强迫同构
Forcing isomorphism
摘要：
一个力迫扩张可以在一阶理论的两个模型之间产生新的同构。对理论的某些模型理论约束和对强迫的其他约束可以防止这种病理。一个可数一阶理论是可分类的，如果它是超稳定的，并且不具有维数序性质或省略型序性质。Shelah [Sh：c]表明，如果理论T是可分类的，那么基数λ的每个模型都由L_{infty, lambda}的句子描述。事实上，这句话可以在L^*_{lambda}中选择。（L^*_{lambda}是通过为每个μ < lambda添加表示依赖结构的维数大于μ的量词来丰富语言L_{infty, beth^+}的结果。）这样的句子的真实性将被任何不坍缩基数<= lambda且不添加lambda的新可数子集的强制所保持。因此，如果幂λ的可分类理论的两个模型是非同构的，那么它们在一个λ-完全强迫之后也是非同构的。在这里，我们表明，强制添加没有新的可数子集的lambda的假设不能被消除。特别是，我们表明，非同构的模型的分类理论不需要保持ccc强迫。
英文摘要：
A forcing extension may create new isomorphisms between two models of a first order theory. Certain model theoretic constraints on the theory and other constraints on the forcing can prevent this pathology. A countable first order theory is classifiable if it is superstable and does not have either the dimensional order property or the omitting types order property. Shelah [Sh:c] showed that if a theory T is classifiable then each model of cardinality lambda is described by a sentence of L_{infty, lambda}. In fact this sentence can be chosen in the L^*_{lambda}. (L^*_{lambda} is the result of enriching the language L_{infty, beth^+} by adding for each mu < lambda a quantifier saying the dimension of a dependence structure is greater than mu .) The truth of such sentences will be preserved by any forcing that does not collapse cardinals <= lambda and that adds no new countable subsets of lambda. Hence, if two models of a classifiable theory of power lambda are non-isomorphic, they are non-isomorphic after a lambda-complete forcing. Here we show that the hypothesis of the forcing adding no new countable subsets of lambda cannot be eliminated. In particular, we show that non-isomorphism of models of a classifiable theory need not be preserved by ccc forcings.
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