Ideas from 'What Numbers Are' by Nicholas P. White [1974], by Theme Structure
[found in 'Philosophy of Mathematics: anthology' (ed/tr Jacquette,Dale) [Blackwell 2002,0-631-21870-x]].
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
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Löwenheim-Skolem says any theory with a true interpretation has a model in the natural numbers
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Full Idea:
The Löwenheim-Skolem theorem tells us that any theory with a true interpretation has a model in the natural numbers.
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From:
Nicholas P. White (What Numbers Are [1974], V)
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
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Finite cardinalities don't need numbers as objects; numerical quantifiers will do
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Full Idea:
Statements involving finite cardinalities can be made without treating numbers as objects at all, simply by using quantification and identity to define numerically definite quantifiers in the manner of Frege.
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From:
Nicholas P. White (What Numbers Are [1974], IV)
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A reaction:
[He adds Quine 1960:268 as a reference]
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