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Single Idea 10301
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
]
Full Idea
The axiom of choice has a troubled history, but is now standard in mathematics. It could be replaced with a principle of comprehension for functions), or one could omit the variables ranging over functions.
Gist of Idea
The axiom of choice is controversial, but it could be replaced
Source
Stewart Shapiro (Higher-Order Logic [2001], n 3)
Book Ref
'Blackwell Guide to Philosophical Logic', ed/tr. Goble,Lou [Blackwell 2001], p.52
The
12 ideas
from 'Higher-Order Logic'
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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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10590
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Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them
[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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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
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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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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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