Local Connectedness and Distance Functions

Morgan, Charles G.

doi:10.1007/3-7643-7692-9_14

Cited by 7 publications

(6 citation statements)

References 19 publications

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“…forcing over a model of CH. This method was extensively analyzed in the context of morasses by Morgan in [38] Having in mind forcing with side conditions in 2-cardinals one can revise the use of stepping up tools for obtaining c.c.c. notions of forcing.…”

Section: Generic Stepping-upmentioning

confidence: 99%

On constructions with 2-cardinals

Koszmider

2017

Arch. Math. Logic

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We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman's neat simplified morasses called 2-cardinals. The theory of related trees, gaps, colorings of pairs and forcing notions is reformulated and sketched from a unifying point of view with the focus on the applicability to constructions of mathematical structures like Boolean algebras, Banach spaces or compact spaces.The paper is dedicated to the memory of Jim Baumgartner whose seminal joint paper [5] with Saharon Shelah provided a critical mass in the theory in question.A new result which we obtain as a side product is the consistency of the existence of a function f : [λ ++ ] 2 → [λ ++ ] ≤λ with the appropriate λ + -version of property ∆ for regular λ ≥ ω satisfying λ <λ = λ.

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Section: Generic Stepping-upmentioning

confidence: 99%

On constructions with 2-cardinals

Koszmider

2017

Arch. Math. Logic

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“…However, he is able to produce a proper forcing which adds such a sequence [27] using his method of side conditions in morasses which is an extension of Todorcevic's method of side conditions in models. More on the method can be found in Morgan's paper [31]. In the context of our approach, this raises the question if it is possible to define something like a countable support iteration along a morass.…”

Section: Theorem 48mentioning

confidence: 99%

Higher-dimensional forcing

Irrgang

2008

Preprint

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We present a method of constructing ccc forcings: Suppose first that a continuous, commutative system of complete embeddings between countable forcings indexed along ω1 is given. Then its direct limit satisfies ccc by a well-known theorem on finite support iterations. However, this limit has size at most ω1. To get larger forcings, we do not consider linear systems but higher-dimensional systems which are indexed along simplified morasses.

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“…Eventually this approach was subsumed by a more powerful one, introduced by Koszmider ([7] -see also [3], [14], [15]). Koszmider's work adapted Todorcevic's elaboration, in terms of ∈-chains of models, of Baumgartner's idea of using side conditions to restrict the possible extensions of forcing conditions, to the structured objects mentioned above.…”

Section: Introductionmentioning

confidence: 99%

Mitchell-style forcing, with small working parts and collections of models as side conditions, and gap-one simplified morasses

Morgan¹

2014

Preprint

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We give a modification of Mitchell's technique ([8]-[11]) for adding objects of size ω 2 with conditions with finite working parts in which the collections of models used as side conditions are very highly structured, arguably making them more wieldy. We use one such forcing (essentially a 'pure side conditions' forcing) to answer affirmatively the question, asked independently by Shelah and Velleman in the late 1980s, as to whether a (κ + , 1)-simplified morass can be added by a forcing with working parts of size < κ.

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Local Connectedness and Distance Functions

Cited by 7 publications

References 19 publications

On constructions with 2-cardinals

On constructions with 2-cardinals

Higher-dimensional forcing

Mitchell-style forcing, with small working parts and collections of models as side conditions, and gap-one simplified morasses

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