扩展、省略类型和标准系统
Expansions, omitting types, and standard systems
摘要：
递归饱和和辉煌是算术模型中的两个重要概念。Kaye、Kossak和Kotlarski引入了算术饱和的概念，并认为递归饱和可能不像最初假设的那样严格。在这篇论文中，我们给出了递归饱和的变化的进一步的例子，所有这些都与类似于辉煌的可扩展性属性有关。然而，可扩展性比辉煌性更强，并且以某种方式暗示着扩展不仅满足一个理论，而且省略了一个类型。我们推测，这种可扩展性的一个特殊版本实际上相当于算术饱和。我们证明了这些性质中的另一个等价于β-饱和。我们还介绍了递归饱和的一个变体，它在标准谓词的上下文中是有意义的，并且相当于一定量的普通饱和。本文还研究了所有忽略某个类型p（x）的模型的理论。我们定义了一个证明系统，它证明一个句子当且仅当它在所有省略类型p（x）的模型中为真。讨论了这类证明系统的复杂性，并给出了在特殊意义下具有高复杂性的一些理论和类型的显式例子。最后，我们对斯科特的问题做了一个小小的评论。在Martin公理的假设下，证明了具有可数链条件的在算术理解下闭的基数<2^{ \aleph_0 }的Scott集是PA的某个模型的标准系.然而，我们不知道是否存在这样的不可数Scott集。
英文摘要：
Recursive saturation and resplendence are two important notions in models of arithmetic. Kaye, Kossak, and Kotlarski introduced the notion of arithmetic saturation and argued that recursive saturation might not be as rigid as first assumed. In this thesis we give further examples of variations of recursive saturation, all of which are connected with expandability properties similar to resplendence. However, the expandability properties are stronger than resplendence and implies, in one way or another, that the expansion not only satisfies a theory, but also omits a type. We conjecture that a special version of this expandability is in fact equivalent to arithmetic saturation. We prove that another of these properties is equivalent to β-saturation. We also introduce a variant on recursive saturation which makes sense in the context of a standard predicate, and which is equivalent to a certain amount of ordinary saturation. The theory of all models which omit a certain type p(x) is also investigated. We define a proof system, which proves a sentence if and only if it is true in all models omitting the type p(x). The complexity of such proof systems are discussed and some explicit examples of theories and types with high complexity, in a special sense, are given. We end the thesis by a small comment on Scott's problem. We prove that, under the assumption of Martin's axiom, every Scott set of cardinality <2^{\aleph_0} closed under arithmetic comprehension which has the countable chain condition is the standard system of some model of PA. However, we do not know if there exists any such uncountable Scott sets.
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