more from Stephen Read

### Single Idea 10980

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

Full Idea

Second-order arithmetic can rule out the non-standard models (with non-standard numbers). Its induction axiom crucially refers to 'any' property, which gives the needed categoricity for the models.

Clarification

'Categoricity' is when all the models are isomorphic to one another

Gist of Idea

Second-order arithmetic covers all properties, ensuring categoricity

Source

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

Book Reference

Read,Stephen: 'Thinking About Logic' [OUP 1995], p.49

Related Idea

Idea 16321
The compactness theorem can prove nonstandard models of PA **[Halbach]**