more from Stephen Read

### Single Idea 10979

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

Full Idea

Second-order arithmetic is categorical - indeed, there is a single formula of second-order logic whose only model is the standard model ω, consisting of just the natural numbers, with all of arithmetic following. It is nevertheless incomplete.

Gist of Idea

Although second-order arithmetic is incomplete, it can fully model normal arithmetic

Source

Stephen Read (Thinking About Logic [1995], Ch.2)

Book Reference

Read,Stephen: 'Thinking About Logic' [OUP 1995], p.47

A Reaction

This is the main reason why second-order logic has a big fan club, despite the logic being incomplete (as well as the arithmetic).