Frege's theorem

Frege's theorem states that, in second-order logic with suitable comprehension principles, Hume’s principle implies the axioms of second-order Peano arithmetic.

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Hume's principle: An abstraction principle equating the number of Fs with the number of Gs exactly when F and G are equinumerous. It supplies the premise from which Frege's theorem derives arithmetic.
Second-order logic: A logical system whose quantifiers can range over properties, relations, or sets as well as individuals. The theorem's strength comes from quantifying over concepts, not individuals alone.
Gottlob Frege: A German philosopher and logician who developed modern quantificational logic and sought to derive arithmetic from logic. Frege formulated the abstraction-based route to arithmetic associated with the theorem.
Neo-logicism: A philosophy of mathematics that seeks to derive arithmetic from logical principles, especially abstraction principles. It treats Frege's theorem as a model for recovering arithmetic after the failure of Frege's original system.
Equinumerosity: A relation between concepts that holds when their instances can be paired one-to-one. Hume's principle identifies concepts as having the same number through this relation.
Second-order Peano arithmetic: The axioms characterizing zero, successor, and induction for the natural numbers using second-order logic. This is the arithmetic system Frege's theorem derives from Hume's principle.
The Foundations of Arithmetic: Frege's 1884 book arguing that arithmetic truths are analytic and grounded in logic. It presents the philosophical program that motivates deriving arithmetic from logical principles.
Hume's principle consistency: The result that Hume's principle is consistent relative to suitable background theories, unlike unrestricted Basic Law V. Consistency helps explain why the theorem survives the collapse of Frege's extension-based system.
Number operator: An operator that assigns a number object to each concept under an abstraction principle. Frege's construction treats numbers as objects assigned to concepts.
Concept (Frege): In Frege's logic, a concept is an unsaturated function-like entity that applies to objects. Concepts are the inputs to the number operator and the domain of comprehension.