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**1 - 3**of**3**### A NORMALIZING INTUITIONISTIC SET THEORY WITH INACCESSIBLE SETS ∗

, 2006

"... Vol. 3 (3:6) 2007, pp. 1–31 ..."

### NORMALIZATION OF IZF WITH REPLACEMENT

, 2007

"... Abstract. IZF is a well investigated impredicative constructive version of Zermelo-Fraenkel set theory. Using set terms, we axiomatize IZF with Replacement, which we call IZFR, along with its intensional counterpart IZF − R. We define a typed lambda calculus λZ corresponding to proofs in IZF − R acc ..."

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Abstract. IZF is a well investigated impredicative constructive version of Zermelo-Fraenkel set theory. Using set terms, we axiomatize IZF with Replacement, which we call IZFR, along with its intensional counterpart IZF − R. We define a typed lambda calculus λZ corresponding to proofs in IZF − R according to the Curry-Howard isomorphism principle. Using realizability for IZF − R, we show weak normalization of λZ. We use normalization to prove the disjunction, numerical existence and term existence properties. An inner extensional model is used to show these properties, along with the set existence property, for full, extensional IZFR. 1.

### TYPES, SETS AND CATEGORIES

"... This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category t ..."

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This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, I have elected to offer detailed technical presentations of just a few important instances. 1 THE ORIGINS OF TYPE THEORY The roots of type theory lie in set theory, to be precise, in Bertrand Russell’s efforts to resolve the paradoxes besetting set theory at the end of the 19 th century. In analyzing these paradoxes Russell had come to find the set, or class, concept itself philosophically perplexing, and the theory of types can be seen as the outcome of his struggle to resolve these perplexities. But at first he seems to have regarded type theory as little more than a faute de mieux.