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]