Skip to main content Accessibility help

Sets constructed from sequences of measures: Revisited

  • William J. Mitchell (a1)

Let U and U′ be measures on a cardinal κ: that is, normal κ-complete ultrafilters on κ. The partial ordering ⊲ was defined in [M74]: we say UU′ if UVκ/U′. A model L(ℱ) was constructed in which this ordering was a well ordering, and it was shown that under proper assumptions the length of this well ordering could be any ordinal up to κ++L(ℱ). In this paper we will revisit this material and show that the coherence required in the construction of ℱ can be greatly weakened. This change simplifies some proofs, weakens the assumption needed for the results stated above (Theorem 1, below), and proves one new result (Theorem 5, below) which is suggestive although its significance is not clear.

We have tried to make this paper self-contained and to that end have repeated some material from [M74]. We begin with some definitions before stating Theorem 1.

The ordering ⊲ is well founded. This may be seen by assuming that it is not and letting κ be least such that ⊲ is not well founded on measures on κ. If U is a measure on κ with an infinite descending chain below it, then the measures on κ in VκU are still not well founded by ⊲, contradicting the fact that iU(κ) is the least cardinal in Vκ/U such that ⊲ is not well founded. Since ⊲ is well founded, we can define O(U) to be the rank of U in the partial ordering ⊲ of measures on κ, and O(κ) to be the rank of this partial ordering.

Hide All
[Ma]Magidor, M., Changing cofinalities of cardinals, Fundamenta Mathematicae, vol. 99, no. 1 (1978), pp. 6171.
[M74]Mitchell, W., Sets constructible from sequences of ultrafilters, this Journal, vol. 39, no. 1 (1974), pp. 5766.
[M?]Mitchell, W., The core model for sequences of measures (to appear).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed