Single Idea 13648

[catalogued under 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems]

Full Idea

The Löwenheim-Skolem theorems mean that no first-order theory with an infinite model is categorical. If Γ has an infinite model, then it has a model of every infinite cardinality. So first-order languages cannot characterize infinite structures.

Clarification

For 'categorical' see Idea 13636

Gist of Idea

The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity

Source

Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)

Book Reference

Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.80


A Reaction

So much of the debate about different logics hinges on characterizing 'infinite structures' - whatever they are! Shapiro is a leading structuralist in mathematics, so he wants second-order logic to help with his project.

Related Idea

Idea 13636 An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]