Single Idea 13701

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

Full Idea

A single second-order sentence has second-order semantic consequences which are all and only the truths of arithmetic, but this is cold comfort because of incompleteness; no axiomatic system draws out the consequences of this axiom.

Gist of Idea

A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically

Source

Theodore Sider (Logic for Philosophy [2010], 5.4.3)

Book Reference

Sider,Theodore: 'Logic for Philosophy' [OUP 2010], p.124