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Ideas for 'Axiomatic Theories of Truth', 'Reference and Definite Descriptions' and '11: Book of Kings 1'

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5 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / n. Pi
17914 He made a molten sea, which was ten cubits across, and thirty cubits round the edge [Anon (Kings)]
     Full Idea: And he made a molten sea, ten cubits from the one brim to the other; it was round all about, and his height was five cubits: and a line of cubits did compass it round about.
     From: Anon (Kings) (11: Book of Kings 1 [c.550 BCE], 7:23)
     A reaction: In the sixth century BCE, this appears to give 3 as the value of Pi, though perhaps it shouldn't be taken too literally!
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
16321 The compactness theorem can prove nonstandard models of PA [Halbach]
     Full Idea: Nonstandard models of Peano arithmetic are models of PA that are not isomorphic to the standard model. Their existence can be established with the compactness theorem or the adequacy theorem of first-order logic.
     From: Volker Halbach (Axiomatic Theories of Truth [2011], 8.3)
16343 The global reflection principle seems to express the soundness of Peano Arithmetic [Halbach]
     Full Idea: The global reflection principle ∀x(Sent(x) ∧ Bew[PA](x) → Tx) …seems to be the full statement of the soundness claim for Peano arithmetic, as it expresses that all theorems of Peano arithmetic are true.
     From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1)
     A reaction: That is, an extra principle must be introduced to express the soundness. PA is, of course, not complete.
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
16312 To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals [Halbach]
     Full Idea: For the reduction of Peano Arithmetic to ZF set theory, usually the set of finite von Neumann ordinals is used to represent the non-negative integers.
     From: Volker Halbach (Axiomatic Theories of Truth [2011], 6)
     A reaction: Halbach makes it clear that this is just one mode of reduction, relative interpretability.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
16308 Set theory was liberated early from types, and recent truth-theories are exploring type-free [Halbach]
     Full Idea: While set theory was liberated much earlier from type restrictions, interest in type-free theories of truth only developed more recently.
     From: Volker Halbach (Axiomatic Theories of Truth [2011], 4)
     A reaction: Tarski's theory of truth involves types (or hierarchies).