Study of first order model theory and its application

我们扩展了Steinhorn的省略类型定理。当理论T是超稳定时，他的定理成立。我们把超稳定的假设弱化为一个较弱的超简单条件。斯坦霍恩最初的证明使用了平均类型的概念。我们的证明没有用这样的概念。我们首先证明了一个相当一般的结果，然后将该结果应用到一个超简单理论模型中，得到了Steinhorn结果的推广作为推论：结果：设L是可数语言，T是用L表述的完全超简单理论，设M是具有无限独立（在分叉意义上）集的可数模型。假设M省略了空集合上的完整类型p。那么M的一个适当初等扩展N也省略了p。

We extended Steinhorn's omitting types theorem. His theorem holds when the theory T is superstable. We weakened the assumption of superstablity to a weaker condition of supersimlicity. Steinhorn's original proof uses the notion of average types. Our proof does not use such a notion. First we proved a quite general result and then, by applying the result to a model of supersimple theory, we get the following extension of Steinhorn's result as a corollary :The result : Let L be a countable language and T a complete supersimple theory formulated in L. Let M be a countable model of Thaving an infinite independent (in the sense of forking) set. Suppose that M omits a complete type p over the empty set. Then there is a proper elementary extension N of M that also omits p.