Introduction Friedman on Interpretations The Friedman Characterization Friedman on Faithful Interpretability Albert Visser OFR, Philosophy, Faculty of Humanities, Utrecht University Honorary Doctorate Harvey Friedman September 5, 2013, Ghent 1
Overview Introduction The Friedman Characterization Introduction Friedman on Faithful Interpretability The Friedman Characterization Friedman on Faithful Interpretability 2
Overview Introduction The Friedman Characterization Introduction Friedman on Faithful Interpretability The Friedman Characterization Friedman on Faithful Interpretability 2
Overview Introduction The Friedman Characterization Introduction Friedman on Faithful Interpretability The Friedman Characterization Friedman on Faithful Interpretability 2
Overview Introduction The Friedman Characterization Introduction Friedman on Faithful Interpretability The Friedman Characterization Friedman on Faithful Interpretability 3
Harvey Friedman Introduction The Friedman Characterization Friedman on Faithful Interpretability 4
The Source Introduction The Friedman Characterization Friedman on Faithful Interpretability Craig Smory´ nski: Nonstandard models and related developments, p179-229, 1985 5
Craig Smory´ nski Introduction The Friedman Characterization Friedman on Faithful Interpretability 6
What is Interpretability? One theory U is interpretable in another theory V if there is a Introduction translation τ such that, for all U -sentences A , if U ⊢ A then V ⊢ A τ . The Friedman Characterization Friedman on What is a translation ? As a first approximation, we can say: Faithful Interpretability anything that commutes with the predicate logical connectives. An n -ary U -predicate P will be translated to a V -formula A ( x 0 , . . . , x n − 1 ) . We allow domain-relativization : ∀ x Bx is translated to ∀ x ( δ ( x ) → B τ x ) . There are more refinements that we blissfully ignore here. We write V ✄ U for V interprets U . 7
Why Interpretatibility? Interpretability is a very good for measuring strength of theories. It is a more refined and trustworthy measure than the popular notion Introduction of conservativity. E.g., according to interpretability GB is precisely The Friedman Characterization one Gödel stronger than ZF, even if GB is conservative over ZF Friedman on w.r.t. the full ZF-language. Faithful Interpretability We will see that interpretability has better properties than verifiable relative consistency. Measuring strength has its natural home in Reverse Mathematics. ◮ ZF ✄ PA ◮ PA interprets a corresponding theory of syntax ◮ Q ✄ ( I ∆ 0 + Ω 1 ) ◮ Euclidean Plane Geometry interprets Hyperbolic Plane Geometry 8
Sequentiality A theory is sequential if it supports a good theory of sequences of Introduction all its objects. The Friedman Characterization Friedman on A theory is sequential iff we can define a predicate ∈ that satisfies Faithful Interpretability Adjunctive Set Theory , AS. The theory AS is a one-sorted theory with a binary relation ∈ . AS1 ⊢ ∃ x ∀ y y �∈ x , AS2 ⊢ ∀ x , y ∃ z ∀ u ( u ∈ z ↔ ( u ∈ x ∨ u = y )) . We need a substantial bootstrap to show that this simple definition gives rise to a theory of sequences (including the numbers to do the projections). 9
Sequential Theories are Everywhere ◮ Adjunctive Set Theory AS. Introduction ◮ PA − , the theory of discretely ordered commutative semirings The Friedman with a least element. Characterization Friedman on ◮ Buss’ theory S 1 2 . Faithful Interpretability ◮ Wilkie and Paris’ theory I ∆ 0 + Ω 1 . ◮ Elementary Arithmetic EA (aka Elementary Function Arithmetic EFA, or I ∆ 0 + exp). ◮ PRA. ◮ I Σ 0 1 . ◮ Peano Arithmetic PA. ◮ ACA 0 . ◮ ZF. ◮ GB 10
Overview Introduction The Friedman Characterization Introduction Friedman on Faithful Interpretability The Friedman Characterization Friedman on Faithful Interpretability 11
Relative Consistency Let some basic theory W be given. For example, we could take I Σ 1 . The theory U is consistent relative to V (w.r.t. W ) iff Introduction W ⊢ con ( V ) → con ( U ) . The Friedman Characterization Friedman on Faithful ◮ In general relative consistency does not coincide with Interpretability interpretability. Relative consistency is complete Σ 1 . Interpretability is complete Σ 3 (Shavrukov).) ◮ Relative consistency is strongly dependent on the chosen axiomatization. We can find an axiomatization β of PA for which (relative to EA) the theory PA is stronger than ZF (with the usual axiomatization). The axiomatization β can be even chosen to be an axiom scheme! In contrast, interpretability is extensional. 12
The Friedman Characterization Friedman ≤ 1975-1980: Suppose A and B are finitely axiomatized and sequential. We have: Introduction ⇔ EA ⊢ con ρ ( A ) ( A ) → con ρ ( B ) ( B ) . The Friedman A ✄ B Characterization Friedman on Faithful We use a standard axiomatization for the finitely axiomatized Interpretability theories here. con ρ ( A ) means consistency for proofs that only contain formulas of the complexity of A . Alternatively we could have used cut-free, tableaux or Herbrand consistency. Even better: A �→ EA + ✸ A ,ρ ( A ) ⊤ is an effective isomorphism between D seq , the interpretability degrees of finitely axiomatized sequential theories and the Π 1 -extensions of EA ordered by derivability. Transfer of information: It follows e.g. that the first-order theory of D seq is not arithmetical, by a result of Shavrukov in 2010. Similarly for other results of Shavrukov and Lindström. 13
Friedman meets Orey-Hájek Introduction The Friedman Characterization Let ✵ ( U ) := S 1 2 + { con n ( U ) | n ∈ ω } . Friedman on Faithful Interpretability Here con n ( U ) is consistency of the axioms of U with Gödelnumer ≤ n for proofs with formulas of complexity ≤ n . Let V be sequential. Then: V ✄ loc U ⇔ ✵ ( V ) ⊢ ✵ ( U ) . 14
Overview Introduction The Friedman Characterization Introduction Friedman on Faithful Interpretability The Friedman Characterization Friedman on Faithful Interpretability 15
Faithful Interpretability Introduction We have: V faithfully interprets U or V ✄ faith U iff, for some The Friedman Characterization translation τ , and, for all U -sentences A , U ⊢ A iff V ⊢ A τ . Friedman on Faithful Interpretability Friedman: If A is consistent, finitely axiomatized and sequential, then, for any U , A ✄ U iff A ✄ faith U . Friedman’s result follows also from the independent results of Jan Krajíˇ cek’s A Note on Proofs of Falsehood of 1987. Example: Suppose e.g. A is I Σ 1 + incon ( I Σ 1 ) , then there is a definable cut J such that A � incon J ( I Σ 1 ) . 16
Improvements Introduction The Friedman Characterization Suppose A is a consistent finitely axiomatized extension of, say Friedman on Faithful S 1 2 , then there is definable cut J and a model M of A plus all P J Interpretability where P is a true Π 1 -sentence. In other words, witnesses of false Σ 1 -sentences are above J . Suppose A is consistent, finitely axiomatized and sequential. Suppose A is mutually interpretable with V . Then, for any U , V ✄ U iff V ✄ faith U . 17
Uses of Friedman’s Result Introduction Friedman’s result implies immediately the well-known result of The Friedman Ryll-Nardzewski that PA is not finitely axiomatizable, since Characterization Friedman on PA + incon ( PA ) has only the trivial definable cut and hence has Faithful Interpretability inconsistencies in every definable cut. Moreover, by the same argument, it implies Montague’s result that no finitely axiomatized theory is inductive. Friedman’s result is a useful tool in the study of degrees of interpretability. Moreover it is a rich source of counterexamples. E.g. we can produce an arithmetical theory U of which the predicate logic Λ( U ) is complete Π 0 2 . 18
An Application U is model-interpretable in V or V ✄ mod U iff for all models M | = V , there is a translation τ such that M | = U τ . In other Introduction words, U is model-interpretable in V iff every model of V has an The Friedman Characterization internal model that satisfies U . Friedman on Faithful Interpretability If U is finitely axiomatized, then interpretability and model-interpretability coincide (exercise in the fat Hodges). Consider e.g. U := I Σ 1 + { incon J ( I Σ 1 ) | J is a definable cut } . By (a strengthening of) Friedman’s result, I Σ 1 � ✄ U . Consider any = incon J ( I Σ 1 ) for every definable cut J , we model M of I Σ 1 . If M | can take τ the identity interpretation. If, for some J ⋆ , = con J ⋆ ( I Σ 1 ) , we have, by the second incompleteness M | = con J ⋆ ( U ) . In this case we can theorem and compactness: M | build the desired internal model as a Henkin model. So I Σ 1 ✄ mod U . 19
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