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Pcf without choice Sh835

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Abstract

We mainly investigate models of set theory with restricted choice, e.g., ZF + DC + the family of countable subsets of \(\lambda \) is well ordered for every \(\lambda \) (really local version for a given \(\lambda \)). 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 \(\mu> \kappa = \textrm{cf}(\mu ) > \aleph _{0},\) then from a well ordering of \({\mathscr {P}}({\mathscr {P}}(\kappa )) \cup {}^{\kappa >} \mu \) we can define a well ordering of \({}^{\kappa } \mu .\)

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Notes

  1. so by actually only \(c \ell {\restriction }[\lambda ]^{\le \kappa }\) count.

  2. Can do somewhat better; we can replace \([\alpha ]^{< \mu _1}\) by \(\{v \subseteq \alpha :\textrm{otp}(v) \subseteq \mu _1\}\)

  3. clearly we can replace \(< \mu \) by \(< \gamma \) for \(\gamma \in (\mu ,\mu ^+)\)

  4. We could have used \(\{t \in Y:f_{\eta ,\alpha }[c \ell ](t) \in c \ell (\textbf{v}(u))\} \ne \emptyset \) mod \(D^{{\mathfrak y}}_2\); also we could have added u to \(c \ell '(u)\) but not necessarily by \(\boxplus _2\).

  5. but, of course, possibly there is no such sequence \(\langle f_\alpha :\alpha < \lambda ^+ \rangle \)

  6. the regular holds many times by 2.13

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Authors and Affiliations

  1. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, 91904, Jerusalem, Israel

    Saharon Shelah

  2. Department of Mathematics, The State University of New Jersey, Hill Center - Busch Campus Rutgers, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Saharon Shelah

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  1. Saharon Shelah
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Correspondence to Saharon Shelah.

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This research was supported by the United States-Israel Binational Science Foundation, and several grants including grants with Maryanthe Malliaris number NFS 2051825, BSF 3013005232, and the ISF 2320/23, The Israel Science Foundation (ISF) (2023–2027). I would like to thank Alice Leonhardt for the beautiful typing. For later versions, the author would like to thank the typist for his work and is also grateful for the generous funding of typing services donated by a person who wishes to remain anonymous. References like [12, Def. 0.4 = Lz15] means the label of Def. 0.4 is z15. The reader should note that the version on my website is usually more updated than the one in the mathematical archive. Submitted to AML 1 July 2005. First Typed - 2004/Jan/20.

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Shelah, S. Pcf without choice Sh835. Arch. Math. Logic (2024). https://doi.org/10.1007/s00153-023-00900-7

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