Combining Texts
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'A Résumé of Metaphysics', 'A Tour through Mathematical Logic' and 'Higher-Order Logic'
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10 ideas
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
13519
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Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
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13531
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Model theory reveals the structures of mathematics [Wolf,RS]
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13532
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Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
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13533
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First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS]
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5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
13537
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An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
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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13539
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The LST Theorem is a serious limitation of first-order logic [Wolf,RS]
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