scholarly journals On the Strength of PFA(ℵ2) in Conjunction with a Precipitous Ideal on ɷ1 and Namba-Like Forcings on Successors of Regular Cardinals

2018 ◽  
Vol 24 (4) ◽  
pp. 456-457
Author(s):  
Dominik Thomas Adolf
Keyword(s):  
Regular Cardinals ◽  
Precipitous Ideal

Two Applications of Inner Model Theory to the Study of Sets

1996 ◽  
Vol 2 (1) ◽  
pp. 94-107 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Los Angeles ◽  
Set Theory ◽  
Model Theory ◽  
Large Cardinals ◽  
Descriptive Set Theory ◽  
Inner Model Theory ◽  
Regular Cardinals ◽  
Inner Model ◽  
Definable Sets ◽  
Descriptive Set

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL(ℝ) are measurable. John Steel's analysis also settled a number of structural questions regarding HODL(ℝ), such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals.

Increasing u2 by a stationary set preserving forcing

2009 ◽  
Vol 74 (1) ◽  
pp. 187-200
Author(s):  
Benjamin Claverie ◽  
Ralf Schindler
Keyword(s):  
Regular Cardinal ◽  
Nonstationary Ideal ◽  
Stationary Set ◽  
Precipitous Ideal ◽  
Countable Structure

AbstractWe show that if I is a precipitous ideal on ω1 and if θ > ω1 is a regular cardinal, then there is a forcing ℙ = ℙ(I, θ) which preserves the stationarity of all I-positive sets such that in Vℙ, ⟨Hθ; ∈, I⟩ is a generic iterate of a countable structure ⟨M; ∈, Ī⟩. This shows that if the nonstationary ideal on ω1 is precipitous and exists, then there is a stationary set preserving forcing which increases . Moreover, if Bounded Martin's Maximum holds and the nonstationary ideal on ω1 is precipitous, then .

Making the hyperreal line both saturated and complete

1991 ◽  
Vol 56 (3) ◽  
pp. 1016-1025 ◽  
Author(s):  
H. Jerome Keisler ◽  
James H. Schmerl
Keyword(s):  
Set Theory ◽  
Nonempty Intersection ◽  
Finite Intersection ◽  
Second Order Arithmetic ◽  
Ordered Field ◽  
Finite Intersection Property ◽  
Regular Cardinals ◽  
Hyperreal Numbers ◽  
Order Arithmetic ◽  
Internal Sets

AbstractIn a nonstandard universe, the κ-saturation property states that any family of fewer than κ internal sets with the finite intersection property has a nonempty intersection. An ordered field F is said to have the λ-Bolzano-Weierstrass property iff F has cofinality λ and every bounded λ-sequence in F has a convergent λ-subsequence. We show that if κ < λ are uncountable regular cardinals and βα < λ whenever α < κ and β < λ then there is a κ-saturated nonstandard universe in which the hyperreal numbers have the λ-Bolzano-Weierstrass property. The result also applies to certain fragments of set theory and second order arithmetic.

scholarly journals Contributions to the Theory of Large Cardinals through the Method of Forcing

2021 ◽  
Vol 27 (2) ◽  
pp. 221-222
Author(s):  
Alejandro Poveda
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Supercompact Cardinal ◽  
Singular Cardinal ◽  
Strong Limit ◽  
Tree Property ◽  
Prikry Forcing ◽  
Regular Cardinals ◽  
Singular Cardinals ◽  
Vopěnka’S Principle

AbstractThe dissertation under comment is a contribution to the area of Set Theory concerned with the interactions between the method of Forcing and the so-called Large Cardinal axioms.The dissertation is divided into two thematic blocks. In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopěnka’s Principle (Part I). In turn, Block II is devoted to the investigation of some problems arising from Singular Cardinal Combinatorics (Part II and Part III).We commence Part I by investigating the Identity Crisis phenomenon in the region comprised between the first supercompact cardinal and Vopěnka’s Principle. As a result, we generalize Magidor’s classical theorems [2] to this higher region of the large-cardinal hierarchy. Also, our analysis allows to settle all the questions that were left open in [1]. Finally, we conclude Part I by presenting a general theory of preservation of $C^{(n)}$ -extendible cardinals under class forcing iterations. From this analysis we derive several applications. For instance, our arguments are used to show that an extendible cardinal is consistent with “ $(\lambda ^{+\omega })^{\mathrm {HOD}}<\lambda ^+$ , for every regular cardinal $\lambda $ .” In particular, if Woodin’s HOD Conjecture holds, and therefore it is provable in ZFC + “There exists an extendible cardinal” that above the first extendible cardinal every singular cardinal $\lambda $ is singular in HOD and $(\lambda ^+)^{\textrm {{HOD}}}=\lambda ^+$ , there may still be no agreement at all between V and HOD about successors of regular cardinals.In Part II and Part III we analyse the relationship between the Singular Cardinal Hypothesis (SCH) with other relevant combinatorial principles at the level of successors of singular cardinals. Two of these are the Tree Property and the Reflection of Stationary sets, which are central in Infinite Combinatorics.Specifically, Part II is devoted to prove the consistency of the Tree Property at both $\kappa ^+$ and $\kappa ^{++}$ , whenever $\kappa $ is a strong limit singular cardinal witnessing an arbitrary failure of the SCH. This generalizes the main result of [3] in two senses: it allows arbitrary cofinalities for $\kappa $ and arbitrary failures for the SCH.In the last part of the dissertation (Part III) we introduce the notion of $\Sigma $ -Prikry forcing. This new concept allows an abstract and uniform approach to the theory of Prikry-type forcings and encompasses several classical examples of Prikry-type forcing notions, such as the classical Prikry forcing, the Gitik-Sharon poset, or the Extender Based Prikry forcing, among many others.Our motivation in this part of the dissertation is to prove an iteration theorem at the level of the successor of a singular cardinal. Specifically, we aim for a theorem asserting that every $\kappa ^{++}$ -length iteration with support of size $\leq \kappa $ has the $\kappa ^{++}$ -cc, provided the iterates belong to a relevant class of $\kappa ^{++}$ -cc forcings. While there are a myriad of works on this vein for regular cardinals, this contrasts with the dearth of investigations in the parallel context of singular cardinals. Our main contribution is the proof that such a result is available whenever the class of forcings under consideration is the family of $\Sigma $ -Prikry forcings. Finally, and as an application, we prove that it is consistent—modulo large cardinals—the existence of a strong limit cardinal $\kappa $ with countable cofinality such that $\mathrm {SCH}_\kappa $ fails and every finite family of stationary subsets of $\kappa ^+$ reflects simultaneously.

scholarly journals SQUARES, ASCENT PATHS, AND CHAIN CONDITIONS

2018 ◽  
Vol 83 (04) ◽  
pp. 1512-1538 ◽  
Author(s):  
CHRIS LAMBIE-HANSON ◽  
PHILIPP LÜCKE
Keyword(s):  
Partial Order ◽  
Set Theory ◽  
Consistency Strength ◽  
Chain Conditions ◽  
Regular Cardinals ◽  
Aronszajn Tree ◽  
Square Principles ◽  
Image Position ◽  
Aronszajn Trees ◽  
Consistency Strengths

AbstractWith the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević’s principle $\square \left( {\kappa ,\lambda } \right)$ implies an indexed version of $\square \left( {\kappa ,\lambda } \right)$, we show that for all infinite, regular cardinals $\lambda < \kappa$, the principle $\square \left( \kappa \right)$ implies the existence of a κ-Aronszajn tree containing a λ-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which ${\aleph _2}$-Aronszajn trees exist and all such trees contain ${\aleph _0}$-ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of $\square \left( \kappa \right)$.

scholarly journals TOWERS IN FILTERS, CARDINAL INVARIANTS, AND LUZIN TYPE FAMILIES

2018 ◽  
Vol 83 (3) ◽  
pp. 1013-1062 ◽  
Author(s):  
JÖRG BRENDLE ◽  
BARNABÁS FARKAS ◽  
JONATHAN VERNER
Keyword(s):  
List Type ◽  
Type Number ◽  
Cardinal Invariants ◽  
Regular Cardinals ◽  
Image Position

AbstractWe investigate which filters onωcan contain towers, that is, a modulo finite descending sequence without any pseudointersection (in${[\omega ]^\omega }$). We prove the following results:(1)Many classical examples of nice tall filters contain no towers (in ZFC).(2)It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well).(3)It is consistent that all towers generate nonmeager filters (this answers a question of P. Borodulin-Nadzieja and D. Chodounský), in particular (consistently) Borel filters do not contain towers.(4)The statement “Every ultrafilter contains towers.” is independent of ZFC (this improves an older result of K. Kunen, J. van Mill, and C. F. Mills).Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters (${\rm{ad}}{{\rm{d}}^{\rm{*}}}\left( {\cal F} \right)$,${\rm{co}}{{\rm{f}}^{\rm{*}}}\left( {\cal F} \right)$,${\rm{no}}{{\rm{n}}^{\rm{*}}}\left( {\cal F} \right)$, and${\rm{co}}{{\rm{v}}^{\rm{*}}}\left( {\cal F} \right)$), and the existence of Luzin type families (of size$\ge {\omega _2}$), that is, if${\cal F}$is a filter then${\cal X} \subseteq {[\omega ]^\omega }$is an${\cal F}$-Luzin family if$\left\{ {X \in {\cal X}:|X \setminus F| = \omega } \right\}$is countable for every$F \in {\cal F}$.

scholarly journals Incompactness in regular cardinals.

1985 ◽  
Vol 26 (3) ◽  
pp. 195-228 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Regular Cardinals

Precipitous ideals

1980 ◽  
Vol 45 (1) ◽  
pp. 1-8 ◽  
Author(s):  
T. Jech ◽  
M. Magidor ◽  
W. Mitchell ◽  
K. Prikry
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Ground Model ◽  
Generic Set ◽  
Precipitous Ideal ◽  
Precipitous Ideals ◽  
The Ideal

The properties of small cardinals such as ℵ1 tend to be much more complex than those of large cardinals, so that properties of ℵ1 may often be better understood by viewing them as large cardinal properties. In this paper we show that the existence of a precipitous ideal on ℵ1 is essentially the same as measurability.If I is an ideal on P(κ) then R(I) is the notion of forcing whose conditions are sets x ∈ P(κ)/I, with x ≤ x′ if x ⊆ x′. Thus a set D R(I)-generic over the ground model V is an ultrafilter on P(κ) ⋂ V extending the filter dual to I. The ideal I is said to be precipitous if κ ⊨R(I)(Vκ/D is wellfounded).One example of a precipitous ideal is the ideal dual to a κ-complete ultrafilter U on κ. This example is trivial since the generic ultrafilter D is equal to U and is already in the ground model. A generic set may be viewed as one that can be worked with in the ground model even though it is not actually in the ground model, so we might expect that cardinals such as ℵ1 that cannot be measurable still might have precipitous ideals, and such ideals might correspond closely to measures.

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