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5. Theory of Logic / J. Model Theory in Logic / 3. L÷wenheim-Skolem Theorems

[group of theorems about models involving infinities]

24 ideas
If a 1st-order proposition is satisfied, it is satisfied in a denumerably infinite domain [Skolem]
The L÷wenheim-Skolem Theorem is close to an antinomy in philosophy of language [Putnam]
The L÷wenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals) [Tharp]
Skolem mistakenly inferred that Cantor's conceptions were illusory [Tharp]
If it is a logic, the L÷wenheim-Skolem theorem holds for it [Hacking]
Down L÷wenheim-Skolem: if a countable language has a consistent theory, that has a countable model [Hodges,W]
Up L÷wenheim-Skolem: if infinite models, then arbitrarily large models [Hodges,W]
L÷wenheim-Skolem says any theory with a true interpretation has a model in the natural numbers [White,NP]
No L÷wenheim-Skolem logic can axiomatise real analysis [Mayberry]
The L÷wenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro]
Upward L÷wenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]
Downward L÷wenheim-Skolem: each satisfiable countable set always has countable models [Shapiro]
Substitutional semantics only has countably many terms, so Upward L÷wenheim-Skolem trivially fails [Shapiro]
Downward L÷wenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
Up L÷wenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
The L÷wenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]
The L÷wenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
Any theory with an infinite model has a model of every infinite cardinality [Shapiro]
Any first-order theory of sets is inadequate [Read]
L÷wenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman]
L÷wenheim-Skolem Theorem, and G÷del's completeness of first-order logic, the earliest model theory [Feferman/Feferman]
The LST Theorem is a serious limitation of first-order logic [Wolf,RS]
The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki]
L÷wenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan]