672 research outputs found
The tree property at first and double successors of singular cardinals with an arbitrary gap
Let and be a supercompact cardinal with
. Assume that there is an increasing and continuous sequence of
cardinals with and
such that, for each , is supercompact. Besides,
assume that is a weakly compact cardinal with
. Let be a cardinal with
. Assuming the , we
construct a generic extension where is strong limit,
, and both
and hold. Further, in this model there is a very
good and a bad scale at . This generalizes the main results of [Sin16a]
and [FHS18]
Robust reflection principles
A cardinal satisfies a property P robustly if, whenever
is a forcing poset and ,
satisfies P in . We study the extent to which certain
reflection properties of large cardinals can be satisfied robustly by small
cardinals. We focus in particular on stationary reflection and the tree
property, both of which can consistently hold but fail to be robust at small
cardinals. We introduce natural strengthenings of these principles which are
always robust and which hold at sufficiently large cardinals, consider the
extent to which these strengthenings are in fact stronger than the original
principles, and investigate the possibility of these strengthenings holding at
small cardinals, particularly at successors of singular cardinals.Comment: 20 page
Partition properties for simply definable colourings
We study partition properties for uncountable regular cardinals that arise by
restricting partition properties defining large cardinal notions to classes of
simply definable colourings. We show that both large cardinal assumptions and
forcing axioms imply that there is a homogeneous closed unbounded subset of
for every colouring of the finite sets of countable ordinals that is
definable by a -formula that only uses the cardinal and
real numbers as parameters. Moreover, it is shown that certain large cardinal
properties cause analogous partition properties to hold at the given large
cardinal and these implications yield natural examples of inaccessible
cardinals that possess strong partition properties for -definable
colourings and are not weakly compact. In contrast, we show that
-definability behaves fundamentally different at by
showing that various large cardinal assumptions and \emph{Martin's Maximum} are
compatible with the existence of a colouring of pairs of elements of
that is definable by a -formula with parameter and has no
uncountable homogeneous set. Our results will also allow us to derive tight
bounds for the consistency strengths of various partition properties for
definable colourings. Finally, we use the developed theory to study the
question whether certain homeomorphisms that witness failures of weak
compactness at small cardinals can be simply definable.Comment: 28 page
The maximality of the core model
If T is an iteration tree on K and F is a countably certified extender that
coheres with the final model of T, then F is on the extender sequence of the
final model of T. Several applications of maximality are proved, including:
o K computes successors of weakly compact cardinals correctly.
o K^c is an iterate of K.
o (with Mitchell) If alpha is a cardinal > aleph_1, then K-restriction-alpha
is universal for mice of height alpha.
Other results in this paper, when combined with work of Woodin, imply:
o If square-kappa-finite fails and kappa is a singular, strong limit
cardinal, then Inductive Determinacy holds.
o If square-kappa-finite fails and kappa is a weakly compact cardinal, then
L(R)-determinacy holds
The tree property at all regular even cardinals
Assuming the existence of a strong cardinal and a measurable cardinal above
it, we construct a model of in which for every singular cardinal
, is strong limit, and the tree
property at holds. This answers a question of Friedman, Honzik
and Stejskalova [8]. We also produce, relative to the existence of a strong
cardinal and two measurable cardinals above it, a model of in which the
tree property holds at all regular even cardinals. The result answers questions
of Friedman-Halilovic [5] and Friedman-Honzik [6].Comment: Comments are welcom
Destructibility of the tree property at
We construct a model in which the tree property holds in and it is destructible under . On the other
hand we discuss some cases in which the tree property is indestructible under
small or closed forcings
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
The tree property at successors of singular cardinals
Assuming some large cardinals, a model of ZFC is obtained in which
aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is
a singular limit of strongly compact cardinals, then lambda^+ carries no
Aronszajn trees
The Jensen covering property
An optimal extension of the Jensen covering lemma, within the limits imposed
by Prikry forcing, is proved. If L[E] is an "iterable" weasel with no
measurable cardinals, then either L[E] has "indiscernibles", or every
uncountable set of ordinals is contained in a set in L[E] of the same
cardinality. (The terms "iterable" and "indiscernibles" are made precise in the
paper.) Most importantly, there is no hypothesis explicitly limiting the large
cardinals which are consistent in L[E]
Contributions to the theory of Large Cardinals through the method of Forcing
[eng] The present dissertation is a contribution to the field of Mathematical Logic and, more particularly, to the subfield of Set Theory. Within Set theory, we are mainly concerned with the interactions between the largecardinal axioms and the method of Forcing. This is the line of research with a deeper impact in the subsequent configuration of modern Mathematics. This area has found many central applications in Topology [ST71][Tod89], Algebra [She74][MS94][DG85][Dug85], Analysis [Sol70] or Category Theory [AR94][Bag+15], among others. The dissertation is divided in two thematic blocks: In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopenka’s Principle (Part I). In Block II we make a contribution to Singular Cardinal Combinatorics (Part II and Part III). Specifically, in Part I we investigate the Identity Crisis phenomenon in the region comprised between the first supercompact cardinal and Vopenka’s Principle. As a result, we settle all the questions that were left open in [Bag12, §5]. Afterwards, we present a general theory of preservation of C(n)– extendible cardinals under class forcing iterations from which we derive many applications. In Part II and Part III we analyse the relationship between the Singular Cardinal Hypothesis (SCH) and other combinatorial principles, such as the tree property or the reflection of stationary sets. In Part II we generalize the main theorems of [FHS18] and [Sin16] and manage to weaken the largecardinal hypotheses necessary for Magidor-Shelah’s theorem [MS96]. Finally, in Part III we introduce the concept of _-Prikry forcing as a generalization of the classical notion of Prikry-type forcing. Subsequently we devise an abstract iteration scheme for this family of posets and, as an application, we prove the consistency of ZFC + ¬SCH_ + Refl([cat] La present tesi és una contribució a l’estudi de la Lògica Matemàtica i més particularment a la Teoria de Conjunts. Dins de la Teoria de Conjunts, la nostra àrea de recerca s’emmarca dins l’estudi de les interaccions entre els Axiomes de Grans Cardinals i el mètode de Forcing. Aquestes dues eines han tigut un impacte molt profund en la configuració de la matemàtica contemporànea com a conseqüència de la resolució de qüestions centrals en Topologia [ST71][Tod89], Àlgebra [She74][MS94][DG85][Dug85], Anàlisi Matemàtica [Sol70] o Teoria de Categories [AR94][Bag+15], entre d’altres. La tesi s’articula entorn a dos blocs temàtics. Al Bloc I analitzem la jerarquia de Grans Cardinals compresa entre el primer cardinal supercompacte i el Principi de Vopenka (Part I), mentre que al Bloc II estudiem alguns problemes de la Combinatòria Cardinal Singular (Part II i Part III). Més precisament, a la Part I investiguem el fenòmen de Crisi d’Identitat en la regió compresa entre el primer cardinal supercompacte i el Principi de Vopenka. Com a conseqüència d’aquesta anàlisi resolem totes les preguntes obertes de [Bag12, §5]. Posteriorment presentem una teoria general de preservació de cardinals C(n)–extensibles sota iteracions de longitud ORD, de la qual en derivem nombroses aplicacions. A la Part II i Part III analitzem la relació entre la Hipòtesi dels Cardinals Singulars (SCH) i altres principis combinatoris, tals com la Propietat de l’Arbre o la reflexió de conjunts estacionaris. A la Part II obtenim sengles generalitzacions dels teoremes principals de [FHS18] i [Sin16] i afeblim les hipòtesis necessàries perquè el teorema de Magidor-Shelah [MS96] siga cert. Finalment, a la Part III, introduïm el concepte de forcing _-Prikry com a generalització de la noció clàssica de forcing del tipus Prikry. Posteriorment dissenyem un esquema d’iteracions abstracte per aquesta família de forcings i, com a aplicació, derivem la consistència de ZFC + ¬SCH_ + Refl
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