Given an inner model $W \subset V$ and a regular cardinal $\kappa$, we consider two alternatives for adding a subset to $\kappa$ by forcing: the Cohen poset $Add(\kappa,1)$, and the Cohen poset of the inner model $Add(\kappa,1)^W$. The forcing from $W$ will be at least as strong as the forcing from $V$ (in the sense that forcing with the former adds a generic for the latter) if and only if the two posets have the same cardinality. On the other hand, a sufficient condition is established for the poset from $V$ to fail to be as strong as that from $W$. The results are generalized to $Add(\kappa,\lambda)$, and to iterations of Cohen forcing where the poset at each stage comes from an arbitrary intermediate inner model.

中文翻译：

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科恩强迫和内部模型

给定一个内部模型 $W \subset V$ 和一个常规的基数 $\kappa$，我们考虑通过强制向 $\kappa$ 添加子集的两种选择：Cohenposet $Add(\kappa,1)$，以及内模型 $Add(\kappa,1)^W$ 的 Cohen 偏序集。当且仅当两个偏序集具有相同的基数时，来自 $W$ 的强制将至少与来自 $V$ 的强制一样强（从某种意义上说，对前者的强制为后者添加了泛型）。另一方面，建立了一个充分条件，使得来自 $V$ 的偏序集不能像 $W$ 那样强。结果被推广到 $Add(\kappa,\lambda)$，以及 Cohen 的迭代，其中每个阶段的偏序集来自任意的中间内部模型。