In a message of the 29 th March 2008 edited on the FOM list "AC and strongly inaccessible cardinals", Robert Solovay shows that the so-called Tarski-Grothendieckset set theory can be equivalently axiomatized as: (1) ZFC + "There exists a proper class of strongly inaccessible ordinals" (2) ZFC + "Every set is member of a Grothendieck universe U" (3) ZFC + "Every set is member of some Tarski class" Moreover, in the case (3), it is possible to show that the added Tarski's axiom A allows to dispense with the three ZFC axioms of Power set, of Infinity and of Choice, so that the Tarski-Grothendieck set theory can be axiomatized using only the following list of axioms: Extensionality, Foundation, Null set, Pair set, Replacement, Union and Axiom A:
First question: Would it be possible to dispense with the Pair set axiom ?
Suppose now that we change our basic set theory from ZFC to MK (Morse-Kelley) set theory. In this context a more natural choice axiom is the Global Choice axiom ("There exists a set-like well-order on the universal class V"), so that:
Question 2: Is it possible to derive the Global Choice axiom from the set theory (4) MK + "There class of strongly inaccessible cardinals is a proper class", where we do not suppose the axiom of choice in MK ? Gérard Lang