Overview Reduction Relations Reduction Relations Two Groups of - PowerPoint PPT Presentation

Reduction Relations P EANO D OWNSTAIRS Two Groups of Theories The Theory PA − Cut-Interpretability Albert Visser The Σ 1 , n -Hierarchy Peano Downstairs and Peano Cellar Department of Philosophy, Faculty of Humanities, Utrecht University Kotlarski-Ratajczyk Conference July 25, 2012, B˛ edlewo 1

Overview Reduction Relations Reduction Relations Two Groups of Theories The Theory PA − Two Groups of Theories Cut-Interpretability The Σ 1 , n -Hierarchy The Theory PA − Peano Downstairs and Peano Cellar Cut-Interpretability The Σ 1 , n -Hierarchy Peano Downstairs and Peano Cellar 2

Overview Reduction Relations Reduction Relations Two Groups of Theories The Theory PA − Two Groups of Theories Cut-Interpretability The Σ 1 , n -Hierarchy The Theory PA − Peano Downstairs and Peano Cellar Cut-Interpretability The Σ 1 , n -Hierarchy Peano Downstairs and Peano Cellar 2

Overview Reduction Relations Reduction Relations Two Groups of Theories The Theory PA − Two Groups of Theories Cut-Interpretability The Σ 1 , n -Hierarchy The Theory PA − Peano Downstairs and Peano Cellar Cut-Interpretability The Σ 1 , n -Hierarchy Peano Downstairs and Peano Cellar 2

Overview Reduction Relations Reduction Relations Two Groups of Theories The Theory PA − Two Groups of Theories Cut-Interpretability The Σ 1 , n -Hierarchy The Theory PA − Peano Downstairs and Peano Cellar Cut-Interpretability The Σ 1 , n -Hierarchy Peano Downstairs and Peano Cellar 2

Overview Reduction Relations Reduction Relations Two Groups of Theories The Theory PA − Two Groups of Theories Cut-Interpretability The Σ 1 , n -Hierarchy The Theory PA − Peano Downstairs and Peano Cellar Cut-Interpretability The Σ 1 , n -Hierarchy Peano Downstairs and Peano Cellar 2

Overview Reduction Relations Reduction Relations Two Groups of Theories The Theory PA − Two Groups of Theories Cut-Interpretability The Σ 1 , n -Hierarchy The Theory PA − Peano Downstairs and Peano Cellar Cut-Interpretability The Σ 1 , n -Hierarchy Peano Downstairs and Peano Cellar 2

Overview Reduction Relations Reduction Relations Two Groups of Theories The Theory PA − Two Groups of Theories Cut-Interpretability The Σ 1 , n -Hierarchy The Theory PA − Peano Downstairs and Peano Cellar Cut-Interpretability The Σ 1 , n -Hierarchy Peano Downstairs and Peano Cellar 3

Reduction Relations ◮ V ✄ U iff, there is a K with K : V ✄ U . Reduction Relations Two Groups of This relation is interpretability . Theories ◮ V ✄ mod U iff, for all models M of V , there is an translation τ The Theory PA − τ ( M ) is a model of U . Cut-Interpretability such that � The Σ 1 , n -Hierarchy This relation is model interpretability . Peano Downstairs ◮ V ✄ loc U iff, for all finitely axiomatized subtheories U 0 of U , and Peano Cellar V ✄ U 0 . This relation is local interpretability . Fact: Suppose A is finitely axiomatized. We have: U ✄ A ⇔ U ✄ mod A . 4

Overview Reduction Relations Reduction Relations Two Groups of Theories The Theory PA − Two Groups of Theories Cut-Interpretability The Σ 1 , n -Hierarchy The Theory PA − Peano Downstairs and Peano Cellar Cut-Interpretability The Σ 1 , n -Hierarchy Peano Downstairs and Peano Cellar 5

Finitely Axiomatized Sequential Theories Reduction Relations S 1 2 , EA, I Σ 1 , ACA 0 , GB. Two Groups of Theories The Theory PA − Let A be a consistent, finitely axiomatized, sequential theory and Cut-Interpretability let N : S 1 2 ✁ A . The Σ 1 , n -Hierarchy ◮ There is a Σ 1 -sentence S such that A ✄ ( A + S N ) and Peano Downstairs and Peano Cellar A ✄ ( A + ¬ S N ) . ◮ Suppose A ⊢ Supexp N . Then the interpretability logic of A w.r.t. N is ILP. ⊢ φ ✄ ψ → ✷ ( φ ✄ ψ ) . ◮ There is a Σ 1 -sound M : S 1 2 ✁ A . 6

Essentially Reflexive Sequential Theories PA, ZF and their extensions in the same language. U is essentially reflexive (w.r.t. N : PA − ✁ U ) iff U proves the full Reduction Relations Two Groups of uniform reflection principle for predicate logic in the signature of U . Theories This implies full induction w.r.t. N . If U is sequential, full induction The Theory PA − Cut-Interpretability w.r.t. N implies full uniform reflection. The Σ 1 , n -Hierarchy Peano Downstairs Let U be consistent, sequential and essentially reflexive w.r.t. N . and Peano Cellar ◮ There is a ∆ 2 -sentence B such that U ✄ ( A + B N ) and A ✄ ( A + ¬ B N ) , but no Σ 1 -sentence has this property. ◮ The interpretability logic of A w.r.t. N is ILM. ⊢ φ ✄ ψ → ( φ ∧ ✷ χ ) ✄ ( ψ ∧ ✷ χ ) . ◮ U + incon N ( U ) is consistent and no M : S 1 2 ✁ ( U + incon N ( U )) is Σ 1 -sound. ◮ U is not locally mutually interpretable with a finitely axiomatized theory. 7

Peano Downstairs en Peano Cellar The theories Peano Downstairs (or PA ↓ ) and Peano Cellar (or PA ↓↓ ) are in many respects like PA: Reduction Relations Two Groups of ◮ They satisfy an induction principle that is in some respects Theories The Theory PA − more like full induction than Σ n -induction. Cut-Interpretability ◮ They are sententially essentially reflexive (w.r.t. restricted The Σ 1 , n -Hierarchy provability). Peano Downstairs and Peano Cellar ◮ They have no consistent finitely axiomatized extension in the same language. So e.g. PA ↓ is not a subtheory of I Σ n . It is a subtheory of PA. On the other hand they are locally weak, i.e. they are locally interpretable (and even cut-interpretable) in PA − . I predict that almost all results of Per Lindstöm’s book Aspects of Incompleteness transfer to extensions of PA ↓↓ / PA ↓ . But what about model theoretic results? This is far less clear. 8

Overview Reduction Relations Reduction Relations Two Groups of Theories The Theory PA − Two Groups of Theories Cut-Interpretability The Σ 1 , n -Hierarchy The Theory PA − Peano Downstairs and Peano Cellar Cut-Interpretability The Σ 1 , n -Hierarchy Peano Downstairs and Peano Cellar 9

The Theory PA − , 1 Reduction Relations Two Groups of The theory PA − is the theory of discretely ordered commutative Theories The Theory PA − semirings with a least element. Cut-Interpretability The Σ 1 , n -Hierarchy The theory is mutually interpretable with Robinson’s Arithmetic Q. Peano Downstairs However, PA − has a more mathematical flavor. Moreover, it has and Peano Cellar the additional good property that it is sequential. This was shown recently by Emil Jeˇ rábek. The theory PA − is given by the following axioms. 10

The Theory PA − , 2 PA − 1 ⊢ x + 0 = x PA − 2 Reduction Relations ⊢ x + y = y + x Two Groups of PA − 3 ⊢ ( x + y ) + z = x + ( y + z ) Theories The Theory PA − PA − 4 ⊢ x · 1 = x Cut-Interpretability PA − 5 ⊢ x · y = y · x The Σ 1 , n -Hierarchy PA − 6 Peano Downstairs ⊢ ( x · y ) · z = x · ( y · z ) and Peano Cellar PA − 7 ⊢ x · ( y + z ) = x · y + x · z PA − 8 ⊢ x ≤ y ∨ y ≤ x PA − 9 ⊢ ( x ≤ y ∧ y ≤ z ) → x ≤ z PA − 10 ⊢ x + 1 �≤ x PA − 11 ⊢ x ≤ y → ( x = y ∨ x + 1 ≤ y ) PA − 12 ⊢ x ≤ y → x + z ≤ y + z PA − 13 ⊢ x ≤ y → x · z ≤ y · z 11

The Theory PA − , 3 Reduction Relations Two Groups of Theories The Theory PA − The subtraction axiom is: Cut-Interpretability sbt ⊢ x ≤ y → ∃ z x + z = y The Σ 1 , n -Hierarchy Peano Downstairs In many presentations the subtraction axiom is part of the axioms and Peano Cellar sbt := PA − + sbt. of PA − . We call PA − sbt is interpretable in PA − on a cut. 12

Overview Reduction Relations Reduction Relations Two Groups of Theories The Theory PA − Two Groups of Theories Cut-Interpretability The Σ 1 , n -Hierarchy The Theory PA − Peano Downstairs and Peano Cellar Cut-Interpretability The Σ 1 , n -Hierarchy Peano Downstairs and Peano Cellar 13

What is a Cut? Reduction Relations Two Groups of We are mostly speaking about definable cuts. A definable cut is a Theories virtual class that is downwards closed w.r.t. ≤ and closed under The Theory PA − successor. Cut-Interpretability The Σ 1 , n -Hierarchy Peano Downstairs If a cut is closed under addition it is an a-cut. If a cut is closed and Peano Cellar under addition and multiplication it is an am-cut. Etc. Solovay’s method of shortening cuts : a definable cut can always be shortened to a definable am-cut. And similarly for closure under the any element of the ω n -hierarchy. 14

Cut-interpretability in PA − A central result: PA − ✄ cut ( I ∆ 0 + Ω 1 ) . Reduction Relations Two Groups of Given that exponentiation is undefined for some n , there is a Theories unique element s , Solovay’s number , such that supexp ( s ) is The Theory PA − defined and supexp ( s + 1 ) is undefined. The following theories are Cut-Interpretability interpretable on a cut: The Σ 1 , n -Hierarchy Peano Downstairs ◮ For k < n : I ∆ 0 + ( Exp ∨ s ≡ k ( mod n )) . and Peano Cellar ◮ I ∆ 0 + (Ω 1 → Exp ) . There are 2 ℵ 0 theories locally cut-interpretable in PA − . To each α : ω → { 0 , 1 } , we assign an extension of I ∆ 0 that says: either Exp or the binary expansion of s ends with . . . α 2 α 1 α 0 . These theories are pairwise incompatible in the sense that their union implies Exp. 15