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.
Partially supported by the Deutsche Forschungsgemeinschaft, grant Ko 490/7-1. Publication no. 506.
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Notes
- 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
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Dedicated to Paul Erdős
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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
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