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  • Cited by 1
  • Print publication year: 1983
  • Online publication date: August 2010

ON THE REGULARITY OF ULTRAFILTERS

Summary

It was shown in that in Gődel's construetible universe L, every uniform ultrafilter over ω1 is regular. This involved a new combinatorial principle stronger that Kurepa's hypothesis. Chang and Jensen have generalized this principle to higher cardinals and extended the regularity result of to all ωn, (n∈ω). Benda and Ketonen have generalized and simplified the result of by making use of a weaker combinatorial principle. In the present note we show how to generalize the Chang-Jensen result along similar lines. Kunen asked in a discussion whether this can be done.

We start by formulating the transversal hypothesis in a customary form:

TH(λ, ν) : There is a family F ⊆ λν such that |F| = λ+ and for all f, g ∈ F, if f ≠ g, then f(γ) ≠ g(γ) for all sufficiently large γ ∈ λ.

The following consistency results are well known.

Theorem 1, (Solovay). In L, TH(κ+, κ) holds for all infinite cardinals κ.

In the opposite direction we only give a special case of Silver's Theorem.

Theorem 2. If ZFC + there is a Ramsey cardinal is consistent, then ZFC + ¬TH(ω1, ω) is consistent.

For regular λ it follows trivially from TH(λ, ν) that λ ≤ ν+. An obvious attempt at generalizing the Benda–Ketonen approach to, say ω2, leads to considering statements like TH(ω2, ω).