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Single Idea 10296
[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
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Full Idea
Both of the Löwenheim-Skolem Theorems fail for second-order languages with a standard semantics
Gist of Idea
The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics
Source
Stewart Shapiro (Higher-Order Logic [2001], 2.3.2)
Book Ref
'Blackwell Guide to Philosophical Logic', ed/tr. Goble,Lou [Blackwell 2001], p.47
The
12 ideas
from 'Higher-Order Logic'
10588
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First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems
[Shapiro]
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10290
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Second-order variables also range over properties, sets, relations or functions
[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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10294
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Second-order logic has the expressive power for mathematics, but an unworkable model theory
[Shapiro]
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10591
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Logicians use 'property' and 'set' interchangeably, with little hanging on it
[Shapiro]
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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]
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10298
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Some say that second-order logic is mathematics, not logic
[Shapiro]
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10299
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If the aim of logic is to codify inferences, second-order logic is useless
[Shapiro]
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10300
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Logical consequence can be defined in terms of the logical terminology
[Shapiro]
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10301
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The axiom of choice is controversial, but it could be replaced
[Shapiro]
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