The injective version of Cantor’s theorem appears in full second-order logic as the inconsistency of the abstraction principle, Frege’s Basic Law V (BLV), an inconsistency easily shown using Russell’s paradox. This incompatibility is akin to others—most notably that of a (Dedekind) infinite universe with the Nuisance Principle (NP) discussed by neo-Fregean philosophers of mathematics. This paper uses the Burali–Forti paradox to demonstrate this incompatibility, and another closely related, without appeal to principles related to the axiom of choice—a result hitherto unestablished. It discusses both the general interest of this result, its interest to neo-Fregean philosophy of mathematics, and the potential significance of the Burali–Fortian method of proof.

中文翻译：

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演绎基数结果和类似滋扰的原则

这内射的康托尔定理的版本出现在完整的二阶逻辑中，作为抽象原理的不一致，弗雷格基本定律 V (BLV)，使用罗素悖论很容易证明不一致性。这种不相容性与其他不相容——最值得注意的是，新弗雷格数学哲学家讨论的具有滋扰原理 (NP) 的 (Dedekind) 无限宇宙。本文使用 Burali-Forti 悖论来证明这种不相容性，以及另一个密切相关的不相容性，而不诉诸与选择公理相关的原则——这一结果迄今尚未得到证实。它讨论了这一结果的普遍意义，它对新弗雷格数学哲学的兴趣，以及 Burali-Fortian 证明方法的潜在意义。