Summary
The pcf theorem (of the possible cofinability theory) was proved for reduced products \(\prod _{i <\kappa }\lambda _{i}/I\), where \(\kappa <\min _{i<\kappa }\lambda _{i}\). Here we prove this theorem under weaker assumptions such as \(\mathrm{wsat}\,(I) <\min _{i<\kappa }\lambda _{i}\), where wsat(I) is the minimalθsuch thatκcannot be divided toθsets ∉I(or even slightly weaker condition). We also look at the existence of exact upper bounds relative to < I ( < I -eub) as well as cardinalities of reduced products and the cardinalsT D (λ).Finally we apply this to the problem of the depth of ultraproducts (and reduced products) of Boolean algebras.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Partially supported by the Deutsche Forschungsgemeinschaft, grant Ko 490/7-1. Publication no. 506.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
actually we do not require p ≤ q ≤ p ⇒ p = q so we should say quasi partial order
- 2.
note if cf (θ) < θ then “θ + -directed” follows from “θ-directed” which follows from “ \(\lim \inf _{{I}^{{\ast}}}(\bar{\lambda }) \geq \theta\) , i.e. first part of clause (β) implies clause (β). Note also that if clause (α) holds then \(\prod \bar{\lambda }/{I}^{{\ast}}\) is θ + -directed (even \((\prod \bar{\lambda },<)\) is θ + -directed), so clause (α) implies clause (β).
- 3.
in fact note that for no \(B_{\varepsilon } \subseteq \kappa (\varepsilon <\theta )\) do we have: \(B_{\varepsilon }\neq B_{\varepsilon +1}\,\mathrm{mod\,}{I}^{{\ast}}\) and \(\varepsilon <\zeta <\theta \Rightarrow B_{\varepsilon } \cap A_{\zeta } \subseteq B_{\zeta }\) where \(A_{\zeta } =\kappa \,\mathrm{mod\,}{I}^{{\ast}}\) (e.g. \(A_{\zeta } = A_{\zeta }^{{\ast}}\))
- 4.
Of course, \(B_{\alpha } =\kappa \,\mathrm{mod\,}J_{<\lambda }(\bar{\lambda })\) , this becomes trivial.
- 5.
Note: if \(\mathrm{otp}(a_{\delta }) =\theta\) and \(\delta =\sup (a_{\delta })\) (holds if δ ∈ S, \(\mu =\theta +1\) and \(\bar{a}\) continuous in S (see below)) and δ ∈ acc(E) then δ is as required.
- 6.
sthe definition of \(B_{i}^{\alpha }\) in the proof of [8, III 2.14(2)] should be changed as in [Sh351, 4.4(2)]
- 7.
≤ s + means here that the right side is a supremum, right bigger than the left or equal but the supremum is obtained
Bibliography
C. C. Chang and H. J. Keisler, Model Theory, North Holland Publishing Company (1973).
F. Galvin and A. Hajnal, Inequalities for cardinal power, Annals of Math., 10 (1975) 491–498.
A. Kanamori, Weakly normal filters and irregular ultra-filter, Trans of A.M.S., 220 (1976) 393–396.
S. Koppelberg, Cardinalities of ultraproducts of finite sets, The Journal of Symbolic Logic, 45 (1980) 574–584.
J. Ketonen, Some combinatorial properties of ultra-filters, Fund Math. VII (1980) 225–235.
J. D. Monk, Cardinal Function on Boolean Algebras, Lectures in Mathematics, ETH Zürich, Bikhäuser, Verlag, Baser, Boston, Berlin, 1990.
S. Shelah, Proper forcing Springer Lecture Notes, 940 (1982) 496+xxix.
S. Shelah, Cardinal Arithmetic, volume 29 of Oxford Logic Guides, General Editors: Dov M. Gabbai, Angus Macintyre and Dana Scott, Oxford University Press, 1994.
S. Shelah, On the cardinality of ultraproduct of finite sets, Journal of Symbolic Logic, 35 (1970) 83–84.
S. Shelah, Products of regular cardinals and cardinal invariants of Boolean Algebra, Israel Journal of Mathematics, 70 (1990) 129–187.
S. Shelah, Cardinal arithmetic for skeptics, American Mathematical Society. Bulletin. New Series, 26 (1992) 197–210.
S. Shelah, More on cardinal arithmetic, Archive of Math Logic, 32 (1993) 399–428.
S. Shelah, Advances in Cardinal Arithmetic, Proceedings of the Conference in Banff, Alberta, April 1991, ed. N. W. Sauer et al., Finite and Infinite Combinatorics, Kluwer Academic Publ., (1993) 355–383.
S. Shelah, Further cardinal arithmetic, Israel Journal of Mathematics, Israel Journal of Mathematics, 95 (1996) 61–114.
M. Magidor and S. Shelah, λ i inaccessible \(>\kappa \prod _{i<\kappa }\lambda _{i}/D\) of order typeμ + , preprint.
S. Shelah, PCF theory: Application, in preparation.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
Dedicated to Paul Erdős
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Shelah, S. (2013). The PCF Theorem Revisited. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős II. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7254-4_26
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7254-4_26
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-7253-7
Online ISBN: 978-1-4614-7254-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)