Set theory without choice: not everything on cofinality is possible

@article{Shelah1997SetTW,
  title={Set theory without choice: not everything on cofinality is possible
},
  author={Saharon Shelah},
  journal={Archive for Mathematical Logic},
  year={1997},
  volume={36},
  pages={81-125}
}
  • S. Shelah
  • Published 15 December 1995
  • Mathematics
  • Archive for Mathematical Logic
Abstract.We prove in ZF+DC, e.g. that: if $\mu=|{\cal H}(\mu)|$ and $\mu>\cf(\mu)>\aleph_0$ then $\mu ^+$ is regular but non measurable. This is in contrast with the results on measurability for $\mu=\aleph_\omega$ due to Apter and Magidor [ApMg]. 
ZF + DC + AX4
TLDR
It is proved that for a sequence δ¯=⟨δs: s∈Y⟩,cf(δ s) large enough compared to Y, the pcf theorem can be proved with minor changes (in particular, using true cofinalities not the pseudo ones).
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
An Easton-like Theorem for Zermelo-Fraenkel Set Theory with the Axiom of Dependent Choice
We show that in the theory ZF + DC + for every cardinal {\lambda}, the set of infinite subsets of {\lambda} is well-ordered (i.e., Shelah's AX4), the {\theta}-function measuring the surjective size
MODEL THEORY WITHOUT CHOICE? CATEGORICITY SH840
We prove Los conjecture = Morley theorem in ZF, with the same char- acterization (of first order countable theories categorical in ℵα for some (equivalently for every ordinal) � > 0. Another central
PCF arithmetic without and with choice
We deal with relatives of GCH which are provable. In particular, we deal with rank version of the revised GCH. Our motivation was to find such results when only weak versions of the axiom of choice
Ordinal definable subsets of singular cardinals
A remarkable result by Shelah states that if κ is a singular strong limit cardinal of uncountable cofinality, then there is a subset x of κ such that HODx contains the power set of κ. We develop a
PCF ARITHMETIC WITHOUT AND WITH CHOICE SH938
We deal with relatives of GCH which are provable. In particular we deal with rank version of the revised GCH. Our motivation was to find such results when only weak versions of the axiom of choice
Applications of PCF theory
  • S. Shelah
  • Mathematics
    Journal of Symbolic Logic
  • 2000
TLDR
Another version of exponentiation is characterized: maximal number of k-branches in a tree with λ nodes, and cardinal invariants for each λ with a pcf restriction are given.
Controlling the number of normal measures at successor cardinals
We examine the number of normal measures a successor cardinal can carry, in universes in which the Axiom of Choice is false. When considering successors of singular cardinals, we establish relative
Applications of Pcf Theory Sh589
Abstract. We deal with several pcf problems; we characterize another version of exponentiation: number of κ-branches in a tree with λ nodes, deal with existence of independent sets in stable
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