Review of Symbolic Logic 14 (3):592-623 (2021)

Sean Ebels-Duggan
Northwestern University
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.
Keywords neo-logicism  abstraction principles  Cantor's theorem  cardinality  second-order logic
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DOI 10.1017/s1755020318000230
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References found in this work BETA

On Some Difficulties in the Theory of Transfinite Numbers and Order Types.Bertrand Russell - 1906 - Proceedings of the London Mathematical Society 4 (14):29-53.
Neo-Fregeanism: An Embarrassment of Riches.Alan Weir - 2003 - Notre Dame Journal of Formal Logic 44 (1):13-48.
Iteration Again.George Boolos - 1989 - Philosophical Topics 17 (2):5-21.
New V, ZF and Abstraction.Stewart Shapiro & Alan Weir - 1999 - Philosophia Mathematica 7 (3):293-321.

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