Trees and -subsets of ω1ω1
Trees and -subsets of ω1ω1
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树和 ω1ω1 的子集
DOI：
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发表时间：
1993
期刊：
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通讯作者：
J. Väänänen
中科院分区：
文献类型：
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作者：
A. Mekler;J. Väänänen
摘要我们研究空间中的描述集合论，让没有不可数分支的树扮演与传统描述集合论中可数序数类似的角色。通过使用这样的树，我们得到，例如，一个覆盖性质的类的集合。我们称一类树的树族为泛树族，如果图中的每一棵树都能有序映射到图中的一棵树。众所周知，没有无限分支的可数树类有一个大小为1的泛族。我们将研究一个泛族的最小基数，这个泛族是基数≤ 101且没有不可数分支的树类。我们证明了这个基数可以是1（在<$CH下）和任何正则基数κ满足（在CH下）。这直接关系到空间的-子集的覆盖性质。我们还研究了可定义子集的可能基数。本文证明了的每个可定义子集的基数<ωn或基数与ZFC（n ≥ 3）和与ZFC加不可达（n = 2）等相容。最后，我们定义了一个类似的概念的Borel集的空间，并证明了一个Souslin-Kleene型定理这个概念。
Abstract We study descriptive set theory in the space by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of -sets of . We call a family of trees universal for a class of trees if ⊆ and every tree in can be order-preservingly mapped into a tree in . It is well known that the class of countable trees with no infinite branches has a universal family of size ℵ1. We shall study the smallest cardinality of a universal family for the class of trees of cardinality ≤ ℵ1 with no uncountable branches. We prove that this cardinality can be 1 (under ¬CH) and any regular cardinal κ which satisfies (under CH). This bears immediately on the covering property of the -subsets of the space . We also study the possible cardinalities of definable subsets of . We show that the statement that every definable subset of has cardinality <ωn or cardinality is equiconsistent with ZFC (if n ≥ 3) and with ZFC plus an inaccessible (if n = 2). Finally, we define an analogue of the notion of a Borel set for the space and prove a Souslin-Kleene type theorem for this notion.