Philosophy of Mathematics

by Øystein Linnebo

What exactly is the arithmetical theory that Frege wished to reduce to pure logic? The target theory is what is now known as second-order Dedekind-Peano arithmetic. This is an arithmetical theory that is formulated in a second-order language with three (for now) nonlogical symbols: a constant ‘0’ for the number zero and predicates ‘Pxy’ and ‘ℕx,’ stating that x immediately precedes y and that x is a natural number, respectively. The theory has the following axioms:13 (1) ℕ0 (2) ℕx ∧ Pxy → Ny (3) ℕx ∧ Pxy ∧ Pxy′ → y = y′ (4) ℕy ∧ Pxy ∧ Px′y → x = x′ (5) ℕx → ∃y Pxy (6) ∀F (F 0 ∧ ∀x∀y(ℕx ∧ Fx ∧ Pxy → Fy) → ∀x(ℕx → Fx)) To reduce this theory to pure logic, Frege’s first task is to provide a logical analysis of the three symbols still left unanalyzed. He begins by observing that zero is the number that applies to concepts under which no objects fall. This suggests that ‘0’ be defined as ‘#x(x ≠ x),’ that is, as the number of objects that are distinct from themselves. Next, what about the relation of one number immediately preceding another? This is simple enough to explain to a young child. Suppose you have some objects F and remove one of them, say a. Then the number of things left immediately precedes the number of things with which you started. This means that ‘Pxy’ can be defined as Finally, there is the concept of being a natural number. Informally, we often say that x is a natural number just in case x = 0, or x = 1, or …. But clearly, the ellipsis ‘…’ would not ha...