of what we prove here is that for every BC w1, either B or a),-B contains arbitrarily long countable closed subsets. Call a set A of ordinals K-stationary if and only if Ac K and A intersects every closed subset of K of power K. We can restate the above wellknown theorem as follows: There is an A such that A and wo1 -A are both wl-stationary.2 We will prove here that every w),-stationary set contains arbitrarily long, countable, closed subsets. Is there a cardinal K such that for all AC K, either A or K-A contains an uncountable closed subset? Is this true for K=aw2? Karel Prikry and the author noticed that, in any case, the statement for K=W)2 cannot be proved true in ZFC.3 THEOREM. Every wl-stationary set contains arbitrarily long countable closed subsets. PROOF. Let A be w),-stationary. We prove by induction on o< a)1 that A has a closed subset of length a. Let the induction hypothesis be that Received by the editors November 30, 1972. AMS (MOS) subject classifications (1970). Primary 04A20; Secondary 02K02, 02K35.