Abstract
In this chapter, a denotes an infinite set of regular cardinals. We will prove the following assertion, due to S. Shelah: If |a|+ < min(a) and λ ∈ pcf(a), then there is a set bλ ∈ T<λ+(a) which generates the ideal T<λ+(a) over T<λ(a), which means that for each set c ∈ T<λ+(a) there is a set d ∈ T<λ(a) such that c ⊑ bλ ⋃ d. Such a set bλ is called a generator of T<λ+(a) over T<λ(a). In Section 5.1 we show under the previous assumptions that there exists a universal sequence (f ξ : ξ < λ). It has the property that, for every ultrafilter D on a satisfying cf (Π a/D) = λ, it is cofinal in Π a modulo D. With the results of Section 4.5, we can prove in the second section the existence of the desired sequence (bλ : λ ∈ pcf(a)) of generators.1
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© 1999 Springer Basel AG
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Holz, M., Steffens, K., Weitz, E. (1999). Generators of T <λ+(a). In: Introduction to Cardinal Arithmetic. Modern Birkhäuser Classics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0346-0330-0_6
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DOI: https://doi.org/10.1007/978-3-0346-0330-0_6
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0346-0327-0
Online ISBN: 978-3-0346-0330-0
eBook Packages: Springer Book Archive