The present volume is intended as an all-round introduction to constructivism. Here constructivism is to be understood in the wide sense, and covers in particular Brouwer's intuitionism, Bishop's constructivism and A.A. Markov's constructive recursive mathematics. The ending "-ism" has ideological overtones: "constructive mathematics is the (only) right mathematics"; we hasten, however, to declare that we do not subscribe to this ideology, and that we do not intend to present our material on such a basis.

Constructive Zermelo–Fraenkel set theory, CZF, can be interpreted in Martin-Löf type theory via the so-called propositions-as-types interpretation. However, this interpretation validates more than what is provable in CZF. We now ask ourselves: is there a reasonably simple axiomatization of the set-theoretic formulae validated in Martin-Löf type theory? The answer is yes for a large collection of statements called the mathematical formulae. The validated mathematical formulae can be axiomatized by suitable forms of the axiom of choice.The paper builds on a self-interpretation (...) of CZF IOS Press, Amsterdam, 2004 ]) that provides an “inner” model of CZF which also validates the so-called -axiom of choice, . The crucial technical step taken in the present paper is to investigate the absoluteness properties of this model under the hypothesis .It is also shown that CZF plus the -axiom of choice possesses the disjunction property, the numerical existence property and the existence property for an important group of formulae. (shrink)

In order to build the collection of Cauchy reals as a set in constructive set theory, the only power set-like principle needed is exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, Constructive Zermelo–Fraenkel set theory with subset collection replaced by exponentiation, in which the Cauchy (...) reals form a set while the Dedekind reals constitute a proper class. (shrink)

Under the assumption that all "rules" are recursive (ECT) the statement $\operatorname{Cont}(N^N,N)$ that all functions from N N to N are continuous becomes equivalent to a statement KLS in the language of arithmetic about "effective operations". Our main result is that KLS is underivable in intuitionistic Zermelo-Fraenkel set theory + ECT. Similar results apply for functions from R to R and from 2 N to N. Such results were known for weaker theories, e.g. HA and HAS. We extend not only (...) the theorem but the method, fp-realizability, to intuitionistic ZF. (shrink)