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Gödel and Formalism freeness Juliette Kennedy

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Bill Tait

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A. Heyting

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Carnap diary entry Dec 1929

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Gödel “On Russell’s Mathematical Logic” 1944

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Kreisel 1972

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Gödel Princeton Bicentennial lecture 1946

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Joint work with M. Magidor and J. Väänänen

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Gödel’s two notions of definability Two canonical inner models: – Constructible sets Model of ZFC Model of GCH – Hereditarily ordinal definable sets Model of ZFC CH? – independent of ZFC

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Constructibility Constructible sets (L):

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Ordinal definability Hereditarily ordinal definable sets (HOD): – A set is ordinal definable if it is of the form {a : φ(a,α 1,…, α n )} where φ(x,y 1,…, y n ) is a first order formula of set theory. – A set is hereditarily ordinal definable if it and all elements of its transitive closure are ordinal definable.

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Myhill-Scott result: Hereditarily ordinal definable sets (HOD) can be seen as the constructible hierarchy with second order logic (in place of first order logic): Chang considered a similar construction with the infinitary logic L ω 1 ω 1 in place of first order logic.

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If V=L, then V=HOD=Chang’s model=L. If there are uncountably many measurable cardinals then AC fails in the Chang model. (Kunen.)

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C( L *) L * any logic. We define C( L *): C( L *) = the union of all L´ α

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Looking ahead: For a variety of logics C( L *)=L – Gödel’s L is robust, not limited to first order logic For a variety of logics C( L *)=HOD – Gödel’s HOD is robust, not limited to second order logic For some logics C( L *) is a potentially interesting new inner model.

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Robustness of L Q 1 xφ(x) {a : φ(a)} is uncountable C( L (Q 1 )) = L. In fact: C( L (Q α )) = L, where – Q α xφ(x) |{a : φ(a)}| ≥ א α Other logics, e.g. weak second order logic, ``absolute” logics, etc.

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Observations: avoiding L C( L ω 1 ω ) = L( R ) C( L ∞ω ) = V

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Quantifiers Q 1 MM xyφ(x,y) there is an uncountable X such that φ(a,b) for all a,b in X – Can express Suslinity. Q 0 cf xyφ(x,y) {(a,b) : φ(a,b)} is a linear order of cofinality ω – Fully compact extension of first order logic.

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Theorems C( L (Q 1 MM )) = L, assuming 0 # C( L (Q 0 cf )) ≠ L, assuming 0 # L µ ⊆ C( L (Q 0 cf )), if L µ exists

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What Myhill-Scott really prove In second order logic L 2 one can quantify over arbitrary subsets of the domain. A more general logic L 2,F : in domain M can quantifier only over subsets of cardinality κ with F(κ) ≤ |M|. F any function, e.g. F(κ)=κ, κ +, 2 κ, ב κ, etc L 2 = L 2,F with F(κ)≡0 Note that if one wants to quantify over subsets of cardinality κ one just has to work in a domain of cardinality at least F(κ).

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Theorem For all F: C( L 2,F )=HOD Third, fourth order, etc logics give HOD.

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Bernays “It seems in no way appropriate that Cantor’s Absolute be identified with set theory formalized in standardized logic, which is considered from a more comprehensive model theory.” -Letter to Gödel, 1961. (Collected Works, vol. 4, Oxford)

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Thank you!

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Robustness of L (contd.) A logic L * is absolute if ``φ ∈ L *” is Σ 1 in φ and ``M ⊨ φ” is Δ 1 in M and φ in ZFC. – First order logic – Weak second order logic – L (Q 0 ): ``there exists infinitely many” – L ω 1 ω, L ∞ω : infinitary logic – L ω 1 G, L ∞G : game quantifier logic – Fragments of the above

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