Gödel and Formalism freeness
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Transcript of Gödel and Formalism freeness
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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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Gödel Princeton Bicentennial lecture 1946
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Joint work with M. Magidor and J. VäänänenJoint work with M. Magidor and J. Väänänen
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Gödel’s two notions of definabilityGödel’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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ConstructibilityConstructibility
• Constructible sets (L):
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Ordinal definabilityOrdinal definability
• Hereditarily ordinal definable sets (HOD):– A set is ordinal definable if it is of the form
{a : φ(a,α1,…, αn)}
where φ(x,y1,…, yn) 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 Lω1ω1
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*)C(L*)
• L* any logic. We define C(L*):
• C(L*) = the union of all L´α
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Looking ahead: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 LRobustness of L
• Q1xφ(x) {a : φ(a)} is uncountable
• C(L(Q1)) = L.
• In fact: C(L(Qα)) = L, where – Qαxφ(x) |{a : φ(a)}| ≥ אα
• Other logics, e.g. weak second order logic, ``absolute” logics, etc.
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Observations: avoiding LObservations: avoiding L
• C(Lω1ω) = L(R)
• C(L∞ω) = V
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QuantifiersQuantifiers
• Q1MMxyφ(x,y) there is an uncountable X such
that φ(a,b) for all a,b in X– Can express Suslinity.
• Q0cfxyφ(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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TheoremsTheorems
• C(L(Q1MM)) = L, assuming 0#
• C(L(Q0cf)) ≠ L, assuming 0#
• Lµ C(⊆ L(Q0cf)), if Lµ exists
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What Myhill-Scott really proveWhat Myhill-Scott really prove
• In second order logic L2 one can quantify over arbitrary subsets of the domain.
• A more general logic L2,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+, 2κ, בκ, etc• L2 = L2,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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TheoremTheorem
• For all F: C(L2,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!Thank you!
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Robustness of L (contd.)Robustness of L (contd.)
• A logic L* is absolute if ``φ∈L*” is Σ1 in φ and ``M⊨φ” is Δ1 in M and φ in ZFC.
– First order logic
–Weak second order logic
– L(Q0): ``there exists infinitely many”
– Lω1ω, L∞ω: infinitary logic
– Lω1G, L∞G: game quantifier logic
– Fragments of the above