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1
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Openly generated Boolean algebras and the Fodor-type reflection principle

100%
Fuchino S., Rinot A.
Fundamenta Mathematicae
|
2011
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tom 212
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nr 3
261-283
EN
We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is ℵ₂-projective. Previously it was known that this characterization of openly generated Boolean algebras follows from Axiom R. Since FRP is preserved by c.c.c. generic extension, we conclude in particular that this characterization is consistent with any set-theoretic assertion forcable by a c.c.c. poset starting from a model of FRP. A crucial step of the proof of the main result is to show that FRP implies Shelah's Strong Hypothesis (SSH). In particular, we show that FRP implies the Singular Cardinals Hypothesis (SCH). Extending a result of the second author, we also establish some new characterizations of SSH in terms of topological reflection theorems.
2
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More set-theory around the weak Freese–Nation property

100%
Fuchino S., Soukup L.
Fundamenta Mathematicae
|
1997
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tom 154
|
nr 2
159-176
EN
We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (Theorem 10) of partial orderings with κ-Freese-Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang's Conjecture for $ℵ_ω$, we can find a counter-example to the characterization (Theorem 12). We then show that, in the model obtained by adding Cohen reals, a lot of ccc complete Boolean algebras of cardinality ≤ λ have the $ℵ_1$-Freese-Nation property provided that $μ^{ℵ_0} = μ$ holds for every regular uncountable μ < λ and the very weak square principle holds for each cardinal $ℵ_0 < μ < λ$ of cofinality ω ((Theorem 15). Finally, we prove that there is no $ℵ_2$-Lusin gap if P(ω) has the $ℵ_1$-Freese-Nation property (Theorem 17)
3
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Some combinatorial principles defined in terms of elementary submodels

100%
Fuchino S., Geschke S.
Fundamenta Mathematicae
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2004
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tom 181
|
nr 3
233-255
EN
We give an equivalent, but simpler formulation of the axiom SEP, which was introduced in [9] in order to capture some of the combinatorial behaviour of models of set theory obtained by adding Cohen reals to a model of CH. Our formulation shows that many of the consequences of the weak Freese-Nation property of 𝒫(ω) studied in [6] already follow from SEP. We show that it is consistent that SEP holds while 𝒫(ω) fails to have the (ℵ₁,ℵ ₀)-ideal property introduced in [2]. This answers a question addressed independently by Fuchino and by Kunen. We also consider some natural variants of SEP and show that certain changes in the definition of SEP do not lead to a different principle, answering a question of Blass.
4
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Coloring ordinals by reals

100%
Brendle J., Fuchino S.
Fundamenta Mathematicae
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2007
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tom 196
|
nr 2
151-195
EN
We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ,λ) for regular κ > ℵ₁ and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of $C^{s}(κ)$ and $F^{s}(κ)$ of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ₂,ℵ₁) (hence also IP(ℵ₂,ℵ₂) as well as HP(ℵ₂)) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ₂,ℵ₂) (hence also HP(ℵ₂)) holds in a generic extension by countable support side-by-side product of Sacks or Prikry-Silver forcing (Corollary 4.8). We also show that the latter result is optimal (Theorem 5.2). Relations between these principles and their influence on the values of the variations $𝔟^{↑}$, $𝔟^{h}$, 𝔟*, 𝔡𝔬 of the bounding number 𝔟 are studied. One of the consequences of HP(κ) besides $C^{s}(κ)$ is that there is no projective well-ordering of length κ on any subset of $^{ω}ω$. We construct a model in which there is no projective well-ordering of length ω₂ on any subset of $^{ω}ω$ (𝔡𝔬 = ℵ₁ in our terminology) while 𝔟* = ℵ₂ (Theorem 6.4).
5
Content available remote

On absolutely divergent series

81%
Fuchino S., Mildenberger H., Shelah S., Vojtáš P.
Fundamenta Mathematicae
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1999
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tom 160
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nr 3
255-268
EN
We show that in the $ℵ_2$-stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(ω)/fin. This complements Vojtáš' result that under $cf(\gc) = \gp$ the two algebras are isomorphic [15].
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