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 THE CONSISTENCY STRENGTH OF LONG PROJECTIVE DETERMINACY

2019 ◽  
Vol 85 (1) ◽  
pp. 338-366 ◽  
Author(s):  
JUAN P. AGUILERA ◽  
SANDRA MÜLLER
Keyword(s):  
Set Theory ◽  
Consistency Strength ◽  
Axiom Of Determinacy ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Image Position ◽  
Woodin Cardinal

AbstractWe determine the consistency strength of determinacy for projective games of length ω2. Our main theorem is that $\Pi _{n + 1}^1 $-determinacy for games of length ω2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn (A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $$A = R$$ and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω2 with payoff in $^R R\Pi _1^1 $ or with σ-projective payoff.

On an Ackermann-type set theory

1973 ◽  
Vol 38 (3) ◽  
pp. 410-412
Author(s):  
John Lake
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Large Cardinal ◽  
Predicate Calculus ◽  
First Order ◽  
Free Variables ◽  
Individual Constant ◽  
Image Position ◽  
Order Predicate Calculus ◽  
Universal Closure

Ackermann's set theory A* is usually formulated in the first order predicate calculus with identity, ∈ for membership and V, an individual constant, for the class of all sets. We use small Greek letters to represent formulae which do not contain V and large Greek letters to represent any formulae. The axioms of A* are the universal closures ofwhere all free variables are shown in A4 and z does not occur in the Θ of A2.A+ is a generalisation of A* which Reinhardt introduced in [3] as an attempt to provide an elaboration of Ackermann's idea of “sharply delimited” collections. The language of A+ is that of A*'s augmented by a new constant V′, and its axioms are A1–A3, A5, V ⊆ V′ and the universal closure ofwhere all free variables are shown.Using a schema of indescribability, Reinhardt states in [3] that if ZF + ‘there exists a measurable cardinal’ is consistent then so is A+, and using [4] this result can be improved to a weaker large cardinal axiom. It seemed plausible that A+ was stronger than ZF, but our main result, which is contained in Theorem 5, shows that if ZF is consistent then so is A+, giving an improvement on the above results.

A WELLORDER OF THE REALS WITH SATURATED

2019 ◽  
Vol 84 (4) ◽  
pp. 1466-1483
Author(s):  
SY-DAVID FRIEDMAN ◽  
STEFAN HOFFELNER
Keyword(s):  
Inner Model ◽  
Nonstationary Ideal ◽  
Image Position ◽  
Woodin Cardinal

AbstractWe show that, assuming the existence of the canonical inner model with one Woodin cardinal $M_1 $ , there is a model of $ZFC$ in which the nonstationary ideal on $\omega _1 $ is $\aleph _2 $-saturated and whose reals admit a ${\rm{\Sigma }}_4^1 $-wellorder.

Proper Forcing and Remarkable Cardinals

2000 ◽  
Vol 6 (2) ◽  
pp. 176-184 ◽  
Author(s):  
Ralf-Dieter Schindler
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinal ◽  
Exact Formulation ◽  
Axiom Of Determinacy ◽  
Slow Pace ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Proper Forcing ◽  
Remarkable Cardinals ◽  
The Universe

The present paper investigates the power of proper forcings to change the shape of the universe, in a certain well-defined respect. It turns out that the ranking among large cardinals can be used as a measure for that power. However, in order to establish the final result I had to isolate a new large cardinal concept, which I dubbed “remarkability.” Let us approach the exact formulation of the problem—and of its solution—at a slow pace.Breathtaking developments in the mid 1980s found one of its culminations in the theorem, due to Martin, Steel, and Woodin, that the existence of infinitely many Woodin cardinals with a measurable cardinal above them all implies that AD, the axiom of determinacy, holds in the least inner model containing all the reals, L(ℝ) (cf. [6[, p. 91). One of the nice things about AD is that the theory ZF + AD + V = L(ℝ) appears as a choiceless “completion” of ZF in that any interesting question (in particular, about sets of reals) seems to find an at least attractive answer in that theory (cf., for example, [5] Chap. 6). (Compare with ZF + V = L!) Beyond that, AD is very canonical as may be illustrated as follows.Let us say that L(ℝ) is absolute for set-sized forcings if for all posets P ∈ V, for all formulae ϕ, and for all ∈ ℝ do we have thatwhere is a name for the set of reals in the extension.

UNIVERSALLY BAIRE SETS AND GENERIC ABSOLUTENESS

2017 ◽  
Vol 82 (4) ◽  
pp. 1229-1251
Author(s):  
TREVOR M. WILSON
Keyword(s):  
Measurable Cardinal ◽  
Consistency Strength ◽  
Woodin Cardinals ◽  
Generic Absoluteness ◽  
Relative Consistency ◽  
Consistency Results ◽  
High Consistency ◽  
Image Position ◽  
General Method

AbstractWe prove several equivalences and relative consistency results regarding generic absoluteness beyond Woodin’s ${\left( {{\bf{\Sigma }}_1^2} \right)^{{\rm{u}}{{\rm{B}}_\lambda }}}$ generic absoluteness result for a limit of Woodin cardinals λ. In particular, we prove that two-step $\exists ^ℝ \left( {{\rm{\Pi }}_1^2 } \right)^{{\rm{uB}}_\lambda } $ generic absoluteness below a measurable limit of Woodin cardinals has high consistency strength and is equivalent, modulo small forcing, to the existence of trees for ${\left( {{\bf{\Pi }}_1^2} \right)^{{\rm{u}}{{\rm{B}}_\lambda }}}$ formulas. The construction of these trees uses a general method for building an absolute complement for a given tree T assuming many “failures of covering” for the models $L\left( {T,{V_\alpha }} \right)$ for α below a measurable cardinal.

VARSOVIAN MODELS I

2018 ◽  
Vol 83 (2) ◽  
pp. 496-528 ◽  
Author(s):  
GRIGOR SARGSYAN ◽  
RALF SCHINDLER
Keyword(s):  
Core Model ◽  
Generic Extension ◽  
Strong Cardinal ◽  
Small Core ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Specific Fragment ◽  
Image Position ◽  
Woodin Cardinal

AbstractLet Msw denote the least iterable inner model with a strong cardinal above a Woodin cardinal. By [11], Msw has a fully iterable core model, ${K^{{M_{{\rm{sw}}}}}}$, and Msw is thus the least iterable extender model which has an iterable core model with a Woodin cardinal. In V, ${K^{{M_{{\rm{sw}}}}}}$ is an iterate of Msw via its iteration strategy Σ.We here show that Msw has a bedrock which arises from ${K^{{M_{{\rm{sw}}}}}}$ by telling ${K^{{M_{{\rm{sw}}}}}}$ a specific fragment ${\rm{\bar{\Sigma }}}$ of its own iteration strategy, which in turn is a tail of Σ. Hence Msw is a generic extension of $L[{K^{{M_{{\rm{sw}}}}}},{\rm{\bar{\Sigma }}}]$, but the latter model is not a generic extension of any inner model properly contained in it.These results generalize to models of the form Ms (x) for a cone of reals x, where Ms (x) denotes the least iterable inner model with a strong cardinal containing x. In particular, the least iterable inner model with a strong cardinal above two (or seven, or boundedly many) Woodin cardinals has a 2-small core model K with a Woodin cardinal and its bedrock is again of the form $L[K,{\rm{\bar{\Sigma }}}]$.

AXIOM I0 AND HIGHER DEGREE THEORY

2015 ◽  
Vol 80 (3) ◽  
pp. 970-1021 ◽  
Author(s):  
XIANGHUI SHI
Keyword(s):  
Critical Point ◽  
Degree Theory ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Generic Absoluteness ◽  
Perfect Set ◽  
Singular Cardinals ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Measurable Cardinals

AbstractIn this paper, we analyze structures of Zermelo degrees via a list of four degree theoretic questions (see §2) in various fine structure extender models, or under large cardinal assumptions. In particular we give a detailed analysis of the structures of Zermelo degrees in the Mitchell model for ω many measurable cardinals. It turns out that there is a profound correlation between the complexity of the degree structures at countable cofinality singular cardinals and the large cardinal strength of the relevant cardinals. The analysis applies to general degree notions, Zermelo degree is merely the author’s choice for illustrating the idea.I0(λ) is the assertion that there is an elementary embedding j : L(Vλ+1) → L(Vλ+1) with critical point < λ. We show that under I0(λ), the structure of Zermelo degrees at λ is very complicated: it has incomparable degrees, is not dense, satisfies Posner–Robinson theorem etc. In addition, we show that I0 together with a mild condition on the critical point of the embedding implies that the degree determinacy for Zermelo degrees at λ is false in L(Vλ+1). The key tool in this paper is a generic absoluteness theorem in the theory of I0, from which we obtain an analogue of Perfect Set Theorem for “projective” subsets of Vλ+1, and the Posner–Robinson follows as a corollary. Perfect Set Theorem and Posner–Robinson provide evidences supporting the analogy between $$AD$$ over L(ℝ) and I0 over L(Vλ+1), while the failure of degree determinacy is one for disanalogy. Furthermore, we conjecture that the failure of degree determinacy for Zermelo degrees at any uncountable cardinal is a theorem of $$ZFC$$.

A universal extender model without large cardinals in V

2004 ◽  
Vol 69 (2) ◽  
pp. 371-386 ◽  
Author(s):  
William Mitchell ◽  
Ralf Schindler
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Inner Model ◽  
Woodin Cardinal

Abstract.We construct, assuming that there is no inner model with a Woodin cardinal but without any large cardinal assumption, a model Kc which is iterable for set length iterations, which is universal with respect to all weasels with which it can be compared, and (assuming GCH) is universal with respect to set sized premice.

THE SOLIDITY AND NONSOLIDITY OF INITIAL SEGMENTS OF THE CORE MODEL

2018 ◽  
Vol 83 (3) ◽  
pp. 920-938
Author(s):  
GUNTER FUCHS ◽  
RALF SCHINDLER
Keyword(s):  
Core Model ◽  
Solid Core ◽  
The Core ◽  
Inner Model ◽  
Cohen Real ◽  
Image Position ◽  
Woodin Cardinal

AbstractIt is shown that $K|{\omega _1}$ need not be solid in the sense previously introduced by the authors: it is consistent that there is no inner model with a Woodin cardinal yet there is an inner model W and a Cohen real x over W such that $K|{\omega _1}\,\, \in \,\,W[x] \setminus W$. However, if ${0^{\rm{\P}}}$ does not exist and $\kappa \ge {\omega _2}$ is a cardinal, then $K|\kappa$ is solid. We draw the conclusion that solidity is not forcing absolute in general, and that under the assumption of $\neg {0^{\rm{\P}}}$, the core model is contained in the solid core, previously introduced by the authors.It is also shown, assuming ${0^{\rm{\P}}}$ does not exist, that if there is a forcing that preserves ${\omega _1}$, forces that every real has a sharp, and increases $\delta _2^1$, then ${\omega _1}$ is measurable in K.

scholarly journals IDEAL PROJECTIONS AND FORCING PROJECTIONS

2014 ◽  
Vol 79 (4) ◽  
pp. 1247-1285 ◽  
Author(s):  
SEAN COX ◽  
MARTIN ZEMAN
Keyword(s):  
Set Theory ◽  
Negative Answer ◽  
Large Cardinals ◽  
Normal Ideal ◽  
List Type ◽  
Type Number ◽  
Inner Model ◽  
Image Position ◽  
Open Question ◽  
Woodin Cardinal

AbstractIt is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove:(1)If${\cal I}$is a normal ideal on$\omega _2 $which satisfiesstationary antichain catching, then there is an inner model with a Woodin cardinal;(2)For any$n \in \omega $, it is consistent relative to large cardinals that there is a normal ideal${\cal I}$on$\omega _n $which satisfiesprojective antichain catching, yet${\cal I}$is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7]).

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