Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

100 the realm of the infiniteMoreover, this one prediction implies all the predictions (of formal consistency) the SetTheorist can currently make based on the entire large cardinal hierarchy as presentlyconceived (in the context of a universe of sets that satisfies the Axiom of Choice). Mypoint is that by appealing to the Skeptic’s Retreat, one could reasonably claim that thetheoryZF + “There is a weak Reinhardt cardinal”is formally consistent, and in making this single claim, one would subsume all theclaims of consistency that the Set Theorist can make on the basis of our currentunderstanding of the universe of sets (without abandoning the Axiom of Choice).Before presenting a potential option to deal with this, I describe an analogous optionof how the Set Theorist can claim that the theoryZF + ADis consistent, even though, as I have indicated, AD also refutes the Axiom of Choice.The explanation requires some definitions, which I shall require anyway. Gödel defineda very special transitive class L ⊆ V and showed that all the axioms of ZFC hold wheninterpreted in L. The definition of L does not require the Axiom of Choice, so oneobtains the seminal result that if the axioms ZF are consistent, then so are the axiomsZFC. Gödel also proved that the continuum hypothesis holds in L, thereby showing thatone cannot formally refute the continuum hypothesis from the axioms ZFC (unless, ofcourse, these axioms are inconsistent).The definition of L is simply given by replacing the operation P(X) in the definitionof V α+1 by the operation P Def (X), which associates to the set X the set of all subsetsY ⊆ X such that Y is logically definable in the structure (X, ∈) from parameters in X.For any infinite set X, P Def (X) ⊂ P(X) and P Def (X) ≠ P(X).Thus, one defines L α by induction on the ordinal α, setting L 0 =∅, settingL α+1 = P Def (L α ),and taking unions at limit stages. The class L is defined as the class of all sets a suchthat a ∈ L α for some ordinal α. It is perhaps important to note that while there mustexist a proper class of ordinals α such thatL α = L ∩ V α ,this is not true for all ordinals α.Relativizing the definition of L to V ω+1 , we obtain the class L(V ω+1 ), which is morecustomarily denoted by L(R); here one definesL 0 (R) = V ω+1and proceeds by induction exactly as above to define L α (R) for all ordinals α. Theclass L(R) is the class of all sets a such that a ∈ L α (R) for some ordinal α.Unlike the case for L, one cannot prove that the Axiom of Choice holds in L(R),although one can show that all of the other axioms of ZFC hold in L(R). The followingtheorem, which is related to Theorem 2, not only establishes the consistency of ZF +AD from simply the existence of large cardinals, but also establishes that L(R) AD