AD and the supercompactness of ℵ1
AD and the supercompactness of ℵ1
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AD 和 ℵ1 的超紧性
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发表时间：
1981
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影响因子：
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通讯作者：
H. Becker
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文献类型：
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作者：
H. Becker
自从60年代初发现强迫以来，很明显，许多自然和有趣的数学问题是不能从集合论的经典公理ZFC中判定的。因此，一些数学家一直在研究更强的集合论假设的后果。两种新类型的公理已成为许多研究的主题是大基数公理和公理断言的确定性可定义的游戏。乍一看，这两者似乎没有关系;最近的研究中最令人惊讶的发现之一是事实并非如此。在本文中，我们将假设确定性公理（AD）加上相关选择公理（DC）。AD是错误的，因为它违背了选择公理。然而，L[R]中的每个集合都是从真实的序数可定义的。我们的公理，可定义的游戏是确定的意味着每个游戏在L[R]是确定的（在V），因为一个战略是一个真实的，它是确定的L[R]。”[10]这是一个[R]的比喻。选择公理蕴含L[R]<$DC。因此，通过将我们自己嵌入L[R]，我们可以假设AD + DC并开始证明定理。这些在L[R]中为真的定理在V中隐含相应的定理，例如将“每一个集合”改为“L[R]中的每一个集合"。关于AD作为公理的更多信息，以及这里涉及的一些观点，读者应该参考[14]，特别是§§ 7 D和8I。在本文中，L[R]将不再被提及。我们就假设剩下的文章都是AD。
Since the discovery of forcing in the early sixties, it has been clear that many natural and interesting mathematical questions are not decidable from the classical axioms of set theory, ZFC. Therefore some mathematicians have been studying the consequences of stronger set theoretic assumptions. Two new types of axioms that have been the subject of much research are large cardinal axioms and axioms asserting the determinacy of definable games. The two appear at first glance to be unrelated; one of the most surprising discoveries of recent research is that this is not the case. In this paper we will be assuming the axiom of determinacy (AD) plus the axiom of dependent choice (DC). AD is false, since it contradicts the axiom of choice. However every set in L[R] is ordinal definable from a real. Our axiom that definable games are determined implies that every game in L[R] is determined (in V), and since a strategy is a real, it is determined in L[R]. That is, L[R] ⊨ AD. The axiom of choice implies L[R] ⊨ DC. So by embedding ourselves in L[R], we can assume AD + DC and begin proving theorems. These theorems true in L[R] imply corresponding theorems in V, by e.g. changing “every set” to “every set in L[R]”. For more information on AD as an axiom, and on some of the points touched on here, the reader should consult [14], particularly §§7D and 8I. In this paper L[R] will no longer even be mentioned. We just assume AD for the rest of the paper.