# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

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

### Tom 212 / 2011

Fundamenta Mathematicae 212 (2011), 261-283 MSC: 03E35, 03E55, 03E65, 03E75, 03G05, 06E05. DOI: 10.4064/fm212-3-4

#### Streszczenie

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 $\aleph _2$-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.

#### Autorzy

• Sakaé FuchinoGraduate School of System Informatics
Kobe University
Rokko-dai 1-1, Nada-ku, Kobe 657-8501, Japan
e-mail
• Assaf RinotDepartment of Mathematics
Ben-Gurion University of the Negev
P.O.B. 653
Be'er Sheva, 84105 Israel
e-mail

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