  Here the following is a typical scenario. Given what are classically two variants of a single set-theoretic principle, their classical proof of equivalence requires  at some point an instance of the excluded middle. However, in general this proof of equivalence will not carry through to an intuitionistic context, and thus what are classically two forms of one principle may result into two distinct principles when working intuitionistically.  Choosing one rather than the other of them may therefore influence the notion of set we thus define.  In the context of constructive set theories like CZF, power set and  separation are  replaced by intuitionistically weaker principles. One reason for this is that  the full strenght of power set and full separation are seen as unnecessary,  since their weaker substitutes seem to  suffice for carrying out constructive mathematics. Another reason is that they  are seen as philosophically problematic, since they may introduce forms of impredicativity within the set theory (see the section on  Predicativity in constructive set theory). The case of replacement versus collection is somehow more complex (see, for example, the articles (Friedman and Scedrov 1985), (Rathjen 2005) and (Rathjen 2012)). It is worth stressing that while adopting the usual formulation of foundation goes against the very assumption of intuitionistic logic as background logic, the principles of separation and power set have no incompatibility with intuitionistic logic at all, so much so that they are integral part of the intuitionistic theory of sets IZF (Friedman 1973a). 