Contributions to the theory of Large Cardinals through the

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Contributions to the theory of Large Cardinals through the

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Contributions to the theory of Large Cardinals through the method of Forcing
Alejandro Poveda Ruzafa
Aquesta tesi doctoral està subjecta a la llicència Reconeixement 4.0. Espanya de Creative Commons. Esta tesis doctoral está sujeta a la licencia Reconocimiento 4.0. España de Creative Commons. This doctoral thesis is licensed under the Creative Commons Attribution 4.0. Spain License.

Universitat de Barcelona
Doctoral Thesis
Contributions to the theory of Large Cardinals through the
method of Forcing

Author: Alejandro Poveda Ruzafa

Supervisor: Dr. Joan Bagaria
Pigrau

A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy
within the program in Mathematics and Computer Science

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“Els matemàtics són una mena de poetes fraudulents que, de fet, intenten l’única poesia possible.”
Joan Fuster

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UNIVERSITAT DE BARCELONA
Abstract
Facultat de Matemàtiques i Informàtica
“Contributions to the theory of Large Cardinals through the method of Forcing”
by Alejandro Poveda Ruzafa
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 Vopěnka’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 Vopěnka’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 Σ-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 + ¬SCHκ + Refl(<ω, κ+), for a strong limit singular cardinal κ with cof(κ) = ω.

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UNIVERSITAT DE BARCELONA
Resum
Facultat de Matemàtiques i Informàtica
“Contributions to the theory of Large Cardinals through the method of Forcing”
per Alejandro Poveda Ruzafa
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 Vopěnka (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 Vˇopenka. 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 Σ-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 + ¬SCHκ + Refl(<ω, κ+), per a κ un cardinal fortament límit i singular amb cof(κ) = ω.

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Contents

Agraïments

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1 Preliminaries

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1.1 Measures and Extenders . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Measures and large cardinals . . . . . . . . . . . . . . . 1

1.1.2 Measures and elementary embeddings . . . . . . . . . . 6

1.1.3 Extenders . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 The C(n)–large cardinals . . . . . . . . . . . . . . . . . . . . . 11

1.3 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.1 Some generalities on Forcing . . . . . . . . . . . . . . . 15

1.3.2 More about Forcing . . . . . . . . . . . . . . . . . . . . 18

1.3.3 Forcing and elementary embeddings . . . . . . . . . . . 21

1.4 -principles and scales . . . . . . . . . . . . . . . . . . . . . . 22

I On the large-cardinal hierarchy between super-

compactness and Vopěnka’s Principle

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2 Woodin’s Extender Embedding Axiom and a question of

Bagaria

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3 Distinguishing C(1)-supercompactness and supercompactness 36
3.1 On another question of Bagaria . . . . . . . . . . . . . . . . . 36 3.1.1 Radin Forcing and ω∗-measurable cardinals . . . . . . . 37
3.2 More on C(1)-supercompactness . . . . . . . . . . . . . . . . . 45

4 The identity crisis phenomenon at C(n)–supercompact cardi-

nals

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4.1 Some preliminary comments . . . . . . . . . . . . . . . . . . . 51

4.2 Magidor Products of Prikry forcing . . . . . . . . . . . . . . . 52

4.3 Preserving C(n)–supercompact cardinals . . . . . . . . . . . . 57

4.4 The Ultimate Identity crisis . . . . . . . . . . . . . . . . . . . 61

4.5 C(n)–supercompactness and Forcing: a new world to explore . 62

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4.5.1 The lifting strategy . . . . . . . . . . . . . . . . . . 63 4.5.2 The extender strategy . . . . . . . . . . . . . . . . 63

5 C(n)–extendibility, Vopěnka’s principle and forcing

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5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 A Magidor-like characterization of C(n)–extendibility . . . . . 68

5.3 Some reflection properties for class forcing iterations . . . . . . 71

5.4 P-Σn-supercompactness . . . . . . . . . . . . . . . . . . . . . . 77

5.5 Suitable iterations . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.6.1 Vopěnka’s principle and suitable iterations . . . . . . . 82

5.6.2 Forcing the GCH and related cardinal configurations . 84

5.6.3 On diamonds . . . . . . . . . . . . . . . . . . . . . . . 87

5.6.4 On weak square sequences . . . . . . . . . . . . . . . . 88

5.6.5 A remark on Woodin’s HOD-Conjecture . . . . . . . . 89

5.7 Non homogeneous suitable iterations and V = HOD . . . . . . 91

5.7.1 C(n)–extendible cardinals, V = HOD and the Ground

Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6 A successor cardinal can be C(n)–extendible in HOD

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II The tree property at the two first successors of a

singular cardinal

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7 Tree Property at Double Successors

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7.1 An introduction to Magidor forcing . . . . . . . . . . . . . . . 109

7.2 The main forcing construction . . . . . . . . . . . . . . . . . . 119

7.3 TP(κ++) holds . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.4 Forcing arbitrary failures of the SCHκ . . . . . . . . . . . . . . 137

8 Tree Property at First and Double Successors

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8.1 Sinapova forcing . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.2 Geometric criterion for genericity . . . . . . . . . . . . . . . . 144

8.2.1 One step extensions and pruned conditions . . . . . . . 145

8.2.2 The Strong Prikry Property . . . . . . . . . . . . . . . 147

8.2.3 The proof of the criterion . . . . . . . . . . . . . . . . 149

8.3 Sinapova sequences and iterated ultrapowers . . . . . . . . . . 152

8.4 The main forcing construction . . . . . . . . . . . . . . . . . . 154

8.5 TP(κ++) holds . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.6 TP(κ+) holds . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

9 Magidor-Shelah Theorem for δ-strong compactness

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III Σ-Prikry forcings and their iterations

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10 The Σ-Prikry framework

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10.1 The axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

10.2 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . 191

10.2.1 Prikry forcing . . . . . . . . . . . . . . . . . . . . . . . 192

10.2.2 Supercompact Prikry forcing . . . . . . . . . . . . . . . 193

10.2.3 Gitik-Sharon forcing . . . . . . . . . . . . . . . . . . . 194

10.2.4 AIM forcing . . . . . . . . . . . . . . . . . . . . . . . . 197

10.2.5 Extender-based Prikry Forcing . . . . . . . . . . . . . . 204

10.2.6 Lottery sum of Σ-Prikry forcings . . . . . . . . . . . . 215

11 Forking projections

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12 Σ-Prikry forcings and simultaneous stationary reflection 223 12.1 Stationary reflection and the SCH . . . . . . . . . . . . . . . . 223 12.2 Simultaneous stationary reflection and Σ-Prikry forcings . . . 224 12.3 Towards a model of Refl(<ω, κ+) . . . . . . . . . . . . . . . . 229

13 Killing one non-reflecting stationary set

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13.1 The poset A(P, T˙ ) . . . . . . . . . . . . . . . . . . . . . . . . . 231

13.2 A(P, T˙ ) and forking projections . . . . . . . . . . . . . . . . . 237

13.3 A(P, T˙ ) is a Σ-Prikry forcing . . . . . . . . . . . . . . . . . . . 241

13.4 The last word about A(P, T˙ ) . . . . . . . . . . . . . . . . . . . 245

14 Iterations of Σ-Prikry forcings

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14.0.1 Defining the iteration . . . . . . . . . . . . . . . . . . . 248

14.0.2 Verification . . . . . . . . . . . . . . . . . . . . . . . . 249

15 Simultaneous reflection and failure of the SCH

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Bibliography

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TheoryCardinalsMethodIterationsTeoria