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Single Idea 10160
[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
]
Full Idea
The Löwenheim-Skolem Theorem, the earliest in model theory, states that if a countable set of sentences in a first-order language has a model, then it has a countable model.
Gist of Idea
Löwenheim-Skolem says if the sentences are countable, so is the model
Source
Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V)
Book Ref
Feferman,S/Feferman,A.B.: 'Alfred Tarski: life and logic' [CUP 2008], p.281
A Reaction
There are 'upward' (sentences-to-model) and 'downward' (model-to-sentences) versions of the theory.
The
24 ideas
with the same theme
[group of theorems about models involving infinities]:
17878
|
If a 1st-order proposition is satisfied, it is satisfied in a denumerably infinite domain
[Skolem]
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9913
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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]
|
10288
|
Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model
[Hodges,W]
|
10289
|
Up Löwenheim-Skolem: if infinite models, then arbitrarily large models
[Hodges,W]
|
17813
|
Löwenheim-Skolem says any theory with a true interpretation has a model in the natural numbers
[White,NP]
|
17790
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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]
|
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]
|
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]
|
10988
|
Any first-order theory of sets is inadequate
[Read]
|
10160
|
Löwenheim-Skolem says if the sentences are countable, so is the model
[Feferman/Feferman]
|
10159
|
Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory
[Feferman/Feferman]
|
13539
|
The LST Theorem is a serious limitation of first-order logic
[Wolf,RS]
|
17748
|
The L-S Theorem says no theory (even of reals) says more than a natural number theory
[Walicki]
|
17929
|
Löwenheim proved his result for a first-order sentence, and Skolem generalised it
[Colyvan]
|