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Published in:
Reconstruction Problems for Digraphs Read chapter Read first chapter

2013 | OriginalPaper | Chapter

The PCF Theorem Revisited

Author : Saharan Shelah

Published in: The Mathematics of Paul Erdős II

Publisher: Springer New York

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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.

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previous chapter On Order-Perfect Lattices
next chapter Paul Erdős: The Master of Collaboration
Footnotes
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
 
Literature
1.
C. C. Chang and H. J. Keisler, Model Theory, North Holland Publishing Company (1973).
2.
3.
A. Kanamori, Weakly normal filters and irregular ultra-filter, Trans of A.M.S., 220 (1976) 393–396.
4.
5.
J. Ketonen, Some combinatorial properties of ultra-filters, Fund Math. VII (1980) 225–235.
6.
J. D. Monk, Cardinal Function on Boolean Algebras, Lectures in Mathematics, ETH Zürich, Bikhäuser, Verlag, Baser, Boston, Berlin, 1990.
7.
S. Shelah, Proper forcing Springer Lecture Notes, 940 (1982) 496+xxix.
8.
S. Shelah, Cardinal Arithmetic, volume 29 of Oxford Logic Guides, General Editors: Dov M. Gabbai, Angus Macintyre and Dana Scott, Oxford University Press, 1994.
9.
10.
S. Shelah, Products of regular cardinals and cardinal invariants of Boolean Algebra, Israel Journal of Mathematics, 70 (1990) 129–187.MathSciNetMATHCrossRef
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13.
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.
14.
S. Shelah, Further cardinal arithmetic, Israel Journal of Mathematics, Israel Journal of Mathematics, 95 (1996) 61–114.MATHCrossRef
15.
M. Magidor and S. Shelah, λ i inaccessible \(>\kappa \prod _{i<\kappa }\lambda _{i}/D\) of order typeμ  + , preprint.
16.
S. Shelah, PCF theory: Application, in preparation.
Metadata
Title
The PCF Theorem Revisited
Author
Saharan Shelah
Copyright Year
2013
Publisher
Springer New York
Book
The Mathematics of Paul Erdős II

Print ISBN: 978-1-4614-7253-7

Electronic ISBN: 978-1-4614-7254-4

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-1-4614-7254-4_26

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