Sequential and distributive forcings without choice
| Autor: | Jonathan Schilhan, Asaf Karagila |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Canadian Mathematical Bulletin. :1-13 |
| ISSN: | 1496-4287 0008-4395 |
| Popis: | In the Zermelo--Fraenkel set theory with the Axiom of Choice a forcing notion is "$\kappa$-distributive" if and only if it is "$\kappa$-sequential". We show that without the Axiom of Choice this equivalence fails, even if we include a weak form of the Axiom of Choice, the Principle of Dependent Choice for $\kappa$. Still, the equivalence may still hold along with very strong failures of the Axiom of Choice, assuming the consistency of large cardinal axioms. We also prove that while a $\kappa$-distributive forcing notion may violate Dependent Choice, it must preserve the Axiom of Choice for families of size $\kappa$. On the other hand, a $\kappa$-sequential can violate the Axiom of Choice for countable families. We also provide a condition of "quasiproperness" which is sufficient for the preservation of Dependent Choice, and is also necessary if the forcing notion is sequential. Comment: 12 pages; accepted version |
| Databáze: | OpenAIRE |
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