A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an arbitrary metric space. The real numbers are characterized as a complete Archimedean Heyting field, a terminal object in the category of Archimedean Heyting fields.

We extend the concept of apartness spaces to the concept of quasi-apartness spaces. We show that there is an adjunction between the category of quasi-apartness spaces and the category of neighbourhood spaces, which indicates that quasi-apartness is a more natural concept than apartness. We also show that there is an adjoint equivalence between the category of apartness spaces and the category of Grayson’s separated spaces.

We extend Bishopʼs concept of function spaces to the concept of pre-function spaces. We show that there is an adjunction between the category of neighbourhood spaces and the category of Φ-closed pre-function spaces. We also show that there is an adjunction between the category of uniform spaces and the category of Ψ-closed pre-function spaces.

The purpose of this paper is to justify the claim that Topos theory and Logic (the latter interpreted in a wide enough sense to include Model theory and Set theory) may interact to the advantage of both fields. Once the necessity of utilizing toposes (other than the topos of Sets) becomes apparent, workers in Topos theory try to make this task as easy as possible by employing a variety of methods which, in the last instance, find their justification in metatheorems (...) from Logic. Some concrete instances of this assertion will be given in the form of simple proofs that certain theorems of Algebra hold in any (Grothendieck) topos, in order to illustrate the various techniques that are used. In the other direction, Topos theory can also be a useful tool in Logic. Examples of this are independence proofs in (classical as well as intuitionistic) Set theory, as well as transfer methods in the presence of a sheaf representation theorem, the latter applied, in particular, to model theoretic properties of certain theories. (shrink)

The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in (...) the famous Hilbert-Brouwer controversy in the 1920s. -/- The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics. (shrink)