Cohen’s first model is a model of Zermelo–Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekind-finite set to some infinite ordinal κ. In the case that we force the function to be injective, it turns out that the resulting model is the same as adding κ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that we do not add any reals, sets of ordinals, or collapse any Dedekind-finite sets. This motivates the question if there is any combinatorial condition on a Dedekind-finite set A which characterises when a forcing will preserve its Dedekind-finiteness or not add new sets of ordinals. We answer this question in the case of ‘Adding a Cohen subset’ by presenting a varied list of conditions each equivalent to the preservation of Dedekind-finiteness. For example, 2A is extremally disconnected, or [A]

中文翻译：

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如何通过忘记如何计算来拥有更多的东西

Cohen 的第一个模型是 Zermelo-Fraenkel 集合论模型，其中存在 Dedekind 有限实数集，这可能是选择公理失败的最著名模型。我们强迫这个模型从这个 Dedekind 有限集添加一个函数到某个无限序数 κ。在我们强制函数为单射的情况下，结果模型与将 κ Cohen 实数添加到地面模型相同，并且我们刚刚添加了规范 Dedekind 有限集的枚举。在函数只是满射的情况下，我们不会添加任何实数、序数集或折叠任何戴德金有限集。这引发了一个问题，如果在 Dedekind 有限集 A 上存在任何组合条件，该条件表征强迫何时将保持其 Dedekind 有限性或不添加新的序数集。在“添加 Cohen 子集”的情况下，我们通过提供各种条件列表来回答这个问题，每个条件都相当于 Dedekind 有限性的保持。例如，2A 极端断开，或 [A]