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Introduction The Friedman Characterization Friedman on Faithful Interpretability

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Friedman on Interpretations

Albert Visser

OFR, Philosophy, Faculty of Humanities, Utrecht University

Honorary Doctorate Harvey Friedman September 5, 2013, Ghent

Introduction The Friedman Characterization Friedman on Faithful Interpretability

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Overview

Introduction The Friedman Characterization Friedman on Faithful Interpretability

Introduction The Friedman Characterization Friedman on Faithful Interpretability

2

Overview

Introduction The Friedman Characterization Friedman on Faithful Interpretability

Introduction The Friedman Characterization Friedman on Faithful Interpretability

2

Overview

Introduction The Friedman Characterization Friedman on Faithful Interpretability

Introduction The Friedman Characterization Friedman on Faithful Interpretability

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Overview

Introduction The Friedman Characterization Friedman on Faithful Interpretability

Introduction The Friedman Characterization Friedman on Faithful Interpretability

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Harvey Friedman

Introduction The Friedman Characterization Friedman on Faithful Interpretability

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The Source

Craig Smory´ nski: Nonstandard models and related developments, p179-229, 1985

Introduction The Friedman Characterization Friedman on Faithful Interpretability

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Craig Smory´ nski

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What is Interpretability?

One theory U is interpretable in another theory V if there is a translation τ such that, for all U-sentences A, if U ⊢ A then V ⊢ Aτ. What is a translation? As a first approximation, we can say: anything that commutes with the predicate logical connectives. An n-ary U-predicate P will be translated to a V-formula A(x0, . . . , xn−1). We allow domain-relativization: ∀x Bx is translated to ∀x (δ(x) → Bτx). There are more refinements that we blissfully ignore here. We write V ✄ U for V interprets U.

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Why Interpretatibility?

Interpretability is a very good for measuring strength of theories. It is a more refined and trustworthy measure than the popular notion

• f conservativity. E.g., according to interpretability GB is precisely
• ne Gödel stronger than ZF, even if GB is conservative over ZF

w.r.t. the full ZF-language. 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 + Ω1) ◮ Euclidean Plane Geometry interprets Hyperbolic Plane

Geometry

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Sequentiality

A theory is sequential if it supports a good theory of sequences of all its objects. A theory is sequential iff we can define a predicate ∈ that satisfies 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).

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Sequential Theories are Everywhere

◮ Adjunctive Set Theory AS. ◮ PA−, the theory of discretely ordered commutative semirings

with a least element.

◮ Buss’ theory S1

2.

◮ Wilkie and Paris’ theory I∆0 + Ω1. ◮ Elementary Arithmetic EA (aka Elementary Function

Arithmetic EFA, or I∆0 + exp).

◮ PRA. ◮ IΣ0

1.

◮ Peano Arithmetic PA. ◮ ACA0. ◮ ZF. ◮ GB

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Overview

Introduction The Friedman Characterization Friedman on Faithful Interpretability

Introduction The Friedman Characterization Friedman on Faithful Interpretability

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Relative Consistency

Let some basic theory W be given. For example, we could take IΣ1. The theory U is consistent relative to V (w.r.t. W) iff W ⊢ con(V) → con(U).

◮ In general relative consistency does not coincide with

interpretability. Relative consistency is complete Σ1. Interpretability is complete Σ3 (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.

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The Friedman Characterization

Friedman ≤ 1975-1980: Suppose A and B are finitely axiomatized and sequential. We have: A ✄ B ⇔ EA ⊢ conρ(A)(A) → conρ(B)(B). We use a standard axiomatization for the finitely axiomatized theories here. conρ(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 Dseq, the interpretability degrees of finitely axiomatized sequential theories and the Π1-extensions of EA ordered by derivability. Transfer of information: It follows e.g. that the first-order theory of Dseq is not arithmetical, by a result of Shavrukov in 2010. Similarly for other results of Shavrukov and Lindström.

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Friedman meets Orey-Hájek

Let ✵(U) := S1

2 + {conn(U) | n ∈ ω}.

Here conn(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).

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Overview

Introduction The Friedman Characterization Friedman on Faithful Interpretability

Introduction The Friedman Characterization Friedman on Faithful Interpretability

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Faithful Interpretability

We have: V faithfully interprets U or V ✄faith U iff, for some translation τ, and, for all U-sentences A, U ⊢ A iff V ⊢ Aτ. 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 IΣ1 + incon(IΣ1), then there is a definable cut J such that A inconJ(IΣ1).

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Improvements

Suppose A is a consistent finitely axiomatized extension of, say S1

2, then there is definable cut J and a model M of A plus all PJ

where P is a true Π1-sentence. In other words, witnesses of false Σ1-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.

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Uses of Friedman’s Result

Friedman’s result implies immediately the well-known result of Ryll-Nardzewski that PA is not finitely axiomatizable, since PA + incon(PA) has only the trivial definable cut and hence has 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 Π0

2.

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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 | = Uτ. In other words, U is model-interpretable in V iff every model of V has an internal model that satisfies U. If U is finitely axiomatized, then interpretability and model-interpretability coincide (exercise in the fat Hodges). Consider e.g. U := IΣ1 + { inconJ(IΣ1) | J is a definable cut }. By (a strengthening of) Friedman’s result, IΣ1 ✄ U. Consider any model M of IΣ1. If M | = inconJ(IΣ1) for every definable cut J, we can take τ the identity interpretation. If, for some J⋆, M | = conJ⋆(IΣ1), we have, by the second incompleteness theorem and compactness: M | = conJ⋆(U). In this case we can build the desired internal model as a Henkin model. So IΣ1 ✄mod U.