13 research outputs found
Weak Indestructibility and Reflection
This work is a part of my upcoming thesis [7]. We establish an
equiconsistency between (1) weak indestructibility for all -degrees
of strength for cardinals in the presence of a proper class of strong
cardinals, and (2) a proper class of cardinals that are strong reflecting
strongs. We in fact get weak indestructibility for degrees of strength far
beyond , well beyond the next inaccessible limit of measurables (of
the ground model). One direction is proven using forcing and the other using
core model techniques from inner model theory. Additionally, connections
between weak indestructibility and the reflection properties associated with
Woodin cardinals are discussed.Comment: 28 page
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
Force to change large cardinal strength
This dissertation includes many theorems which show how to change large
cardinal properties with forcing. I consider in detail the degrees of
inaccessible cardinals (an analogue of the classical degrees of Mahlo
cardinals) and provide new large cardinal definitions for degrees of
inaccessible cardinals extending the hyper-inaccessible hierarchy. I showed
that for every cardinal , and ordinal , if is
-inaccerssible, then there is a forcing that
which preserves that -inaccessible but destorys that is
-inaccessible. I also consider Mahlo cardinals and degrees of Mahlo
cardinals. I showed that for every cardinal , and ordinal ,
there is a notion of forcing such that is still
-Mahlo in the extension, but is no longer -Mahlo.
I also show that a cardinal which is Mahlo in the ground model can
have every possible inaccessible degree in the forcing extension, but no longer
be Mahlo there. The thesis includes a collection of results which give forcing
notions which change large cardinal strength from weakly compact to weakly
measurable, including some earlier work by others that fit this theme. I
consider in detail measurable cardinals and Mitchell rank. I show how to change
a class of measurable cardinals by forcing to an extension where all measurable
cardinals above some fixed ordinal have Mitchell rank below
Finally, I consider supercompact cardinals, and a few theorems about strongly
compact cardinals. Here, I show how to change the Mitchell rank for
supercompactness for a class of cardinals
Generic Large Cardinals and Systems of Filters
We introduce the notion of -system of filters, generalizing the
standard definitions of both extenders and towers of normal ideals. This
provides a framework to develop the theory of extenders and towers in a more
general and concise way. In this framework we investigate the topic of
definability of generic large cardinals properties.Comment: 36 page
Capturing sets of ordinals by normal ultrapowers
We investigate the extent to which ultrapowers by normal measures on
can be correct about powersets for . We
consider two versions of this questions, the capturing property
and the local capturing property
. holds if there is
an ultrapower by a normal measure on which correctly computes
. is a weakening of
which holds if every subset of is
contained in some ultrapower by a normal measure on . After examining
the basic properties of these two notions, we identify the exact consistency
strength of . Building on results of Cummings,
who determined the exact consistency strength of
, and using a forcing due to Apter and Shelah, we
show that can hold at the least measurable
cardinal.Comment: 20 page
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
I0 and rank-into-rank axioms
This is a survey about I0 and rank-into-rank axioms, with some previously unpublished proofs