Pcf without choice Sh835

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...

Full description

Saved in:
Bibliographic Details
Published in:Archive for mathematical logic 2024, Vol.63 (5-6), p.623-654
Main Author: Shelah, Saharon
Format: Article
Language:English
Subjects:
Algebra
Combinatorial analysis
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Set theory
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 ) > ℵ 0 , then from a well ordering of P ( P ( κ ) ) ∪ κ > μ we can define a well ordering of κ μ .
ISSN:0933-5846
1432-0665
DOI:10.1007/s00153-023-00900-7