Axiomatic classes of models in modal logics Evgeny Zolin Moscow State University Workshop on Proof Theory, Modal Logic and Reflection Principles (Wormshop 2017) Steklov Mathematical Institute, Moscow October 20, 2017 Evgeny Zolin Axiomatic classes October 20, 2017 1 / 18
To worm up... Evgeny Zolin Axiomatic classes October 20, 2017 2 / 18
To worm up... Theorem (Keisler, 1961) Let K be any class of first-order models (of a fixed signature). K is axiomatizable ⇐ ⇒ K is closed under ≡ FO and ultraproducts Elementary equivalence: M ≡ FO N means: � � for every formula A ∈ FO we have M | = A ⇐ ⇒ N | = A Evgeny Zolin Axiomatic classes October 20, 2017 2 / 18
To worm up... Theorem (Keisler, 1961) Let K be any class of first-order models (of a fixed signature). K is axiomatizable ⇐ ⇒ K is closed under ≡ FO and ultraproducts K is finitely axiomatizable ⇐ ⇒ both K and its complement K are closed under ≡ FO and ultraproducts Elementary equivalence: M ≡ FO N means: � � for every formula A ∈ FO we have M | = A ⇐ ⇒ N | = A Evgeny Zolin Axiomatic classes October 20, 2017 2 / 18
To worm up... Theorem (Keisler, 1961) Let K be any class of first-order models (of a fixed signature). K is axiomatizable ⇐ ⇒ K is closed under ≡ FO and ultraproducts K is finitely axiomatizable ⇐ ⇒ both K and its complement K are closed under ≡ FO and ultraproducts Closure conditions Both K and K K K K is axiomatizable ≡ FO uProd K is finitely axiomatizable ≡ FO uProd uProd Evgeny Zolin Axiomatic classes October 20, 2017 2 / 18
To worm up... Theorem (Keisler, 1961) Let K be any class of first-order models (of a fixed signature). K is axiomatizable ⇐ ⇒ K is closed under ≡ FO and ultraproducts K is finitely axiomatizable ⇐ ⇒ both K and its complement K are closed under ≡ FO and ultraproducts Closure conditions Both K and K K K K is axiomatizable ≡ FO uProd K is finitely axiomatizable ≡ FO uProd uProd Why non-symmetric? Evgeny Zolin Axiomatic classes October 20, 2017 2 / 18
To worm up... Theorem (Keisler, 1961) Let K be any class of first-order models (of a fixed signature). K is axiomatizable ⇐ ⇒ K is closed under ≡ FO and ultraproducts K is finitely axiomatizable ⇐ ⇒ both K and its complement K are closed under ≡ FO and ultraproducts Closure conditions Both K and K K K K is axiomatizable ≡ FO uProd K is finitely axiomatizable ≡ FO uProd uProd Why non-symmetric? Because this is not the whole story! Evgeny Zolin Axiomatic classes October 20, 2017 2 / 18
The 4 “species” of classes Definition. For a class of models K we write: K ∈ L if K = Models ( A ) , for some formula A ∈ FO. Evgeny Zolin Axiomatic classes October 20, 2017 3 / 18
The 4 “species” of classes Definition. For a class of models K we write: K ∈ L if K = Models ( A ) , for some formula A ∈ FO. K ∈ ⩀ L if K = Models (Γ) , for some set of formulas Γ ⊆ FO. Evgeny Zolin Axiomatic classes October 20, 2017 3 / 18
The 4 “species” of classes Definition. For a class of models K we write: K ∈ L if K = Models ( A ) , for some formula A ∈ FO. K ∈ ⩀ L if K = Models (Γ) , for some set of formulas Γ ⊆ FO. K = � Equivalently: if K i for some classes K i ∈ L . i ∈ I Evgeny Zolin Axiomatic classes October 20, 2017 3 / 18
The 4 “species” of classes Definition. For a class of models K we write: K ∈ L if K = Models ( A ) , for some formula A ∈ FO. K ∈ ⩀ L if K = Models (Γ) , for some set of formulas Γ ⊆ FO. K = � Equivalently: if K i for some classes K i ∈ L . i ∈ I K ∈ ⊍ L if K = � K i for some classes K i ∈ L . i ∈ I Evgeny Zolin Axiomatic classes October 20, 2017 3 / 18
The 4 “species” of classes Definition. For a class of models K we write: K ∈ L if K = Models ( A ) , for some formula A ∈ FO. K ∈ ⩀ L if K = Models (Γ) , for some set of formulas Γ ⊆ FO. K = � Equivalently: if K i for some classes K i ∈ L . i ∈ I K ∈ ⊍ L if K = � K i for some classes K i ∈ L . i ∈ I K ∈ ⊍⩀ L if K = � � K i , j for some classes K i , j ∈ L . i ∈ I j ∈ J i Evgeny Zolin Axiomatic classes October 20, 2017 3 / 18
The 4 “species” of classes Definition. For a class of models K we write: K ∈ L if K = Models ( A ) , for some formula A ∈ FO. K ∈ ⩀ L if K = Models (Γ) , for some set of formulas Γ ⊆ FO. K = � Equivalently: if K i for some classes K i ∈ L . i ∈ I K ∈ ⊍ L if K = � K i for some classes K i ∈ L . i ∈ I K ∈ ⊍⩀ L if K = � � K i , j for some classes K i , j ∈ L . i ∈ I j ∈ J i A somewhat “oldish” terminology: K ∈ L — an elementary class of models ( finitely axiomatizable ) K ∈ ⩀ L — a ∆ -elementary class of models ( axiomatizable ) K ∈ ⊍ L — a Σ -elementary class of models ( co-axiomatizable? ) K ∈ ⊍⩀ L — a Σ∆ -elementary class of models Evgeny Zolin Axiomatic classes October 20, 2017 3 / 18
The hierarchy of the 4 species of classes ⊍⩀ L = ⩀⊍ L ⩀ L ⊍ L L Evgeny Zolin Axiomatic classes October 20, 2017 4 / 18
The hierarchy of the 4 species of classes ⊍⩀ L = ⩀⊍ L ⩀ L ⊍ L L In L : the classes of all groups (rings, fields) In ⩀ L : infinite groups (rings, fields), algebraically closed fields In ⊍ L : finite groups (rings, fields) In ⊍⩀ L : infinite fields of characteristic p > 0; infinite finitely dimensional vector spaces Not even in ⊍⩀ L : well-ordered sets, periodic groups, simple groups Evgeny Zolin Axiomatic classes October 20, 2017 4 / 18
First-order language | Criteria for the 4 species Theorem (Keisler, 1961) Both K K K ∈ ⊍⩀ L ≡ FO K ∈ ⊍ L ≡ FO uProd K ∈ ⩀ L ≡ FO uProd K ∈ L ≡ FO uProd uProd Legend: uProd = ultraproduct Evgeny Zolin Axiomatic classes October 20, 2017 5 / 18
First-order language | Criteria for the 4 species Theorem (Keisler, 1961) (Keisler, 1961; Shelah, 1971) Both Both K K K K ∼ K ∈ ⊍⩀ L ≡ FO = uPow uPow ∼ K ∈ ⊍ L ≡ FO uProd = uPow uProd ∼ K ∈ ⩀ L ≡ FO uProd = uProd uPow ∼ K ∈ L ≡ FO uProd uProd = uProd uProd Legend: uProd = ultraproduct uPow = ultrapower Evgeny Zolin Axiomatic classes October 20, 2017 5 / 18
First-order language | Criteria for the 4 species Theorem (Keisler, 1961) (Keisler, 1961; Shelah, 1971) Both Both K K K K ∼ K ∈ ⊍⩀ L ≡ FO = uPow uPow ∼ K ∈ ⊍ L ≡ FO uProd = uPow uProd ∼ K ∈ ⩀ L ≡ FO uProd = uProd uPow ∼ K ∈ L ≡ FO uProd uProd = uProd uProd Legend: uProd = ultraproduct uPow = ultrapower Main reason for the symmetry in the tables: M � | = A ⇐ ⇒ M | = ¬ A Evgeny Zolin Axiomatic classes October 20, 2017 5 / 18
Modal language | Kripke semantics Formulas: p i | ¬ A | ( A ∧ B ) | ( A ∨ B ) | ( A → B ) | � A Evgeny Zolin Axiomatic classes October 20, 2017 6 / 18
Modal language | Kripke semantics Formulas: p i | ¬ A | ( A ∧ B ) | ( A ∨ B ) | ( A → B ) | � A Kripke semantics: Kripke model: M = ( W , R , V ) , where W � = ∅ — a nonempty set of worlds R ⊆ W × W — a accessibility relation between worlds V ( p i ) ⊆ W — a valuation of variables Evgeny Zolin Axiomatic classes October 20, 2017 6 / 18
Modal language | Kripke semantics Formulas: p i | ¬ A | ( A ∧ B ) | ( A ∨ B ) | ( A → B ) | � A Kripke semantics: Kripke model: M = ( W , R , V ) , where W � = ∅ — a nonempty set of worlds R ⊆ W × W — a accessibility relation between worlds V ( p i ) ⊆ W — a valuation of variables Truth of a formula is defined in a pointed model ( M , x ) : M , x | = p i ⇌ x ∈ V ( p i ) M , x | = ¬ A M , x � | = A ⇌ M , x | = A ∧ B M , x | = A and M , x | = B ⇌ M , x | = A ∨ B ⇌ M , x | = A or M , x | = B M , x | = A → B M , x | = A ⇒ M , x | = B ⇌ M , x | = � A ⇌ for every y ∈ W ( x R y ⇒ M , y | = A ) Truth of a formula in a model: M | = A ⇌ ∀ x ∈ W M , x | = A . Evgeny Zolin Axiomatic classes October 20, 2017 6 / 18
Modal language | Relations between (pointed) models Modal equivalence of two (pointed) Kripke models M ≡ ML N ⇌ for every formula A ∈ ML: M | = A ⇐ ⇒ N | = A Evgeny Zolin Axiomatic classes October 20, 2017 7 / 18
Modal language | Relations between (pointed) models Modal equivalence of two (pointed) Kripke models M ≡ ML N ⇌ for every formula A ∈ ML: M | = A ⇐ ⇒ N | = A Bisimulation between two pointed Kripke models M , a ≃ N , b — • respects the valuation of variables • every step in M is “simulated” by some step in N • every step in N is “simulated” by some step in M Evgeny Zolin Axiomatic classes October 20, 2017 7 / 18
Modal language | Relations between (pointed) models Modal equivalence of two (pointed) Kripke models M ≡ ML N ⇌ for every formula A ∈ ML: M | = A ⇐ ⇒ N | = A Bisimulation between two pointed Kripke models M , a ≃ N , b — • respects the valuation of variables • every step in M is “simulated” by some step in N • every step in N is “simulated” by some step in M Global bisimulation between Kripke models M : ≃ : N — bisimulation that covers the whole models M and N Evgeny Zolin Axiomatic classes October 20, 2017 7 / 18
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