more from Peter Smith

Single Idea 10616

[catalogued under 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic]

Full Idea

Going second-order in arithmetic enables us to prove new first-order arithmetical sentences that we couldn't prove before.

Gist of Idea

Second-order arithmetic can prove new sentences of first-order


Peter Smith (Intro to Gödel's Theorems [2007], 23.4)

Book Reference

Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.206

A Reaction

The wages of Satan, perhaps. We can prove things about objects by proving things about their properties and sets and functions. Smith says this fact goes all the way up the hierarchy.