Combining Philosophers

Ideas for Eubulides, Geoffrey Hellman and Stewart Shapiro

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16 ideas

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
13644 Semantics for models uses set-theory [Shapiro]
10239 The central notion of model theory is the relation of 'satisfaction' [Shapiro]
10240 Model theory deals with relations, reference and extensions [Shapiro]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
13670 Categoricity can't be reached in a first-order language [Shapiro]
10214 Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro]
10238 The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro]
13636 An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
10590 Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
10296 The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [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]
10297 The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
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]
10292 Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
10234 Any theory with an infinite model has a model of every infinite cardinality [Shapiro]