Topoi: The Categorial Analysis of LogicA classic introduction to mathematical logic from the perspective of category theory, this text is suitable for advanced undergraduates and graduate students and accessible to both philosophically and mathematically oriented readers. Its approach moves always from the particular to the general, following through the steps of the abstraction process until the abstract concept emerges naturally. Beginning with a survey of set theory and its role in mathematics, the text proceeds to definitions and examples of categories and explains the use of arrows in place of set-membership. The introduction to topos structure covers topos logic, algebra of subobjects, and intuitionism and its logic, advancing to the concept of functors, set concepts and validity, and elementary truth. Explorations of categorial set theory, local truth, and adjointness and quantifiers conclude with a study of logical geometry. |
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algebra arrow f assigns axioms bijection Bn(I Boolean C-arrows C-object category theory characterisation classical co-product codomain colimits collection component construction corresponding defined definition denoted diagram commute disjoint domain epic epimorphic equaliser equivalent EXAMPLE EXERCISE exists extensional finite formula function f functor F ƒ and g geometric morphism germs given hence Heyting Heyting algebra identity arrow implies inclusion interpretation intuitionistic inverse L-formula L-model lattice left adjoint Lemma logic mathematical monic monoid N-Set natural numbers object natural transformation notion open sets pair partial elements poset pr₁ pre-order precisely preserves PROOF pseudo-complement pullback relation satisfies semantics sentence set theory set-theoretic Sh(C sheaf sheaves Show singleton stalk structure Sub(d subobject classifier subset symbol terminal object Theorem Top(I topoi topological space topology true truth truth-values unique arrow v₁ valid variables well-pointed