Geometric Logic and Classifying Topoi

  • Saunders Mac Lane
  • Ieke Moerdijk
Part of the Universitext book series (UTX)


A first-order formula q5(xl,xn)is called“geometric”if it is built up from atomic formulas by using conjunction,disjunction,and existential quantification,Geometric logic is the logic of the implications between geometric formulas:
$$\forall x(\phi (x) \to \psi (x))$$
where the arrow here is for “implication” and and z/) are geometric. Many mathematical structures can be axiomatized by formulas of this form (1).For instance, local rings are axiomatized by the usual equations for a commutative ring with unit, together with the axiom
$$\forall x,y \in R(x + y = 1 \to \exists z(x \cdot z = 1) \vee \exists z(y \cdot z = 1))$$
which states that the ring is local; this axiom (2) is indeed of the form (1).


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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Saunders Mac Lane
    • 1
  • Ieke Moerdijk
    • 2
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Mathematical InstituteUniversity of UtrechtUtrechtThe Netherlands

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