Topoi: The Categorial Analysis of Logic
Courier Corporation, Jan 1, 2006 - Mathematics - 551 pages
A 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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6-arrow adjoint algebra applied arrow assigns associated axioms Boolean called Chapter classical collection commutes complete component composite concept condition consider construction continuous corresponding covers deﬁned deﬁnition denoted described determined developed diagram domain elements epic equal equaliser equivalent EXAMPLE EXERCISE exists expression f and g fact ﬁnite ﬁrst follows formula function functor geometric given gives hence holds identity iﬁ implies inclusion interpretation isomorphic language lattice logic mathematical means monic morphism natural notion object obtain operation pair partial particular poset precisely preserves principle PROOF prove pullback relation result rules satisﬁes sentence set theory sheaf sheaves Show space square statement structure subobject subset symbol Theorem theory topoi topology topos transitive true truth truth-values unique universal valid variables