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

#### [group of theorems about models involving infinities]

24 ideas
 17878 If a 1st-order proposition is satisfied, it is satisfied in a denumerably infinite domain [Skolem]
 9913 The Löwenheim-Skolem Theorem is close to an antinomy in philosophy of language [Putnam]
 10773 The Löwenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals) [Tharp]
 10777 Skolem mistakenly inferred that Cantor's conceptions were illusory [Tharp]
 13843 If it is a logic, the Löwenheim-Skolem theorem holds for it [Hacking]
 10289 Up Löwenheim-Skolem: if infinite models, then arbitrarily large models [Hodges,W]
 10288 Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model [Hodges,W]
 17813 Löwenheim-Skolem says any theory with a true interpretation has a model in the natural numbers [White,NP]
 17790 No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry]
 13648 The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro]
 13659 Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]
 13658 Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [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]
 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]
 10297 The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
 10234 Any theory with an infinite model has a model of every infinite cardinality [Shapiro]