A-The addition of the Grothendieck Universe Axiom (for every set x, there exists a set y that is a universe and contains x as member element) to ZFC (ZFC+GU) is considered as giving an almost good solution permitting the insertion of Category theory inside Set theory. It is known that we obtain an equivalent theory (ZFC+TA) by replacing GU with the Axiom A of Tarski (for every set x, there exists a set y that is an A Tarski set and contains x as a member element), or also (ZFC+IN) by replacing GU by the axiom IN (for every ordinal x, there exists a strongly inaccessible ordinal y containing x as a member element.
1-In each three cases, we can prove (using foundation) that "there exists a proper class of (GU sets, TA sets, inaccessible cardinals)" is a theorem. But is it possible to replace the axioms GU, TA and IN by the formulation with proper classes (this seems possible for IN and GU ) ?
B-If we replace ZFC by ZF, the situation seems much more involved. As it is provable that AC is a consequence of TA, ZF+TA is equivalent to ZFC+TA. As it is provable that some GU sets cannot be well-ordered, AC is not a consequence of ZF+GU, that cannot be equivalent to ZFC+GU. For the third case, this is depending of how we define an inaccessible without AC; but if we take the "correct one", Ac is not derivable from ZF+IN'. These question are very thoroughly presented in a message of R. Solovay to FOM "AC and stongly inaccessible cardinals" (29/02/2008). But, in fact, the power set axiom and the Infinity axiom can also be derived from the Tarski a axiom. So that one could think that the theory: Extensionality+Replacement+AT+Union+Foundation is equivalent to ZFC+GU. But when you try to developp such a theory, it seems that you are obliged to also introduce the Pair Axiom before introducing AT that needs a definition of functions.
2-Is it in fact possible to dispense of the axiom of the pair within this theory ?
C-Tarski's A axiom is given inside his paper (auf deutsch) "Über unerreichbare Kardinalzahlen", Fund Math 1938, page 84. 3-On the same page, Tarski gives another axiom, named A'with four conditions (as in the case of A) and writes ""Übrigens sind vershiedene âquivalente Unformung dieses Axioms [A] bekannt. Man kann Z. B. Bedingungen A-1-A4 beziehungsweise durch folgenden Bedingungen [A'1-A'4] ersetzen (und zwar jede Bedingung unhabhängig von denen anderen). Does anyone completely understand what is exactly meant here by Tarski, and how is this proved ?