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Single Idea 17748
[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
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Full Idea
The L-S Theorem is ...a shocking result, since it implies that any consistent formal theory of everything - even about biology, physics, sets or the real numbers - can just as well be understood as being about natural numbers. It says nothing more.
Gist of Idea
The L-S Theorem says no theory (even of reals) says more than a natural number theory
Source
Michal Walicki (Introduction to Mathematical Logic [2012], History E.2)
Book Ref
Walicki,Michal: 'Introduction to Mathematical Logic' [World Scientific 2012], p.34
A Reaction
Illuminating. Particularly the point that no theory about the real numbers can say anything more than a theory about the natural numbers. So the natural numbers contain all the truths we can ever express? Eh?????
The
24 ideas
with the same theme
[group of theorems about models involving infinities]:
17878
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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]
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10773
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The Löwenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals)
[Tharp]
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10777
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Skolem mistakenly inferred that Cantor's conceptions were illusory
[Tharp]
|
13843
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If it is a logic, the Löwenheim-Skolem theorem holds for it
[Hacking]
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10289
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Up Löwenheim-Skolem: if infinite models, then arbitrarily large models
[Hodges,W]
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10288
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Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model
[Hodges,W]
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17813
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Löwenheim-Skolem says any theory with a true interpretation has a model in the natural numbers
[White,NP]
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17790
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No Löwenheim-Skolem logic can axiomatise real analysis
[Mayberry]
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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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13659
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Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes
[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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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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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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10297
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The Löwenheim-Skolem theorem seems to be a defect of first-order logic
[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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10988
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Any first-order theory of sets is inadequate
[Read]
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10160
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Löwenheim-Skolem says if the sentences are countable, so is the model
[Feferman/Feferman]
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10159
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Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory
[Feferman/Feferman]
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13539
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The LST Theorem is a serious limitation of first-order logic
[Wolf,RS]
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17748
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The L-S Theorem says no theory (even of reals) says more than a natural number theory
[Walicki]
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17929
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Löwenheim proved his result for a first-order sentence, and Skolem generalised it
[Colyvan]
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