Geodesic
Openly generated Boolean algebras and the Fodor-type reflection principle
Sakaé Fuchino 1   ; Assaf Rinot 2
1 Graduate School of System Informatics Kobe University Rokko-dai 1-1, Nada-ku, Kobe 657-8501, Japan
2 Department of Mathematics Ben-Gurion University of the Negev P.O.B. 653 Be'er Sheva, 84105 Israel
Fundamenta Mathematicae, Tome 212 (2011) no. 3, pp. 261-283 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/fm212-3-4
Keywords: prove fodor type reflection principle frp equivalent assertion boolean algebra openly generated only aleph projective previously known characterization openly generated boolean algebras follows axiom since frp preserved generic extension conclude particular characterization consistent set theoretic assertion forcable c poset starting model frp crucial step proof main result frp implies shelahs strong hypothesis ssh particular frp implies singular cardinals hypothesis sch extending result second author establish characterizations ssh terms topological reflection theorems

Sakaé Fuchino  1   ; Assaf Rinot  2

1 Graduate School of System Informatics Kobe University Rokko-dai 1-1, Nada-ku, Kobe 657-8501, Japan
2 Department of Mathematics Ben-Gurion University of the Negev P.O.B. 653 Be'er Sheva, 84105 Israel 
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Sakaé Fuchino; Assaf Rinot. Openly generated Boolean algebras and the Fodor-type reflection principle. Fundamenta Mathematicae, Tome 212 (2011) no. 3, pp. 261-283. doi: 10.4064/fm212-3-4

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