The von Neumann hierarchy of sets is heavily used as a basic tool in classical set theory, being an underlying ingredient in many proofs and concepts. In constructive set theories like without the powerset axiom however, it loses much of its potency by ceasing to be a hierarchy of sets as its single stages become only classes. This article proposes an alternative cumulative hierarchy which does not have this drawback and provides examples of how it can be used to prove (...) new theorems in. (shrink)

This is the first in a series of papers on Predicative Algebraic Set Theory, where we lay the necessary groundwork for the subsequent parts, one on realizability [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory II: Realizability, Theoret. Comput. Sci. . Available from: arXiv:0801.2305, 2008], and the other on sheaves [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory III: Sheaf models, 2008 ]. We introduce the notion of a predicative category with small (...) maps and show that it provides a sound and complete semantics for constructive set theories like IZF and CZF. The main technical contribution of this paper is that it shows in detail that such categories can always be conservatively embedded in categories that are exact. These exactness properties play a crucial rôle in showing that predicative categories with small maps contain models of set theory and that they are closed under sheaves and realizability. We will prove the former statement in this paper as well, leaving a proof of the closure properties to the papers on realizability and sheaves as mentioned above. (shrink)

The paper studies how the localic notion of sublocale transfers to formal topology. For any formal topology (not necessarily with positivity predicate) we define a sublocale to be a cover relation that includes that of the formal topology. The family of sublocales has set-indexed joins. For each set of base elements there are corresponding open and closed sublocales, boolean complements of each other. They generate a boolean algebra amongst the sublocales. In the case of an inductively generated formal topology, the (...) collection of inductively generated sublocales has coframe structure. Overt sublocales and weakly closed sublocales are described, and related via a new notion of "rest closed" sublocale to the binary positivity predicate. Overt, weakly closed sublocales of an inductively generated formal topology are in bijection with "lower powerpoints", arising from the impredicative theory of the lower powerlocale. Compact sublocales and fitted sublocales are described. Compact fitted sublocales of an inductively generated formal topology are in bijection with "upper powerpoints", arising from the impredicative theory of the upper powerlocale. (shrink)

We develop a number of variants of Lifschitz realizability for $\mathbf {CZF}$ by building topological models internally in certain realizability models. We use this to show some interesting metamathematical results about constructive set theory with variants of the lesser limited principle of omniscience including consistency with unique Church’s thesis, consistency with some Brouwerian principles and variants of the numerical existence property.

We present an alternative solution to the problem of inductive generation of covers in formal topology by using a restricted form of type universes. These universes are at the same time constructive analogues of regular cardinals and sets of infinitary formulae. The technique of regular universes is also used to construct canonical positivity predicates for inductively generated covers.

We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory. The original interpretation treated logic in Martin-Löf type theory via the propositions-as-types interpretation. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in type theories allows us to study reinterpretations of logic, such as the double-negation translation.

Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Earlier work showed all of these implications to be strict. Here we reprove one of the strictness results, using very different arguments. The technique used is a mixture of realizability, forcing in the guise of Heyting-valued models, and Kripke models.

In this paper a method to construct Kripke models for subtheories of constructive set theory is introduced that uses constructions from classical model theory such as constructible sets and generic extensions. Under the main construction all axioms except the collection axioms can be shown to hold in the constructed Kripke model. It is shown that by carefully choosing the classical models various instances of the collection axioms, such as exponentiation, can be forced to hold as well. The paper does not (...) contain any deep results. It consists of first observations on the subject, and is meant to introduce some notions that could serve as a foundation for further research. (shrink)

We prove a version of the associated sheaf functor theorem in Algebraic Set Theory. The proof is established working within a Heyting pretopos equipped with a system of small maps satisfying the axioms originally introduced by Joyal and Moerdijk. This result improves on the existing developments by avoiding the assumption of additional axioms for small maps and the use of collection sites.

In intuitionistic generalized predicative systems as constructive set theory, or constructive type theory, two categories have been proposed to play the role of the category of locales: the category FSp of formal spaces, and its full subcategory FSpi of inductively generated formal spaces. Considered in impredicative systems as the intuitionistic set theory IZF, FSp and FSpi are both equivalent to the category of locales. However, in the mentioned predicative systems, FSp fails to be closed under basic constructions such as that (...) of finite products, while FSpi is complete , but does not contain some naturally occurring classes of formal spaces. Moreover, completeness of FSpi only holds under the assumption of strong principles for the existence of inductively defined sets.In this paper, working in the context of constructive set theory, the category ImLoc of imaginary locales is introduced. ImLoc is complete already over weak formulations of constructive set theory, contains FSp and FSpi as full subcategories, and, as FSp and FSpi, is equivalent to the category of locales in IZF.Applications of the concept of imaginary locale to a proof of a point-free version of Tychonoff embedding theorem, and to set-representation theorems, are then presented. (shrink)

This paper presents several independence results concerning the topos-valid and the intuitionistic predicative theory of locales. In particular, certain consequences of the consistency of a general form of Troelstra's uniformity principle with constructive set theory and type theory are examined.

We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.

We introduce a new axiom scheme for constructive set theory, the Relation Reflection Scheme . Each instance of this scheme is a theorem of the classical set theory ZF. In the constructive set theory CZF–, when the axiom scheme is combined with the axiom of Dependent Choices , the result is equivalent to the scheme of Relative Dependent Choices . In contrast to RDC, the scheme RRS is preserved in Heyting-valued models of CZF– using set-generated frames. We give an application (...) of the scheme to coinductive definitions of classes. (shrink)

Constructive and intuitionistic Zermelo-Fraenkel set theories are axiomatic theories of sets in the style of Zermelo-Fraenkel set theory (ZF) which are based on intuitionistic logic. They were introduced in the 1970's and they represent a formal context within which to codify mathematics based on intuitionistic logic. They are formulated on the basis of the standard first order language of Zermelo-Fraenkel set theory and make no direct use of inherently constructive ideas. In working in constructive and intuitionistic ZF we can thus (...) to some extent rely on our familiarity with ZF and its heuristics. -/- Notwithstanding the similarities with classical set theory, the concepts of set defined by constructive and intuitionistic set theories differ considerably from that of the classical tradition; they also differ from each other. The techniques utilised to work within them, as well as to obtain metamathematical results about them, also diverge in some respects from the classical tradition because of their commitment to intuitionistic logic. In fact, as is common in intuitionistic settings, a plethora of semantic and proof-theoretic methods are available for the study of constructive and intuitionistic set theories. -/- The entry introduces the main features of Constructive and Intuitionistic ZF and offers links to the relevant bibliography. (shrink)