The λσ-calculus is a refinement of the λ-calculus where substitutions are manipulated explicitly. The λσ-calculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical λ-calculus and concrete implementations.

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The standard construction of quotient spaces in topology uses full separation and power sets. We show how to make this construction using only the (generalised) predicative methods available in constructive type theory and constructive set theory. MSC: primary 03F65, 54B15, secondary 03B15, 03E70. Keywords: quotient spaces, constructive topology, Martin-Löf type

In this paper we show that the usual intuitionistic characterization of the decidability of the propositional function B(x) prop [x: A], i.e. to require that the predicate (∀x ∈ A) B(x)∨¬B(x) is provable, is equivalent, when working within the framework of Martin-Löf’s Intuitionistic Type Theory, to require that there exists a decision function φ: A → Boole such that (∀x ∈ A) (φ(x) =Boole true) ↔ B(x). Since we will also show that the proposition x =Boole true [x: Boole] is decidable, we can alternatively say that the main result of this paper is a proof that the decidability of the predicate B(x) prop [x: A] can be effectively reduced by a function φ ∈ A → Boole to the decidability of the predicate φ(x) =Boole true [x: A]. All the proofs are carried out within the Intuitionistic Type Theory and hence the decision function φ, together with a proof of its correctness, is effectively constructed as a function of the proof of (∀x ∈ A) B(x)∨¬B(x). 1 The basic lemmas

The theory of formal spaces and the more recent theory of apartness spaces have a priori not much more in common than that each of them was initiated as a constructive approach to general topology. We nonetheless try to do the first steps in relating these competing theories to each other. Formal topology was put forward in the mid 1980s by Sambin [8] in order to make available to Martin–Löf’s type theory [7] the concepts of classical topology that are worth keeping to such a constructive and predicative framework. In the meantime formal topology has proved a fairly universal setting for doing topology in a point–free way. We refer to [9] for a recent and exhaustive survey of formal topology. The theory of apartness spaces was started by Bridges and Vîță [4] nearly twenty years later to reformulate set–theoretic topology as an extension of Bishop’s constructive analysis [2, 3]. The subsequent development of the theory of apartness spaces has also shed some light on its classical counterpart, the theory of proximity or nearness spaces. A comprehensive overview will be available soon [5]. In formal topology ‘basic neighbourhood ’ is a primitive concept, whereas ‘point ’ is a derived notion; as sets of basic neighbourhoods, points have to be handled with particular care to meet the needs of a predicative framework like Martin–Löf type theory. In the theory of apartness spaces, it is the other way round: as in classical topology, points are given as such, and (basic) neighbourhoods are sets of points. Since, however, it is hard to detect any truly impredicative move in the practice of Bishop’s constructive mathematics in general, we dare to undertake the following attempt to link formal topology and the theory of apartness spaces to each other. 1 Basic definitions We recall the standard definitions associated with formal topologies and morphisms between them (approximable mappings).

Abstract. We consider an extensional version, called qmTT, of the intensional Minimal Type Theory mTT, introduced in a previous paper with G. Sambin, enriched with proof-irrelevance of propositions and effective quotient sets. Then, by using the construction of total setoid à la Bishop we build a model of qmTT over mTT. The design of an extensional type theory with quotients and its interpretation in mTT is a key technical step in order to build a two level system to serve as a minimal foundation for constructive mathematics as advocated in the mentioned paper about mTT.

GF (Grammatical Framework) is a Logical Framework enriched with concrete syntax specifications.

Abstract. We explain in detail why the notion of list-arithmetic pretopos should be taken as the general categorical definition for the construction of arithmetic universes introduced by André Joyal to give a categorical proof of Gödel’s incompleteness results. We motivate this definition for three reasons: first, Joyal’s arithmetic universes are listarithmetic pretopoi; second, the initial arithmetic universe among Joyal’s constructions is equivalent to the initial list-arithmetic pretopos; third, any list-arithmetic pretopos enjoys the existence of free internal categories and diagrams as required to prove Gödel’s incompleteness. In doing our proofs we make an extensive use of the internal type theory of the categorical structures involved in Joyal’s constructions. The definition of list-arithmetic pretopos is equivalent to the general one that I came to know in a recent talk by André Joyal. 1.

We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the set-theoretic version introduced in [MS05] with collections. The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model. This two-level theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the “proofs-as-programs ” paradigm and acts as a programming language.

Introduction. In 1937 E. Čech and M.H. Stone independently introduced the maximal compactification of a completely regular topological space, thereafter called Stone-Čech compactification [8, 18]. In the introduction of [8] the non-constructive character of this result is so described: “it must be emphasized that β(S) [the Stone-Čech compactification of S] may be