The following lemma is from this introduction to cardinals.
Lemma 2.7. Let ωα be a limit cardinal. Then α is a limit ordinal and cof(ωα)=cof(α).
I understand the proof of this lemma, including the proof of relevant lemma 2.4 (beware, there is a typo in that proof: g instead of h).
Now I want to use it to make the following conclusion:
cof(ωλ)2.7=cof(λ)≤λ<ωλ
However, in the same paper I'm being told that the existence of regular limit cardinals is independent of ZFC! So there must be a flaw in my conclusion.
I can think of two things that would undermine my argument:
- λ<ωλ is not always true.
- For a nonzero limit ordinal λ, ωλ need not be a limit cardinal.
But I don't see how either 1. or 2. is possible.
Answer
It’s the first alternative: it’s not always true that λ<ωλ. Let λ0=ω, and for n∈ω let λn+1=ωλn. Let λ=sup; then
\lambda=\sup_n\lambda_n=\sup_n\omega_{\lambda_n}=\omega_{\sup_n\lambda_n}=\omega_\lambda\;.
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