more from Stewart Shapiro

### Single Idea 13675

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

Full Idea

The Upward Löwenheim-Skolem theorem fails (trivially) with substitutional semantics. If there are only countably many terms of the language, then there are no uncountable substitution models.

Gist of Idea

Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails

Source

Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4)

Book Reference

Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.245

A Reaction

Better and better. See Idea 13674. Why postulate more objects than you can possibly name? I'm even suspicious of all real numbers, because you can't properly define them in finite terms. Shapiro objects that the uncountable can't be characterized.

Related Idea

Idea 13674
We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models **[Shapiro]**