Abstract Assume ZF + AD +V=L(ℝ) and letκ< Θ be an uncountable cardinal. We show thatκis Jónsson, and that if cof (κ) = ω thenκis Rowbottom. We also establish some other partition properties.

Abstract In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and and are generic extensions ofVsatisfying CH then and agree on all Σ12-statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for Σ12. Moreover, CH is the unique Σ12-statement with this feature in the sense that any other Σ12-statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axiomsAsuch that relative to large cardinal axioms ZFC +Ais Ω-complete for all of third-order arithmetic. Going further, for each specifiable segmentVλof the universe of sets (for example, one might takeVλto be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axiomsAsuch that relative to large cardinal axioms ZFC +Ais Ω-complete for the theory ofVλ. If such theories exist, extend one another, and are unique in the sense that any other such theoryBwith the same level of Ω-completeness asAis actually Ω-equivalent toAover ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies ¬-CH.

The Inner Model Hypothesis (IMH) and the Strong Inner Model Hypothesis (SIMH) were introduced in [4]. In this article we establish some upper and lower bounds for their consistency strength. We repeat the statement of the IMH, as presented in [4]. A sentence in the language of set theory is internally consistent iff it holds in some (not necessarily proper) inner model. The meaning of internal consistency depends on what inner models exist: If we enlarge the universe, it is possible that more statements become internally consistent. The Inner Model Hypothesis asserts that the universe has been maximised with respect to internal consistency: The Inner Model Hypothesis (IMH): If a statement φ without parameters holds in an inner model of some outer model of V (i.e., in some model compatible with V), then it already holds in some inner model of V. Equivalently: If φ is internally consistent in some outer model of V then it is already internally consistent in V. This is formalised as follows. Regard V as a countable model of Gödel-Bernays class theory, endowed with countably many sets and classes. Suppose that V* is another such model, with the same ordinals as V. Then V* is an outer model of V (V is an inner model of V*) iff the sets of V* include the sets of V and the classes of V* include the classes of V. V* is compatible with V iff V and V* have a common outer model.

The Jensen covering lemma says that either L has a club class of indiscernibles, or else, for every uncountable set A of ordinals, there is a set B ∈ L with A ⊆ B and card (B) = card(A). One might hope to extend Jensen's covering lemma to richer core models, which for us will mean to inner models of the form L[ ] where is a coherent sequence of extenders of the kind studied in Mitchell-Steel [8], The papers [8], [12], [10] and [1] show how to construct core models with Woodin cardinals and more. But, as Prikry forcing shows, one cannot expect too direct a generalization of Jensen's covering lemma to core models with measurable cardinals. Recall from [8] that if L[ ] is a core model and α is an ordinal, then either Eα = ∅, or else Eα is an extender over As in [8], we assume here that if Eα is an extender, then Eα is below superstrong type in the sense that the set of generators of Eα is bounded in (crit(Eα)). Let us say that L[ ] is a lower-part core model iff for every ordinal α, Eα is not a total extender over L[ ]. In other words, if L[ ] is a lower-part core model, then no cardinal in L[ ] is measurable as witnessed by an extender on . Other than the “below superstrong” hypothesis, we impose no bounds on the large cardinal axioms true in the levels of a lower-part core model.

Some of the most striking results in modern set theory have emerged from the study of simply-definable sets of real numbers. Indeed, simple questions like: what are the posible cardinalities?, are they measurable?, do they have the property of Baire?, etc., cannot be answered in ZFC. When one restricts the attention to the analytic sets, i.e., the continuous images of Borel sets, then ZFC does provide an answer to these questions. But this is no longer true for the projective sets, i.e., all the sets of reals that can be obtained from the Borel sets by taking continuous images and complements. In this paper we shall concentrate on particular projective classes, the , and using forcing constructions we will produce models of ZFC where, for some n, all , sets have some specified property. For the definition and basic facts about the projective classes , and , as well as the Kleene (or lightface) classes , and , we refer the reader to Moschovakis [19]. The first part of the paper is about measure and category. Early in this century, Luzin [16] and Luzin-Sierpiński [17] showed that all analytic (i.e., ) sets of reals are Lebesgue measurable and have the property of Baire.

Abstract It is proven that the following statement: “there exists a club C ⊆ κ such that every α ∈ C is an inaccessible cardinal in L and, for every δ a limit point of C, C ∩ δ is almost contained in every club of δ of L” is equiconsistent with a weakly compact cardinal if δ = ℵ1, and with a weakly compact cardinal of order 1 if δ = ℵ2.

We prove several independence results relevant to an old question in the folklore of set theory. These results complement those in [Sh, Chapter XIII, §4]. The question is the following. Suppose V ⊨ “ZFC + CH” and r is a real not in V. Must V[r] ⊨ CH? To avoid trivialities assume = . We answer this question negatively. Specifically we find pairs of models (W, V) such that W ⊨ ZFC + CH, V = W[r], r a real, = and V ⊨ ¬CH. Actually we find a spectrum of such pairs using ZFC up to “ZFC + there exist measurable cardinals”. Basically the nicer the pair is as a solution, the more we need to assume in order to construct it. The relevant results in [Sh, Chapter XIII] state that if a pair (of inner models) (W, V) satisfies (1) and (2) then there is an inaccessible cardinal in L; if in addition V ⊨ 2ℵ0 > ℵ2 then 0# exists; and finally if (W, V) satisfies (1), (2) and (3) with V ⊨ 2ℵ0 > ℵω, then there is an inner model with a measurable cardinal. Definition 1. For a pair (W, V) we shall consider the following conditions: (1) V = W[r], r a real, = , W ⊨ ZFC + CH but CH fails in V. (2) W ⊨ GCH. (3) W and V have the same cardinals.