A minimal model for set theory
A minimal model for set theory
复制标题
集合论的最小模型
DOI：
10.1090/s0002-9904-1963-10989-1
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发表时间：
1963
期刊：
影响因子：
--
通讯作者：
P. J. Cohen
中科院分区：
文献类型：
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作者：
P. J. Cohen
为了证明连续假设的一致性和与集合理论的其他公理的选择公理，Godel [L]引入了可构造集的概念，并表明该构造集构成了集合理论的模型。这些集合是可以通过几个简单操作的转限序列来达到的那些集合。然后，他表明，在可构造集的集合中，选择的公理和连续假设。如果原始集合的原始宇宙足够丰富，那么每个集合都是可构造的，在这种情况下，我们说满足了构造性的公理。该公理意味着前面提到的两个公理。但是，从一种角度来看，这种构造性的概念似乎与建设性的含义并不相对，因为可能发生宇宙中的所有集合都是建设性的。在本文中，我们表明，“构造”的更受限制的概念将产生一类集合，这些集合构成了集合理论的最低模型。通过这种方式，我们证明了更强的构造性公理形式的一致性。我们观察到，满足某些公理的对象的最小收集的想法在数学中是众所周知的，例如，在小组理论中，人们经常考虑由元素集合产生的子组，而在衡量理论中，我们将Borel集定义为最小的集合。 y）持有，然后存在一个B集，该集由y依靠y组成。由于我们陈述的理论的许多证据都非常紧密地遵循[L]的论点，因此我们将很简短。我们的主要结果是
In the proof of the consistency of the Continuum Hypothesis and the Axiom of Choice with the other axioms of set theory, Godel [ l ] introduced the notion of a constructible set and showed that the constructible sets form a model for set theory. These sets are intuitively those which can be reached by means of a transfinite sequence of several simple operations. He then showed that the Axiom of Choice and Continuum Hypothesis held in the collection of constructible sets. If the original universe of sets is sufficiently rich in ordinal numbers, it will follow that every set is constructible, in which case we say that the Axiom of Constructibility is satisfied. This axiom implies the two axioms previously mentioned. However, from one point of view it may seem that this notion of constructibility does not intuitively correspond to what is meant by constructive since it may happen that all sets in the universe are constructive. In this paper we show that a more restricted notion of "construction" will yield a class of sets which form a minimal model for set theory. In this manner we prove the consistency of a stronger form of the Axiom of Constructibility. We observe that the idea of a minimal collection of objects satisfying certain axioms is well known in mathematics, for example, in group theory one often considers the subgroup generated by a collection of elements, and in measure theory we define the Borel sets as the smallest y) holds, then there exists a set B consisting of precisely those y. Since much of the proofs of the theorems we state follow quite closely the arguments of [ l ] , we shall be rather brief. Our main result is