标题： 一个类似东斯顿定理的策梅洛-弗兰克尔集合论无选择公理的情况 显示英文标题

标题： An Easton-like theorem for Zermelo-Fraenkel Set Theory without Choice

摘要： 我们证明，在没有选择公理的策梅洛-弗兰克尔集合论中，一个通过满射修改的连续统函数$\theta(\kappa)$可以在所有无限基数上几乎取任意值。 这个没有选择公理版本的Easton定理与ZFC中的情况形成鲜明对比，在ZFC中，对于奇异基数$\kappa$，连续统函数$2^\kappa$的值受到其下方连续统函数行为的强烈影响。 我们的构造大致如下：在一个基数上有“合理”函数$F: Card \rightarrow Card$的基模型$V \models ZFC + GCH$中，引入了一个类强迫${\mathbb P}$，它根据$F$将所有基数的幂集放大。 最终模型 $N \models ZF$是由${\mathbb P}$的对称扩展，使得 $\theta^N(\kappa) = F(\kappa)$对所有$\kappa$成立。 显示英文摘要

摘要： We show that in Zermelo-Fraenkel Set Theory without the Axiom of Choice a surjectively modified continuum function $\theta(\kappa)$ can take almost arbitrary values for all infinite cardinals. This choiceless version of Easton's Theorem is in sharp contrast to the situation in ZFC, where for singular cardinals $\kappa$, the value of $2^\kappa$ is strongly influenced by the behaviour of the continuum function below. Our construction can roughly be described as follows: In a ground model $V \models ZFC + GCH$ with a "reasonable" function $F: Card \rightarrow Card$ on the infinite cardinals, a class forcing ${\mathbb P}$ is introduced, which blows up the power sets of all cardinals according to $F$ . The eventual model $N \models ZF$ is a symmetric extension by ${\mathbb P}$ such that $\theta^N(\kappa) = F(\kappa)$ holds for all $\kappa$.