Creature forcing and five cardinal characteristics in Cichoń’s diagram

Fischer, Arthur; Goldstern, Martin; Kellner, Jakob; Shelah, Saharon

doi:10.1007/s00153-017-0553-8

Cited by 14 publications

(55 citation statements)

References 17 publications

(20 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…). Quite recently, A. Fischer, Goldstern, Kellner and Shelah used this technique to prove that 5 cardinal invariants on the right of Cichoń's diagram are pairwise different. Such method would also work to solve the dual of Question , that is, Question Is there a model of

add false(I_{f} false) < non false(I_{f} false) < cov false(I_{f} false) < cof false(I_{f} false)

for all increasing

f : ω \to ω

?…”

Section: Discussion and Open Questionsmentioning

confidence: 97%

“…Other way to attack Question 6.1 is by large products of creature forcing (cf. [9,20,21]). Quite recently, A. Fischer, Goldstern, Kellner and Shelah [9] used this technique to prove that 5 cardinal invariants on the right of Cichoń's diagram are pairwise different.…”

Section: Discussion and Open Questionsmentioning

confidence: 99%

“…[9,20,21]). Quite recently, A. Fischer, Goldstern, Kellner and Shelah [9] used this technique to prove that 5 cardinal invariants on the right of Cichoń's diagram are pairwise different. Such method would also work to solve the dual of Question 6.1, that is, The same question for M is also open.…”

Section: Discussion and Open Questionsmentioning

confidence: 99%

See 2 more Smart Citations

On cardinal characteristics of Yorioka ideals

Cardona

Mejía

2019

Mathematical Logic Qtrly

View full text Add to dashboard Cite

Yorioka introduced a class of ideals (parametrized by reals) on the Cantor space to prove that the relation between the size of the continuum and the cofinality of the strong measure zero ideal on the real line cannot be decided in sans-serifZFC. We construct a matrix iteration of c.c.c. posets to force that, for many ideals in that class, their associated cardinal invariants (i.e., additivity, covering, uniformity and cofinality) are pairwise different. In addition, we show that, consistently, the additivity and cofinality of Yorioka ideals does not coincide with the additivity and cofinality (respectively) of the ideal of Lebesgue measure zero subsets of the real line.

show abstract

add false(I_{f} false) < non false(I_{f} false) < cov false(I_{f} false) < cof false(I_{f} false)

for all increasing

f : ω \to ω

?…”

Section: Discussion and Open Questionsmentioning

confidence: 97%

Section: Discussion and Open Questionsmentioning

confidence: 99%

See 1 more Smart Citation

On cardinal characteristics of Yorioka ideals

Cardona

Mejía

2019

Mathematical Logic Qtrly

View full text Add to dashboard Cite

show abstract

“…These two hypotheses together prove a certain partition property P . 7 If we substitute B = Col(ω, <ω 2 ), then from this hypothesis we can prove ¬P . Although the existence of any ℵ 2 -saturated normal ideal on [ω 2 ] ω1 is not known to be consistent relative to large cardinals, Foreman notes that the same argument can be carried out on the basis of the conjunction of CH with two weaker ideal hypotheses known to be individually consistent (with GCH) relative to conventional large cardinals.…”

Section: Inconsistenciesmentioning

confidence: 96%

Generic Large Cardinals as Axioms

Eskew

2019

The Review of Symbolic Logic

View full text Add to dashboard Cite

We argue against Foreman's proposal to settle the continuum hypothesis and other classical independent questions via the adoption of generic large cardinal axioms.Shortly after proving that the set of all real numbers has a strictly larger cardinality than the set of integers, Cantor conjectured his Continuum Hypothesis (CH): that there is no set of a size strictly in between that of the integers and the real numbers [1]. A resolution of CH was the first problem on Hilbert's famous list presented in 1900 [19]. Gödel made a major advance by constructing a model of the Zermelo-Frankel (ZF) axioms for set theory in which the Axiom of Choice and CH both hold, starting from a model of ZF. This showed that the axiom system ZF, if consistent on its own, could not disprove Choice, and that ZF with Choice (ZFC), a system which suffices to formalize the methods of ordinary mathematics, could not disprove CH [16]. It remained unknown at the time whether models of ZFC could be found in which CH was false, but Gödel began to suspect that this was possible, and hence that CH could not be settled on the basis of the normal methods of mathematics. Gödel remained hopeful, however, that new mathematical axioms known as "large cardinals" might be able to give a definitive answer on CH [17].The independence of CH from ZFC was finally solved by Cohen's invention of the method of forcing [2]. Cohen's method showed that ZFC could not prove CH either, and in fact could not put any kind of bound on the possible number of cardinals between the sizes of the integers and the reals. Lévy and Solovay further developed the forcing machinery, and noticed that it also destroyed Gödel's hopes for large cardinals. Forcing allowed one to manipulate the cardinal value of the set of reals, passing from one model of ZFC to another giving a different answer on CH, without disturbing any large cardinals in the process [22].This was not the last word on CH from the community of set theorists. Several programs to develop acceptable axioms that settle CH have been put forward. Matthew Foreman, a leading set theorist, has suggested a solution to CH via axioms called "generic large cardinals." Our goal here is to critically examine Foreman's proposal. First, we describe the goals these axioms are supposed to meet and the kinds of considerations in their favor, highlighting the claim that the favorable considerations for traditional large cardinals transfer to the generic ones. Second, we discuss many technical difficulties in accommodating generic large cardinals in a single axiomatic framework, and present some new "mutual inconsistency results" that raise troubles for the program. Third, we examine the considerations in favor of traditional large cardinals and argue that they do not have the same import for the generic variety. Finally, we consider an alternative take on these kinds of axiomsThe author would like to thank Sean Walsh and Neil Barton for many helpful discussions.

show abstract

“…5.1(d)]). On the other hand, many examples can be obtained by creature forcing constructions as in [KS09,KS12,FGKS17], for instance, in the latter reference it is proved that the right side of Cichoń's diagram can be divided into 5 different values where 4 of them are singular. However, all these constructions are ω ω -bounding, so they force d = ℵ 1 and do not allow separation of cardinal invariants below d.…”

Section: Introductionmentioning

confidence: 99%

Matrix Iterations with Vertical Support Restrictions

Mejía

2019

Proceedings of the 14th and 15th Asian Logic Conferences

View full text Add to dashboard Cite

We use coherent systems of FS iterations on a power set, which can be seen as matrix iteration that allows restriction on arbitrary subsets of the vertical component, to prove general theorems about preservation of certain type of unbounded families on definable structures and of certain mad families (like those added by Hechler's poset for adding an a.d. family) regardless of the cofinality of their size. In particular, we define a class of posets called σ-Frechet-linked and show that they work well to preserve mad families, and unbounded families on ω ω .As applications of this method, we show that a large class of FS iterations can preserve the mad family added by Hechler's poset (regardless of the cofinality of its size), and the consistency of a constellation of Cichoń's diagram with 7 values where two of these values are singular.2010 Mathematics Subject Classification. 03E17, 03E15, 03E35, 03E40. Key words and phrases. Coherent system of finite support iterations, Frechet-linked, preservation of unbounded families, preservation of mad families, Cichoń's diagram.

show abstract

Creature forcing and five cardinal characteristics in Cichoń’s diagram

Abstract: We use a (countable support) creature construction to show that consistentlyThe same method shows the consistency of

Cited by 14 publications

References 17 publications

On cardinal characteristics of Yorioka ideals

On cardinal characteristics of Yorioka ideals

Generic Large Cardinals as Axioms

Matrix Iterations with Vertical Support Restrictions

Contact Info

Product

Resources

About