Starkes und überstarkes Auswahlaxiom

Abstract

It is well known that equivalence holds between the weak axiom of choice (AC) and the well ordering principle (WOP) for sets, resp. between strong AC and WOP for classes. It will be shown that in a theory PC^* with inpredicative classes (i. e. with no restriction of quantification in the defining formula) the super-strong AC used by the informally working mathematician is equivalent to a superstrong WOP. The equivalence between strong AC and super-strong AC is implied by a conditionC refutable in PC^* but provable in PC which is PC^* with predicative classes only and with the general ordered pair axiom. PC^* [super-strong AC] is inconsistent because the super-strong AC impliesC. Therefore the application of choice functions to non-empty classes generally makes a predicative definition of these classes necessary. Connected with these problems is a statement equivalent to the conjunction of the axioms of power set and foundation based on a function which coincides with the von Neumann-function under the assumption of one of the mentioned axioms.