These comparisons have involved fragments of arithmetic, fragments of real analysis, set theories, and type theories.
Intuitionistic versions of Zermelo-Fraenkel set theories were introduced in the early 1970s by Friedman and Myhill.
But, as is well known, set theories such as ZF, NBG and the like were in various ways ad hoc.
The most famous example here is probably the continuum hypothesis (CH), which is undecidable in currently accepted set theories, e.g., Zermelo-Frankel set theory (ZF).
This gives rise to set theories such as Intuitionistic Zermelo-Fraenkel, IZF, a variant of which was introduced as early as in (Friedman 1973a).
Of these set theories, only NFU with Infinity, Choice and possibly further strong axioms of infinity (of which more anon) is really mathematically serviceable.
It has been argued that question meanings as construed by proposition set theories are all too coarse-grained to account for certain linguistic phenomena.
The study of alternative set theories can dispel a facile identification of "set theory" with "Zermelo-Fraenkel set theory"; they are not the same thing.
In the following we shall make use of a convention which is often in place today, according to which the adjective "intuitionistic" refers to those set theories, such as IZF, which are impredicative, while "constructive" refers to set theories, such as CZF, which comply with a form of predicativity.
We observe, though, that there are other set theories in which nonstandard objects are deliberately provided (we will review some of these below), and some of the applications of those set theories to "nonstandard analysis" might be duplicated in suitable versions of NFU.
