Infinitary first-order categorical logic Christian Esp´ ındola Stockholm University August 11th, 2016 Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 1 / 16
Classical infinitary logics Described and studied extensively by Carol Karp (1964) Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 2 / 16
Classical infinitary logics Described and studied extensively by Carol Karp (1964) The language L κ,κ is a two-fold generalization of the finitary case. Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 2 / 16
Classical infinitary logics Described and studied extensively by Carol Karp (1964) The language L κ,κ is a two-fold generalization of the finitary case. Let φ, { φ α : α < γ } (for each γ < κ ) be formulas. Then the following are also formulas: Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 2 / 16
Classical infinitary logics Described and studied extensively by Carol Karp (1964) The language L κ,κ is a two-fold generalization of the finitary case. Let φ, { φ α : α < γ } (for each γ < κ ) be formulas. Then the following are also formulas: 1 � � φ α , φ α α<γ α<γ Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 2 / 16
Classical infinitary logics Described and studied extensively by Carol Karp (1964) The language L κ,κ is a two-fold generalization of the finitary case. Let φ, { φ α : α < γ } (for each γ < κ ) be formulas. Then the following are also formulas: 1 � � φ α , φ α α<γ α<γ 2 ∀ x γ φ, ∃ x γ φ (where x γ = { x α : α < γ } ) Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 2 / 16
Classical infinitary logics Hilbert-style system enough to derive a completeness theorem for S et -valued models. Featuring the following axiom schemata, for each γ < κ : Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 3 / 16
Classical infinitary logics Hilbert-style system enough to derive a completeness theorem for S et -valued models. Featuring the following axiom schemata, for each γ < κ : Classical distributivity: 1 � � � � ψ ij → ψ if ( i ) i <γ j <γ f ∈ γ γ i <γ Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 3 / 16
Classical infinitary logics Hilbert-style system enough to derive a completeness theorem for S et -valued models. Featuring the following axiom schemata, for each γ < κ : Classical distributivity: 1 � � � � ψ ij → ψ if ( i ) i <γ j <γ f ∈ γ γ i <γ Classical dependent choice up to γ ( DC γ ): 2 � � ∀ β<α x β ∃ x α ψ α → ∃ α<γ x α ψ α α<γ α<γ (for disjoint x α and such that no variable in x α is free in ψ β for β < α ). Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 3 / 16
Classical infinitary logics Hilbert-style system enough to derive a completeness theorem for S et -valued models. Featuring the following axiom schemata, for each γ < κ : Classical distributivity: 1 � � � � ψ ij → ψ if ( i ) i <γ j <γ f ∈ γ γ i <γ Classical dependent choice up to γ ( DC γ ): 2 � � ∀ β<α x β ∃ x α ψ α → ∃ α<γ x α ψ α α<γ α<γ (for disjoint x α and such that no variable in x α is free in ψ β for β < α ). Completeness theorem proved using Boolean algebraic methods and thus relies heavily in the use of the excluded middle axiom. Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 3 / 16
κ -regular logic Within infinite-quantifier languages, Makkai (1990) provides a partial answer by considering infinitary regular theories. Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 4 / 16
κ -regular logic Within infinite-quantifier languages, Makkai (1990) provides a partial answer by considering infinitary regular theories. These are a generalization of regular theories admitting the use of infinitary conjunction and infinitary existential quantification. Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 4 / 16
κ -regular logic Within infinite-quantifier languages, Makkai (1990) provides a partial answer by considering infinitary regular theories. These are a generalization of regular theories admitting the use of infinitary conjunction and infinitary existential quantification. Makkai identified the correct type of categories corresponding to κ -regular logic, the so called κ -regular categories. Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 4 / 16
κ -regular logic Within infinite-quantifier languages, Makkai (1990) provides a partial answer by considering infinitary regular theories. These are a generalization of regular theories admitting the use of infinitary conjunction and infinitary existential quantification. Makkai identified the correct type of categories corresponding to κ -regular logic, the so called κ -regular categories. Definition (Makkai) A κ -regular category is a regular category that has κ -limits (i.e., limits of κ -small diagrams) and satisfies further an exactness property of S et corresponding to the axioms DC γ of dependent choice up to γ for each γ < κ . Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 4 / 16
κ -regular logic Consider a κ -chain in a category C with κ -limits, i.e., a diagram Γ : γ op → C specified by morphisms ( h β,α : C β → C α ) α ≤ β<γ with the following condition: Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 5 / 16
κ -regular logic Consider a κ -chain in a category C with κ -limits, i.e., a diagram Γ : γ op → C specified by morphisms ( h β,α : C β → C α ) α ≤ β<γ with the following condition: the restriction Γ | β is a limit diagram for every limit ordinal β . Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 5 / 16
κ -regular logic Consider a κ -chain in a category C with κ -limits, i.e., a diagram Γ : γ op → C specified by morphisms ( h β,α : C β → C α ) α ≤ β<γ with the following condition: the restriction Γ | β is a limit diagram for every limit ordinal β . We say that the morphisms h β,α compose transfinitely, and take the limit projection f β, 0 to be the transfinite composite of h α + 1 ,α for α < β . Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 5 / 16
κ -regular logic Consider a κ -chain in a category C with κ -limits, i.e., a diagram Γ : γ op → C specified by morphisms ( h β,α : C β → C α ) α ≤ β<γ with the following condition: the restriction Γ | β is a limit diagram for every limit ordinal β . We say that the morphisms h β,α compose transfinitely, and take the limit projection f β, 0 to be the transfinite composite of h α + 1 ,α for α < β . Then the exactness condition reads that if all maps h β,α are epimorphisms, so is f β, 0 . Loosely speaking we say that the transfinite composition of epimorphisms is itself an epimorphism. Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 5 / 16
Generalizations Goals: Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 6 / 16
Generalizations Goals: 1 Generalize κ -regular categories to κ -coherent categories, adding κ -disjunctions to the language Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 6 / 16
Generalizations Goals: 1 Generalize κ -regular categories to κ -coherent categories, adding κ -disjunctions to the language 2 Investigate infinitary-first-order categorical logic by coding κ -first-order theories via Morleyization Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 6 / 16
Generalizations Goals: 1 Generalize κ -regular categories to κ -coherent categories, adding κ -disjunctions to the language 2 Investigate infinitary-first-order categorical logic by coding κ -first-order theories via Morleyization Connections with large cardinal axioms: Christian Esp´ ındola (Stockholm University) Infinitary first-order categorical logic August 11th, 2016 6 / 16
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