Constructive set theory started with Myhill's seminal 1975 article [8]. This paper will be concerned with axiomatizations of the natural numbers in constructive set theory discerned in [3], clarifying the deductive relationships between these axiomatizations and the strength of various weak constructive set theories.

Gödel interpreted Heyting arithmetic HA in a “logic-free” fragment T 0 of his theory T of primitive recursive functionals of finite types by his famous Dialectica-translation D . This works because the logic of HA is extremely simple. If the logic of the interpreted system is different—in particular more complicated—, it forces us to look for different and more complicated functional translations. We discuss the arising logical problems for arithmetical and set theoretical systems from HA to CZF . We want (...) to test the thesis: While the functionals take care of the proof theoretic strength of the interpreted system, it is the functional translation that has to cope with the logical complexities of the system. (shrink)

Gödel's dialectica interpretation of Heyting arithmetic HA may be seen as expressing a lack of confidence in our understanding of unbounded quantification. Instead of formally proving an implication with an existential consequent or with a universal antecedent, the dialectica interpretation asks, under suitable conditions, for explicit 'interpreting' instances that make the implication valid. For proofs in constructive set theory CZF-, it may not always be possible to find just one such instance, but it must suffice to explicitly name a set (...) consisting of such interpreting instances. The aim of eliminating unbounded quantification in favor of appropriate constructive functionals will still be obtained, as our ∧-interpretation theorem for constructive set theory in all finite types CZFω- shows. By changing to a hybrid interpretation ∧q, we show closure of CZFω- under rules that – in stronger forms – have already been studied in the context of Heyting arithmetic. In a similar spirit, we briefly survey modified realizability of CZFω- and its hybrids. Central results of this paper have been proved by Burr 2000a and Schulte 2006, however, for different translations. We use a simplified interpretation that goes back to Diller and Nahm 1974. A novel element is a lemma on absorption of bounds which is essential for the smooth operation of our translation. (shrink)

The Dialectica-style functional interpretation of Kripke-Platek set theory with infinity ( $\hbox{\sf KP} \omega$ ) given in [1] uses a choice functional (which is not a definable set function of ( $hbox{\sf KP} \omega$ ). By means of a Diller-Nahm-style interpretation (cf. [4]) it is possible to eliminate the choice functional and give an interpretation by set functionals primitive recursive in $x\mapsto\omega$ . This yields the following characterization: The class of $\Sigma$ -definable set functions of $\hbox{\sf KP} \omega$ coincides with (...) the collection of set functionals of type 1 primitive recursive in $x\mapsto \omega$. (shrink)