Projective well-orderings and bounded forcing axioms

2005 ◽  
Vol 70 (2) ◽  
pp. 557-572
Author(s):  
Andrés Eduardo Caicedo
Keyword(s):  
Large Cardinal ◽  
Forcing Axioms ◽  
Woodin Cardinals ◽  
Inner Models

AbstractIn the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.

Fine structure for tame inner models

1996 ◽  
Vol 61 (2) ◽  
pp. 621-639 ◽  
Author(s):  
E. Schimmerling ◽  
J. R. Steel
Keyword(s):  
Fine Structure ◽  
Large Cardinal ◽  
Structure Theory ◽  
Uniqueness Problem ◽  
Covering Property ◽  
Strong Cardinal ◽  
Woodin Cardinals ◽  
Inner Models ◽  
Normal Measure ◽  
Consistency Results

In this paper, we solve the strong uniqueness problem posed in [St2]. That is, we extend the full fine structure theory of [MiSt] to backgrounded models all of whose levels are tame (defined in [St2] and below). As a consequence, more powerful large cardinal properties reflect to fine structural inner models. For example, we get the following extension to [MiSt, Theorem 11.3] and [St2, Theorem 0.3].Suppose that there is a strong cardinal that is a limit of Woodin cardinals. Then there is a good extender sequence such that(1) every level of is a sound, tame mouse, and(2) ⊨ “There is a strong cardinal that is a limit of Woodin cardinals”.Recall that satisfies GCH if all its levels are sound. Another consequence of our work is the following covering property, an extension to [St1, Theorem 1.4] and [St3, Theorem 1.10].Suppose that fi is a normal measure on Ω and that all premice are tame. Then Kc, the background certified core model, exists and is a premouse of height Ω. Moreover, for μ-almost every α < Ω.Ideas similar to those introduced here allow us to extend the fine structure theory of [Sch] to the level of tame mice. The details of this extension shall appear elsewhere. From the extension of [Sch] and Theorem 0.2, new relative consistency results follow. For example, we have the following application.If there is a cardinal κ such that κ is κ+-strongly compact, then there is a premouse that is not tame.

scholarly journals The Jensen Covering Property

2001 ◽  
Vol 66 (4) ◽  
pp. 1505-1523 ◽  
Author(s):  
E. Schimmerling ◽  
W. H. Woodin
Keyword(s):  
Large Cardinal ◽  
Core Model ◽  
Covering Property ◽  
Prikry Forcing ◽  
Covering Lemma ◽  
Woodin Cardinals ◽  
Inner Models ◽  
Coherent Sequence ◽  
Measurable Cardinals

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.

Splitting number at uncountable cardinals

1997 ◽  
Vol 62 (1) ◽  
pp. 35-42 ◽  
Author(s):  
Jindřich Zapletal
Keyword(s):  
Large Cardinal ◽  
Inner Models ◽  
Splitting Number ◽  
Uncountable Cardinal ◽  
Uncountable Cardinals

AbstractWe study a generalization of the splitting number s to uncountable cardinals. We prove that 𝔰(κ) > κ+ for a regular uncountable cardinal κ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption 𝔰(ℵω) > ℵω+1 has a considerable large cardinal strength as well.

Σ1(κ)-DEFINABLE SUBSETS OF H(κ+)

2017 ◽  
Vol 82 (3) ◽  
pp. 1106-1131 ◽  
Author(s):  
PHILIPP LÜCKE ◽  
RALF SCHINDLER ◽  
PHILIPP SCHLICHT
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinal ◽  
Woodin Cardinals ◽  
Perfect Set ◽  
Nonstationary Ideal ◽  
Definable Sets ◽  
Image Position ◽  
Stationary Subsets ◽  
Woodin Cardinal ◽  
Generic Ultrapowers

AbstractWe study Σ1(ω1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1(ω1)-definable, the set of all stationary subsets of ω1 is not Σ1(ω1)-definable and the complement of every Σ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω1)-definable well-ordering of H(ω2) and the existence of a Δ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1(ω1)-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1(ω1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κκ, in the case where κ itself has certain large cardinal properties.

scholarly journals HOD IN INNER MODELS WITH WOODIN CARDINALS

2021 ◽  
pp. 1-30
Author(s):  
SANDRA MÜLLER ◽  
GRIGOR SARGSYAN
Keyword(s):  
Woodin Cardinals ◽  
Inner Models

Woodin's axiom (*), bounded forcing axioms, and precipitous ideals on ω1

2012 ◽  
Vol 77 (2) ◽  
pp. 475-498 ◽  
Author(s):  
Benjamin Claverie ◽  
Ralf Schindler
Keyword(s):  
Model Operator ◽  
Forcing Axioms ◽  
Woodin Cardinals ◽  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
Reflection Principles ◽  
Precipitous Ideals ◽  
Forcing Axiom ◽  
The Way

AbstractIf the Bounded Proper Forcing Axiom BPFA holds, then Mouse Reflection holds at ℵ2 with respect to all mouse operators up to the level of Woodin cardinals in the next ZFC-model. This yields that if Woodin's ℙmax axiom (*) holds, then BPFA implies that V is closed under the “Woodin-in-the-next-ZFC-model” operator. We also discuss stronger Mouse Reflection principles which we show to follow from strengthenings of BPFA, and we discuss the theory BPFA plus “NSω1 is precipitous” and strengthenings thereof. Along the way, we answer a question of Baumgartner and Taylor, [2, Question 6.11].

Proper forcing and L(ℝ)

2001 ◽  
Vol 66 (2) ◽  
pp. 801-810 ◽  
Author(s):  
Itay Neeman ◽  
Jindřich Zapletal
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Consistency Strength ◽  
Woodin Cardinals ◽  
Proper Forcing

AbstractWe present two ways in which the model L(ℝ) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing: we show further that a set of ordinals in V cannot be added to L(ℝ) by small forcing. The large cardinal needed corresponds to the consistency strength of ADL(ℝ): roughly ω Woodin cardinals.

The well-foundedness of the Mitchell order

1993 ◽  
Vol 58 (3) ◽  
pp. 931-940 ◽  
Author(s):  
J. R. Steel
Keyword(s):  
Critical Point ◽  
Elementary Proof ◽  
Large Cardinal ◽  
Canonical Embedding ◽  
Inner Models ◽  
Normal Ultrafilter ◽  
Well Foundedness

Let E ⊲ F iff E and F are extenders and E ∈ Ult(V, F). Intuitively, E ⊲ F implies that E is weaker—embodies less reflection—than F. The relation ⊲ was first considered by W. Mitchell in [M74], where it arises naturally in connection with inner models and coherent sequences. Mitchell showed in [M74] that the restriction of ⊲ to normal ultrafilters is well-founded.The relation ⊲ is now known as the Mitchell order, although it is not actually an order. It is irreflexive, and its restriction to normal ultrafilters is transitive, but under mild large cardinal hypotheses, it is not transitive on all extenders. Here is a counterexample. Let κ be (λ + 2)-strong, where λ > κ and λ is measurable. Let E be an extender with critical point κ and let U be a normal ultrafilter with critical point λ such that U ∈ Ult(V, E). Let i: V → Ult(V, U) be the canonical embedding. Then i(E) ⊲ U and U ⊲ E, but by 3.11 of [MS2], it is not the case that i(E) ⊲ E. (The referee pointed out the following elementary proof of this fact. Notice that i ↾ Vλ+2 ∈ Ult(V, E) and X ∈ Ea ⇔ X ∈ i(E)i(a). Moreover, we may assume without loss of generality that = support(E). Thus, if i(E) ∈ Ult(V, E), then E ∈ Ult(V, E), a contradiction.)By going to much stronger extenders, one can show the Mitchell order is not well-founded. The following example is well known. Let j: V → M be elementary, with Vλ ⊆ M for λ = joω(crit(j)). (By Kunen, Vλ+1 ∉ M.) Let E0 be the (crit(j), λ) extender derived from j, and let En+1 = i(En), where i: V → Ult(V, En) is the canonical embedding. One can show inductively that En is an extender over V, and thereby, that En+1 ⊲ En for all n < ω. (There is a little work in showing that Ult(V, En+1) is well-founded.)

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