more from E Reck / M Price

### Single Idea 10164

#### [catalogued under 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order]

Full Idea

A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction.

Gist of Idea

Peano Arithmetic can have three second-order axioms, plus '1' and 'successor'

Source

E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2)

A Reaction

This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'!