ABSOLUTENESS FOR UNIVERSALLY BAIRE SETS AND THE UNCOUNTABLE II
ABSOLUTENESS FOR UNIVERSALLY BAIRE SETS AND THE UNCOUNTABLE II
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普遍贝雷集和不可数 II 的绝对性
DOI：
10.1142/9789812796554_0009
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发表时间：
2008
影响因子：
1.8
通讯作者：
M. Magidor
中科院分区：
文献类型：
--
作者：
I. Farah;R. Ketchersid;P. Larson;M. Magidor
康托的连续统假设被证明独立于Gödel和Cohen通常的集合论ZFC公理。科恩为此开发的强迫方法在接下来的几十年里导致了大量的独立结果。Borel猜想、Whitehead问题和Banach代数的自动连续性等许多关于无限集的陈述都被证明是独立的，这可能会给人留下这样一种印象，即大多数关于无限集的非平凡陈述在ZFC中既不能证明也不能反驳。此外，一些经典命题蕴含着ZFC与更强理论的一致性，而利用Gödel的不完全性定理，这些命题与ZFC的一致性只能用无穷强公理，即所谓的大基数公理来证明。一个经典的例子是巴纳赫的“勒贝格度量对所有的领域集都有一个σ可加扩展”。虽然很容易找到一个模型，其中这是假的，并且没有已知的ZFC证明其否定，但要证明这一陈述的一致性，需要假设存在一个可测的基数([32])。Shoenfield([31])证明了一个显著的结果：形式为(∃x∈R)(∀y∈R)φ(x，y)的任何语句，其中φ中的所有量化都在自然数上且其所有参数都是实数，则在传递且包含所有可数序数的ZFC模型之间是绝对的。在这种形式下，肖恩菲尔德定理是最有可能的，因为它甚至不能通过增加一个范围在R上的量词的替换来改进。然而，一个推论是，任何Σ2语句(即，上述句法形式之一)的真实性不能通过强迫来改变，事实证明，它容易受到影响深远的概括的影响。现代集合论中一个比较显著的结果是，适当的大基数的存在意味着内模L(R)的理论(ZF的最小内模，没有选择公理的集合论的通常公理，包含所有实数)不能通过集合强迫来改变(见[13，21])。具体地说，具有实参且量词在R上有任意次数变化的句子具有固定的真值，不能通过强制来改变。大基数对实数集的影响远远超出了L(R)的范围，它暗示了某些标准实数集的绝对性，泛贝尔集([11])，定义如下。一个值得注意的结果是，大基数的存在直接意味着L(R)中的所有实数集，实际上是所有泛Baire集，都共享Borel集的所有经典正则性性质，如勒贝格可测性。
Cantor’s Continuum Hypothesis was proved to be independent from the usual ZFC axioms of Set Theory by Gödel and Cohen. The method of forcing, developed by Cohen to this end, has lead to a profusion of independence results in the following decades. Many other statements about infinite sets, such as the Borel Conjecture, Whitehead’s problem, and automatic continuity for Banach Algebras, were proved independent, perhaps leaving an impression that most nontrivial statements about infinite sets can be neither proved nor refuted in ZFC. Moreover, some classical statements imply the consistency of ZFC and stronger theories, and by Gödel’s incompleteness theorems the consistency of these statements with ZFC can be proved only by using strong axioms of infinity, so-called large cardinal axioms. A classical example is Banach’s ‘Lebesgue measure has a σ-additive extension to all sets of reals.’ While it is fairly easy to find a model in which this is false and there is no known ZFC-proof of its negation, proving the consistency of this statement requires assuming the existence of a measurable cardinal ([32]). A remarkable result was proved by Shoenfield ([31]): every statement of the form (∃x ∈ R)(∀y ∈ R)φ(x, y), where all quantification in φ is over the natural numbers and all of its parameters are real numbers, is absolute between models of ZFC that are transitive and contain all countable ordinals. In this form Shoenfield’s theorem is best possible, as it cannot even be improved by adding one more alternation of quantifiers ranging over R. However, a corollary that the truth of any Σ2 statement (i.e., one of the above syntactical form) cannot be changed by forcing turned out to be susceptible to far-reaching generalizations. One of the more striking results in modern set theory is that the existence of suitable large cardinals implies that the theory of the inner model L(R) (the smallest inner model of ZF, the usual axioms of Set Theory without the Axiom of Choice, containing all real numbers) cannot be changed by set forcing (see [13, 21]). In particular, a sentence with real parameters and any number of alterations of quantifiers ranging over R, has a fixed truth value that cannot be changed by forcing. The impact of large cardinals on sets of reals goes well beyond L(R) to imply absoluteness for certain canonical sets of reals, the universally Baire sets ([11]), as defined below. A remarkable consequence is that the existence of large cardinals outright implies that all sets of reals in L(R), and indeed all universally Baire sets, share all the classical regularity properties of Borel sets such as Lebesgue measurability.