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 U ⊲ U′ if U ∈ Vκ/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.
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