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 ?


Gérard LANG



About Grothendieck Universe and Tarski's A and A' Axioms
The answer to question 1 is affirmative for GU and IN but
negative for TA. That is, the proper class formulation of
TA is not equivalent to TA, unless both are inconsistent.


Asserting that every ordinal is below an inaccessible
cardinal is clearly equivalent in ZF to asserting that the
inaccessible cardinals form a proper class. And since the
Grothendieck universes are exactly the sets $V\_\kappa$ for
an inaccessible cardinal $\kappa$, the theory GU over ZFC
is equivalent to the assertion that there are a proper
class of universes. So those cases are relatively straightforward, as you had guessed.


But the case of the Tarski axiom is different, since his
universes are not transitive sets. In fact, if there is a
single inaccessible cardinal $\kappa$, one can already form
a proper class of Tarski sets. To see this, suppose that
$\kappa$ is inaccessible and $x$ is any set. Build a Tarski
set $U$ as follows: 


* $U\_0=\{\{x\}\}$,
* $U\_{\alpha=1}=P(U\_\alpha)$ and
* $U\_\lambda=\bigcup\_\alpha
U\_\alpha$ at limit ordinals.


That is, we perform the usual cumulative
hiearchy, but starting with object {x} instead of nothing.
The resulting set $U=U\_\kappa$ contains {x} as an element,
has size $\kappa$ and is closed under subsets, power sets
and small unions, so it is a Tarski universe. (Note that if
$y\in U\_{\alpha+1}$, then all subsets of $y$ are also in
$U\_{\alpha+1}$, and so $P(y)$ is added on the next step.
And {x} has only two subsets, which are both in $U$.) But
for sufficiently large $x$, the resulting $U$ will not be
transitive, since $x$ itself will never be added as an
element. Furthermore, since $x$ was arbitrary, from a
single inaccessible cardinal we can build a proper class of
Tarski universes. And so the proper class formulation of TA
will not be equivalent to TA, unless both theories are
inconsistent, because if the existence of an inaccessible
cardinal is consistent with ZFC, then it is consistent that
there is exactly one such cardinal, by chopping the
universe off at the second one. The resulting model will
satisfy the proper class formulation of TA but not TA
itself.


