Combining Philosophers
Ideas for William Lyons, Alexander Baumgarten and Stewart Shapiro
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16 ideas
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
13644
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Semantics for models uses set-theory [Shapiro]
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10239
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The central notion of model theory is the relation of 'satisfaction' [Shapiro]
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10240
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Model theory deals with relations, reference and extensions [Shapiro]
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5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
13670
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Categoricity can't be reached in a first-order language [Shapiro]
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10214
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Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro]
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10238
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The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro]
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13636
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An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
10590
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Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
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10296
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The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]
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13658
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Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro]
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13659
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Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]
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10297
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The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
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13648
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The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro]
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13675
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Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro]
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10292
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Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
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10234
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Any theory with an infinite model has a model of every infinite cardinality [Shapiro]
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