Abstract
HP asserts that the number of As is the same as the number of Bs just in case the As can be correlated one-one with the Bs. Fregean arithmetics are thus theories of cardinal numbers. Since HP quantifies over relations, the logic of a Fregean arithmetic has to be some sort of second-order logic, and how much arithmetic we can interpret will depend upon how strong that logic is. Now, the strength of second-order logic derives from the so-called comprehension axioms,2 each of which states, in effect, that a given formula defines a ‘concept’ or a ‘relation’: something in the domain of the second-order variables. These axioms take the form:
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