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 \({}^{\kappa } \mu .\)
Similar content being viewed by others
Notes
so by actually only \(c \ell {\restriction }[\lambda ]^{\le \kappa }\) count.
Can do somewhat better; we can replace \([\alpha ]^{< \mu _1}\) by \(\{v \subseteq \alpha :\textrm{otp}(v) \subseteq \mu _1\}\)
clearly we can replace \(< \mu \) by \(< \gamma \) for \(\gamma \in (\mu ,\mu ^+)\)
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 \(\boxplus _2\).
but, of course, possibly there is no such sequence \(\langle f_\alpha :\alpha < \lambda ^+ \rangle \)
the regular holds many times by 2.13
References
Apter, A., Magidor, M.: Instances of dependent choice and the measurability of \(\aleph _{\omega + 1}\). Ann. Pure Appl. Logic 74, 203–219 (1995)
Easton, W.B.: Powers of regular cardinals. Annals Math. Logic 1, 139–178 (1970)
Galvin, F., Hajnal, A.: Inequalities for cardinal powers. Ann. Math. 101, 491–498 (1975)
Gitik, M.: All uncountable cardinals can be singular. Israel J. Math. 35, 61–88 (1980)
Paul, B.: Larson and Saharon Shelah, Splitting stationary sets from weak forms of choice. MLQ Math. Log. Q. 55(3), 299–306 (2009). arXiv: 1003.2477
Shelah, S. et al.: Tba, In preparation. Preliminary number: Sh:F1039
Shelah, S.: A note on cardinal exponentiation. J. Symbolic Logic 45(1), 56–66 (1980)
Shelah, S.: Cardinal arithmetic, Oxford Logic Guides, vol. 29. The Clarendon Press, Oxford University Press, New York (1994)
Shelah, S.: Set theory without choice: not everything on cofinality is possible. Arch. Math. Logic 36(2), 81–125 (1997). arXiv: math/9512227
Shelah, S.: Applications of PCF theory. J. Symbolic Logic 65(4), 1624–1674 (2000). arXiv:math/9804155
Shelah, S.: PCF arithmetic without and with choice. Israel J. Math. 191(1), 1–40 (2012). arXiv: 0905.3021
Shelah, S.: Pseudo PCF. Israel J. Math. 201(1), 185–231 (2014). arXiv: 1107.4625
Shelah, S.: ZF + DC + AX\(_4\). Arch. Math. Logic 55(1–2), 239–294 (2016). arXiv: 1411.7164
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Shelah, S. Pcf without choice Sh835. Arch. Math. Logic (2024). https://doi.org/10.1007/s00153-023-00900-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00153-023-00900-7