We define and investigate constructibility in higher order arithmetics. In particular we get an interpretation ofn-order arithmetic inn-order arithmetic without the scheme of choice such that ∈ and the property “to be a well-ordering” are absolute in it and such that this interpretation is minimal among such interpretations.

We show an axiom A such that there is no nontrivial interpretation of the alternative set theory (AST) inAST+A keeping ∈, sets and the class of all “standard” natural numbers. Furthermore, there is no interpretation ofAST inAST without the prolongation axiom, but there is an interpretation ofAST in the theory having the prolongation axiom and the basic set-theoretical axioms only.

Proceeding in the theory with extensionality, comprehension for classes, existence of the empty set and the assumption the addition of one element to a set makes again a set we show a week assumption which guarantees existence of a saturated elementary extension of the system of hereditarily finite sets.