Set Theory Without Choice: Not Everything on Cofinality Is Possible
Sh:497 Arch. Math. Logic (1997) 36: 81–125 c Springer-Verlag 1997 Set theory without choice: not everything on cofinality is possible Saharon Shelah1,2,? 1 Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel∗∗ 2 Department of Mathematics, Rutgers University, New Brunswick, NJ, USA Received May 6, 1993 / Revised December 11, 1995 Abstract. We prove in ZF+DC, e.g. that: if µ = H (µ) and µ>cf(µ) > 0 then µ+ is regular but non measurable. This is in| contrast| with the resultsℵ on measurability for µ = ω due to Apter and Magidor [ApMg]. ℵ Annotated content 0 Introduction [In addition to presenting the results and history, we gave some basic definitions and notation.] 1 Exact upper bound (A∗) [We define some variants of least upper bound (lub, eub)in( Ord,<D) for D a filter on A∗. We consider <D -increasing sequence indexed by ordinals δ or indexed by sufficiently directed partial orders I , of members of (A∗)Ord or of members of (A∗)Ord/D (and cases in the middle). We give sufficient conditions for existence involving large cofinality (of δ or I ) and some amount of choice. Mostly we look at what the ZFC proof gives without choice. Note in particular 1.8, which assumes only DC (ZF is not mentioned of course), the filter is 1- complete and cofinality of δ large and we find an eub only after extendingℵ the filter.] 2 hpp [We look at various ways to measure the size of the set f (a)/D, like a A ∈ ∗ supremum length of <D -increasing sequence in f (a) (calledQ ehppD ), or a A ∈ ∗ Q ? Research supported by “The Israel Science Foundation” administered by The Israel Academy of Sciences and Humanities.
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