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 inconsis-tency 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 princi-ples 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.
ASJC Scopus subject areas
- Mathematics (miscellaneous)