Stone duality for infinitary logic Ronnie Chen University of Illinois at Urbana{Champaign BLAST, May 21, 2019 syntax ! semantics Duality: ∗ algebra A ! dual space A where A∗ := Hom(A; K) for some fixed K For a theory T in a logic, syntactical algebra hT i ! space of models Mod(T ) Duality as completeness Duality theorem= strong completeness theorem for a logic. syntax ! semantics For a theory T in a logic, syntactical algebra hT i ! space of models Mod(T ) Duality as completeness Duality theorem= strong completeness theorem for a logic. Duality: ∗ algebra A ! dual space A where A∗ := Hom(A; K) for some fixed K syntax ! semantics Duality as completeness Duality theorem= strong completeness theorem for a logic. Duality: ∗ algebra A ! dual space A where A∗ := Hom(A; K) for some fixed K For a theory T in a logic, syntactical algebra hT i ! space of models Mod(T ) Duality as completeness Duality theorem= strong completeness theorem for a logic. Duality: ∗ algebra A ! dual space A where A∗ := Hom(A; K) for some fixed K For a theory T in a logic, syntactical algebra hT i ! space of models Mod(T ) syntax ! semantics Main examples I propositional logic L!0 (Stone duality) I first-order logic L!! (Makkai duality) I infinitary first-order logic L!1! =: hL j T i∗ Propositional logic L!0: formulas built from L using finite ^; _; : T : L!0-theory (set of L!0-formulas) Model of T : M : L! 2 = f0; 1g s.t. every φ 2 T 7! 1 (M j= T ) Mod(L; T ) := fmodels of T g ⊆ 2L Lindenbaum{Tarski algebra hL j T i := fL!0-formulasg=∼ where φ ∼ :() T ` φ , = Boolean algebra presented by generators L, relations T (= 1) Hence, Mod(L; T ) =∼ Bool(hL j T i; 2) (where C(A; B) := fmorphisms A ! B in category Cg) Propositional logic L : propositional language (set of atomic propositions) =: hL j T i∗ T : L!0-theory (set of L!0-formulas) Model of T : M : L! 2 = f0; 1g s.t. every φ 2 T 7! 1 (M j= T ) Mod(L; T ) := fmodels of T g ⊆ 2L Lindenbaum{Tarski algebra hL j T i := fL!0-formulasg=∼ where φ ∼ :() T ` φ , = Boolean algebra presented by generators L, relations T (= 1) Hence, Mod(L; T ) =∼ Bool(hL j T i; 2) (where C(A; B) := fmorphisms A ! B in category Cg) Propositional logic L : propositional language (set of atomic propositions) Propositional logic L!0: formulas built from L using finite ^; _; : =: hL j T i∗ Model of T : M : L! 2 = f0; 1g s.t. every φ 2 T 7! 1 (M j= T ) Mod(L; T ) := fmodels of T g ⊆ 2L Lindenbaum{Tarski algebra hL j T i := fL!0-formulasg=∼ where φ ∼ :() T ` φ , = Boolean algebra presented by generators L, relations T (= 1) Hence, Mod(L; T ) =∼ Bool(hL j T i; 2) (where C(A; B) := fmorphisms A ! B in category Cg) Propositional logic L : propositional language (set of atomic propositions) Propositional logic L!0: formulas built from L using finite ^; _; : T : L!0-theory (set of L!0-formulas) =: hL j T i∗ Lindenbaum{Tarski algebra hL j T i := fL!0-formulasg=∼ where φ ∼ :() T ` φ , = Boolean algebra presented by generators L, relations T (= 1) Hence, Mod(L; T ) =∼ Bool(hL j T i; 2) (where C(A; B) := fmorphisms A ! B in category Cg) Propositional logic L : propositional language (set of atomic propositions) Propositional logic L!0: formulas built from L using finite ^; _; : T : L!0-theory (set of L!0-formulas) Model of T : M : L! 2 = f0; 1g s.t. every φ 2 T 7! 1 (M j= T ) Mod(L; T ) := fmodels of T g ⊆ 2L =: hL j T i∗ Hence, Mod(L; T ) =∼ Bool(hL j T i; 2) (where C(A; B) := fmorphisms A ! B in category Cg) Propositional logic L : propositional language (set of atomic propositions) Propositional logic L!0: formulas built from L using finite ^; _; : T : L!0-theory (set of L!0-formulas) Model of T : M : L! 2 = f0; 1g s.t. every φ 2 T 7! 1 (M j= T ) Mod(L; T ) := fmodels of T g ⊆ 2L Lindenbaum{Tarski algebra hL j T i := fL!0-formulasg=∼ where φ ∼ :() T ` φ , = Boolean algebra presented by generators L, relations T (= 1) =: hL j T i∗ Propositional logic L : propositional language (set of atomic propositions) Propositional logic L!0: formulas built from L using finite ^; _; : T : L!0-theory (set of L!0-formulas) Model of T : M : L! 2 = f0; 1g s.t. every φ 2 T 7! 1 (M j= T ) Mod(L; T ) := fmodels of T g ⊆ 2L Lindenbaum{Tarski algebra hL j T i := fL!0-formulasg=∼ where φ ∼ :() T ` φ , = Boolean algebra presented by generators L, relations T (= 1) Hence, Mod(L; T ) =∼ Bool(hL j T i; 2) (where C(A; B) := fmorphisms A ! B in category Cg) Propositional logic L : propositional language (set of atomic propositions) Propositional logic L!0: formulas built from L using finite ^; _; : T : L!0-theory (set of L!0-formulas) Model of T : M : L! 2 = f0; 1g s.t. every φ 2 T 7! 1 (M j= T ) Mod(L; T ) := fmodels of T g ⊆ 2L Lindenbaum{Tarski algebra hL j T i := fL!0-formulasg=∼ where φ ∼ :() T ` φ , = Boolean algebra presented by generators L, relations T (= 1) Hence, Mod(L; T ) =∼ Bool(hL j T i; 2) =: hL j T i∗ (where C(A; B) := fmorphisms A ! B in category Cg) and we have a canonical evaluation map ηA : A −! a 7−! (x 7! x(a)): Hence for A 2 Bool, A∗ := Bool(A; 2) ⊆ 2A 2 Top; ∗ A∗∗ := Top(A∗; 2) ⊆ 2A 2 Bool Theorem (Stone duality { algebraic) ∗∗ For every A 2 Bool, ηA : A ! A is an isomorphism. Stone duality 2 is a dualizing object, i.e., has two commuting structures: I 2 2 Bool; I 2 2 Top (= topological spaces); I these two structures on 2 commute: ^; _; : are continuous. and we have a canonical evaluation map ηA : A −! a 7−! (x 7! x(a)): Theorem (Stone duality { algebraic) ∗∗ For every A 2 Bool, ηA : A ! A is an isomorphism. Stone duality 2 is a dualizing object, i.e., has two commuting structures: I 2 2 Bool; I 2 2 Top (= topological spaces); I these two structures on 2 commute: ^; _; : are continuous. Hence for A 2 Bool, A∗ := Bool(A; 2) ⊆ 2A 2 Top; ∗ A∗∗ := Top(A∗; 2) ⊆ 2A 2 Bool Theorem (Stone duality { algebraic) ∗∗ For every A 2 Bool, ηA : A ! A is an isomorphism. Stone duality 2 is a dualizing object, i.e., has two commuting structures: I 2 2 Bool; I 2 2 Top (= topological spaces); I these two structures on 2 commute: ^; _; : are continuous. Hence for A 2 Bool, A∗ := Bool(A; 2) ⊆ 2A 2 Top; and we have a canonical evaluation map ∗∗ ∗ A∗ ηA : A −! A := Top(A ; 2) ⊆ 2 2 Bool a 7−! (x 7! x(a)): Stone duality 2 is a dualizing object, i.e., has two commuting structures: I 2 2 Bool; I 2 2 Top (= topological spaces); I these two structures on 2 commute: ^; _; : are continuous. Hence for A 2 Bool, A∗ := Bool(A; 2) ⊆ 2A 2 Top; and we have a canonical evaluation map ∗∗ ∗ A∗ ηA : A −! A := Top(A ; 2) ⊆ 2 2 Bool a 7−! (x 7! x(a)): Theorem (Stone duality { algebraic) ∗∗ For every A 2 Bool, ηA : A ! A is an isomorphism. I injectivity: for any two L!0-formulas φ, , if φ , in all models of T , then T ` φ , (completeness theorem) I surjectivity: any clopen set of models is named by an L!0-formula (definability theorem) Strong completeness for L!0 When A = hL j T i for a propositional L!0-theory T , Stone duality becomes: Theorem (Stone duality { logical) For every L!0-theory T , we have an isomorphism ∗∗ ∼ ∗ ∼ ηT : hL j T i −! hL j T i = Mod(L; T ) = Clopen(Mod(L; T )) [φ] 7−! fM 2 Mod(L; T ) j M j= φg: I surjectivity: any clopen set of models is named by an L!0-formula (definability theorem) Strong completeness for L!0 When A = hL j T i for a propositional L!0-theory T , Stone duality becomes: Theorem (Stone duality { logical) For every L!0-theory T , we have an isomorphism ∗∗ ∼ ∗ ∼ ηT : hL j T i −! hL j T i = Mod(L; T ) = Clopen(Mod(L; T )) [φ] 7−! fM 2 Mod(L; T ) j M j= φg: I injectivity: for any two L!0-formulas φ, , if φ , in all models of T , then T ` φ , (completeness theorem) Strong completeness for L!0 When A = hL j T i for a propositional L!0-theory T , Stone duality becomes: Theorem (Stone duality { logical) For every L!0-theory T , we have an isomorphism ∗∗ ∼ ∗ ∼ ηT : hL j T i −! hL j T i = Mod(L; T ) = Clopen(Mod(L; T )) [φ] 7−! fM 2 Mod(L; T ) j M j= φg: I injectivity: for any two L!0-formulas φ, , if φ , in all models of T , then T ` φ , (completeness theorem) I surjectivity: any clopen set of models is named by an L!0-formula (definability theorem) The adjunction unit is the evaluation map ηA : A ! Top(Bool(A; 2); 2), an isomorphism. By a general fact about adjunctions, this is equivalent to: Theorem (Stone duality { algebraic, II) The functor Bool(−; 2) : Bool ! Topop is fully faithful, i.e., a bijection Bool(A; B) −!∼ Top(B∗; A∗) on each homset.