Abstract
In the Galvin-Hajnal theorem, the question of whether it holds for singular cardinals with countable cofinality was left open. Let ℵδ be singular and v be a cardinal. If δ = ℵδ, then \(X_\delta ^\nu = {\left| \delta \right|^\nu } < {X_{\left( {{{\left| \delta \right|}^v}} \right) + }}\) , and thus the Galvin-Hajnal theorem holds for the case that δ is a fixed point of the aleph function. Let us now turn to the case that δ < ℵδ. Then | δ | < ℵδ, and for every regular cardinal λ with | δ | < λ < ℵδ, the set a := [λ, ℵδ)reg is an interval of regular cardinals satisfying |a| < min(a), since |a| ≤ |δ|. Shelah defines an operator pcf which assigns to each set a of regular cardinals and each cardinal μ a set pcf μ (a) of regular cardinals satisfying the following properties for μ ≥ 1:
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a)
a ⊆ pcfμ (a).
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b)
min(a) = min pcfμ (a).
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c)
If |a| < min(a), then |pcfμ (a)| ≤ |a|μ.
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d)
If a is an interval of regular cardinals with |a| < min(a), then pcfµ(a) is an interval of regular cardinals.
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© 1999 Springer Basel AG
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Holz, M., Steffens, K., Weitz, E. (1999). Ordinal Functions. In: Introduction to Cardinal Arithmetic. Modern Birkhäuser Classics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0346-0330-0_4
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DOI: https://doi.org/10.1007/978-3-0346-0330-0_4
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0346-0327-0
Online ISBN: 978-3-0346-0330-0
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