scholarly journals Weak Vopěnka's Principle does not imply Vopěnka's Principle

2020 ◽  
Vol 363 ◽  
pp. 106986
Author(s):  
Trevor M. Wilson
Keyword(s):  
Vopěnka’S Principle

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.

IDENTITY CRISIS BETWEEN SUPERCOMPACTNESS AND VǑPENKA’S PRINCIPLE

2020 ◽  
pp. 1-32
Author(s):  
YAIR HAYUT ◽  
MENACHEM MAGIDOR ◽  
ALEJANDRO POVEDA
Keyword(s):  
Identity Crisis ◽  
Vopěnka’S Principle

Vopěnka's principle and compact logics

1985 ◽  
Vol 50 (1) ◽  
pp. 42-48 ◽  
Author(s):  
J. A. Makowsky
Keyword(s):  
Finitely Generated ◽  
Compact Cardinal ◽  
Hanf Number ◽  
Vopěnka’S Principle

AbstractWe study the effects of Vopěnka's principle on properties of model theoretic logics. We show that Vopěnka's principle is equivalent to the assumption that every finitely generated logic has a compact cardinal. We show also that it is equivalent to the assumption that every such logic has a global Hanf number.

The large cardinal strength of weak Vopenka’s principle

2021 ◽  
pp. 2150024
Author(s):  
Trevor M. Wilson
Keyword(s):  
Lower Bound ◽  
Large Cardinal ◽  
Proper Class ◽  
Strong Cardinal ◽  
Elementary Embeddings ◽  
Vopěnka’S Principle ◽  
Measurable Cardinals

We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class [Formula: see text] there is a [Formula: see text]-strong cardinal. Weak Vopěnka’s Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing [Formula: see text]-strongness of some cardinal.

scholarly journals Indestructibility of Vopěnka’s Principle

2011 ◽  
Vol 50 (5-6) ◽  
pp. 515-529 ◽  
Author(s):  
Andrew D. Brooke-Taylor
Keyword(s):  
Vopěnka’S Principle

Consequences of Vopěnka’s Principle over weak set theories

2016 ◽  
Vol 235 (2) ◽  
pp. 127-152
Author(s):  
Athanassios Tzouvaras
Keyword(s):  
Vopěnka’S Principle
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