[Footnote 1] Too technical to define here. It is important only for systems in which there is possible a very deep nesting of sets, and so has little effect on ordinary mathematics.

[Footnote 2] An "intangible" is an object whose existence can be proved in conventional mathematics (ZF + AC) but not in the weaker quasiconstructive mathematics (ZF + DC).

[Footnote 3] I am thinking of the incident in which a revolutionary new calculus book that featured nonstandard analysis was delivered to a rabid constructivist to review for Mathematical Reviews. I thought the book was adventurous and exciting; the reviewer, however, succumbed to territoriality and gave the book a thrashing.

[Footnote 4] If I were presenting this material to a class, I would have defined a dissection of [a,b] as a set of partition points {} plus a set of intermediate points {}. Then v is a Henstock integral of f if for each number there exists a positive function for which (1) holds for all dissections satisfying .