Introduction to the constructive point of view in the foundations of mathematics, in particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation and formalization of logic is described. For constructive (intuitionistic) arithmetic, Kleene’s realizability interpretation is given; this provides an example of the possibility of a constructive mathematical practice which diverges from classical mathematics. The crucial notion in intuitionistic analysis, choice sequence, is briefly described and some principles which are valid for choice sequences are discussed. The second half of the article deals with some aspects of proof theory, i.e., the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions are outlined: his introduction of the so-called Gentzen systems which use sequents instead of formulas and his result on first-order arithmetic showing that (suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in first-order arithmetic.

Abstract. In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters. The Stone-Yosida representation theorem for Riesz spaces [LZ71, Zaa83] shows how to embed every Riesz space into the Riesz space of continuous functions on its spectrum. Theorem 1. [Stone-Yosida] Let R be an Archimedean Riesz space (vector lattice) with unit. Let Σ be its (compact Hausdorff) space of representations. Define the continuous function ˆr(σ): = σ(r) on Σ. Then r ↦ → ˆr is a Riesz embedding of R into C(Σ,). The theorem is very convenient, but sometimes better avoided, since it leads out of the theory of Riesz spaces. To quote Zaanen [Zaa97]: