关于Kappa-Resplant模型的谱
On Spectrum of kappa-Resplendent Models
摘要：
我们证明了一些自然的“外”属性是等价的（一阶类）是稳定的。对于一个模型来说，璀璨是对卡帕饱和的强化。将我们自己限制在透明的情况kappa > |T|，模型M是kappa-辉煌的意思是：当我们将M扩展<kappa单个常数（c_i：i < alpha）时，如果（M，c_i）_ {i<alpha}具有可扩展为T'的模型的基本扩展，其中Th（（M，c_i）_{i<alpha}）subseteq T'，|T'|< kappa则（M，c_i）_{i<α}已经可以扩展到T '的模型。平凡地，基数λ的任何饱和模型都是λ-辉煌的。我们问：基数λ的（一阶完备）理论T的卡帕辉煌模型是如何存在的？自然地，我们将自己限制为基数lambda=lambda^ kappa + 2^ { |T| }。然后我们得到了一个完整而令人满意的答案：这仅仅取决于T是稳定的还是不稳定的。在这种情况下，证明对于稳定的T，我们得到的很少，并不难;事实上，T的每一个辉煌的模型都是饱和的，因此由它的基数直到同构决定。逆函数的问题更大，因为我们必须使用带有任意alpha<kappa位置的Skolem函数。通常我们使用相关的划分定理（拉姆齐定理或Erdens-Rado定理），但在我们的情况下，相关的划分定理失败了，所以我们必须小心。
英文摘要：
We prove that some natural "outside" property is equivalent (for a first order class) to being stable. For a model, being resplendent is a strengthening of being kappa-saturated. Restricting ourselves to the case kappa > |T| for transparency, a model M is kappa-resplendent means: when we expand M by <kappa individual constants (c_i:i < alpha), if (M, c_i)_ {i<alpha} has an elementary extension expandable to be a model of T' where Th((M,c_i)_{i<alpha}) subseteq T', |T'| < kappa then already (M,c_i)_{i<α} can be expanded to a model of T'. Trivially any saturated model of cardinality lambda is lambda-resplendent. We ask: how may kappa-resplendent models of a (first order complete) theory T of cardinality lambda are there? Naturally we restrict ourselves to cardinals lambda=lambda^ kappa + 2^ {|T|}. Then we get a complete and satisfying answer: this depends just on T being stable or unstable. In this case proving that for stable T we get few, is not hard; in fact every resplendent model of T is saturated hence determined by its cardinality up to isomorphism. The inverse is more problematic because naturally we have to use Skolem functions with any alpha<kappa places. Normally we use relevant partition theorems (Ramsey theorem or Erdős-Rado theorem), but in our case the relevant partitions theorems fails so we have to be careful.
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