Hume's principle

Hume's principle or HP says that, given two collections of objects ${\displaystyle {\mathcal {F}}}$ and ${\displaystyle {\mathcal {G}}}$ with properties ${\displaystyle F}$ and ${\displaystyle G}$ respectively, the number of objects with property ${\displaystyle F}$ is equal to the number of objects with property ${\displaystyle G}$ if and only if there is a one-to-one correspondence (a bijection) between ${\displaystyle {\mathcal {F}}}$ and ${\displaystyle {\mathcal {G}}}$. In other words, that bijections are the "correct" way of measuring size.

HP can be stated formally in systems of second-order logic. It is named for the Scottish philosopher David Hume and was coined by George Boolos. The principle plays a central role in Gottlob Frege's philosophy of mathematics. Frege shows that HP and suitable definitions of arithmetical notions entail all axioms of what we now call second-order arithmetic. This result is known as Frege's theorem, which is the foundation for a philosophy of mathematics known as neo-logicism.