Combining Texts

Ideas for 'Symposium', 'Foundations without Foundationalism' and 'Ordinatio'

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7 ideas

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models

13644

Semantics for models uses set-theory [Shapiro]

5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms

13636

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

13670

Categoricity can't be reached in a first-order language [Shapiro]

5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems

13648

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

13675

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

13658

Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro]

13659

Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]