Universal graphs at the successor of a singular cardinal

Džamonja, Mirna; Shelah, Saharon

doi:10.2178/jsl/1052669056

Cited by 28 publications

(52 citation statements)

References 18 publications

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“…Work by Džamonja and Shelah in [11] gives, modulo a supercompact cardinal, the consistency of a forcing axiom at a supercompact cardinal that will be made to have cofinality ω by a certain Prikry extension. The axiom has a different form than the ones that we have seen so far, as it applies in the universe before we do the Prikry extension, rather than the final universe.…”

Section: Away From Propernessmentioning

confidence: 99%

Forcing Axioms, Finite Conditions and Some More

Džamonja

2013

Logic and Its Applications

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Abstract. We survey some classical and some recent results in the theory of forcing axioms, aiming to present recent breakthroughs and interest the reader in further developing the theory. The article is written for an audience of logicians and mathematicians not necessarily familiar with set theory. IntroductionWe shall work within the axioms of the Zermelo-Fraenkel set theory with Choice (ZFC). These axioms were introduced basically starting from 1908 and improving to a final version in the 1920s as an attempt to axiomatize the foundations of mathematics. There have been other such attempts at about the same time and later, but it is fair to say that for the purposes of much of modern mathematics the axioms of ZFC represent the accepted foundation (see [13] for a detailed discussion of foundational issues in set theory). Gödel's Incompleteness theorems [16] prove that for any consistent theory T which implies the Peano Axioms and whose axioms are presentable as a recursively enumerable set of sentences, so for any reasonable theory one would say, there is a sentence ϕ in the language of T such that T does not prove or disprove ϕ. In some sense the discussion of which axioms to use is made less interesting by these theorems, which can be interpreted as saying that a perfect choice of axioms does not exists. We therefore do like the most, we concentrate on the axioms that correctly model most of mathematics, and for the rest, we try to understand the limits and how we can improve them. For us ZFC is a basis for a foundation which in some circumstances can be extended to a larger set of axioms which provide an insight into various parts of mathematics. In here we concentrate on the forcing axioms (and their negations).

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Section: Away From Propernessmentioning

confidence: 99%

Forcing Axioms, Finite Conditions and Some More

Džamonja

2013

Logic and Its Applications

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“…We provide here a complete proof using a different modification of [6] and further study the values of other natural generalizations of classical cardinal characteristics in our model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.…”

Section: Introductionmentioning

confidence: 99%

“…[10] provides a sketch of how to obtain such a model by modifying the construction in [6]. We provide here a complete proof using a different modification of [6] and further study the values of other natural generalizations of classical cardinal characteristics in our model.…”

Section: Introductionmentioning

confidence: 99%

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Cardinal characteristics at κ in a small u(κ) model

Brooke-Taylor

Fischer²,

Friedman³

et al. 2017

Annals of Pure and Applied Logic

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We provide a model where u(κ) = κ + < 2 κ for a supercompact cardinal κ.[10] provides a sketch of how to obtain such a model by modifying the construction in [6]. We provide here a complete proof using a different modification of [6] and further study the values of other natural generalizations of classical cardinal characteristics in our model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.

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“…For example, in a long series of papers (see e.g. [21,18,15,5,25] and the references contained therein), it was determined for various cardinals κ whether there is a universal graph of size κ (i.e. a graph such that all other graphs of size κ embed into it) and, in the negative case, E-mail address: luca.motto.ros@math.uni-freiburg.de.…”

Section: Introductionmentioning

confidence: 99%

The descriptive set-theoretical complexity of the embeddability relation on models of large size

Ros

2013

Annals of Pure and Applied Logic

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We show that if κ is a weakly compact cardinal then the embeddability relation on (generalized) trees of size κ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space κ 2 there is an L κ + κ -sentence ϕ such that the embeddability relation on its models of size κ, which are all trees, is Borel bi-reducible (and, in fact, classwise Borel isomorphic) to R. In particular, this implies that the relation of embeddability on trees of size κ is complete for analytic quasi-orders on κ 2. These facts generalize analogous results for κ = ω obtained in Louveau and Rosendal (2005) [17] and Friedman and Motto Ros (2011) [6], and it also partially extends a result from Baumgartner (1976) [3] concerning the structure of the embeddability relation on linear orders of size κ.

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Universal graphs at the successor of a singular cardinal

Cited by 28 publications

References 18 publications

Forcing Axioms, Finite Conditions and Some More

Forcing Axioms, Finite Conditions and Some More

Cardinal characteristics at κ in a small u(κ) model

The descriptive set-theoretical complexity of the embeddability relation on models of large size

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