Abstract
We mainly investigate models of set theory with restricted choice, e.g., ZF + DC + the family of countable subsets of λ is well ordered for every λ (really local version for a given λ). We think that in this frame much of pcf theory, (and combinatorial set theory in general) can be generalized. We prove here, in particular, that there is a proper class of regular cardinals, every large enough successor of singular is not measurable and we can prove cardinal inequalities. Solving some open problems, we prove that if μ>κ=cf(μ)>ℵ0, then from a well ordering of P(P(κ))∪κ>μ we can define a well ordering of κμ.
Original language | English |
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Pages (from-to) | 623-654 |
Number of pages | 32 |
Journal | Archive for Mathematical Logic |
Volume | 63 |
Issue number | 5-6 |
DOIs | |
State | Published - Jul 2024 |
Bibliographical note
Publisher Copyright:© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.
Keywords
- 03E50
- Pcf
- Primary 03E17
- Secondary: 03E05
- Set theory
- Weak axiom of choice