论文标题
关于ZF和ZFC渐近表达性的注释
A note on the asymptotic expressiveness of ZF and ZFC
论文作者
论文摘要
我们研究了在Zermelo-Fraenkel Set理论ZF及其扩展ZFC中可证明的定理的渐近密度，包括选择的公理。假设公式的规范表示，我们在ZF中无法证明渐近的句子，但在ZFC中可证明。此外，我们将ZFC定理的渐近密度与ZFC本身的可证明的一致性联系起来。因此，如果ZFC一致，则不可能反驳ZFC定理在ZFC内的渐近密度的存在。这两个结果都涉及Zaionc关于ZF和ZFC的渐近等效性的最新问题。
We investigate the asymptotic densities of theorems provable in Zermelo-Fraenkel set theory ZF and its extension ZFC including the axiom of choice. Assuming a canonical De Bruijn representation of formulae, we construct asymptotically large sets of sentences unprovable within ZF, yet provable in ZFC. Furthermore, we link the asymptotic density of ZFC theorems with the provable consistency of ZFC itself. Consequently, if ZFC is consistent, it is not possible to refute the existence of the asymptotic density of ZFC theorems within ZFC. Both these results address a recent question by Zaionc regarding the asymptotic equivalence of ZF and ZFC.
