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4 ideas
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. |
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. |
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). |