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I am reading a paper by Goldberg, and he uses a theorem by Ketonen, which is highlighted in red below: enter image description here

Do you know where can I find the theorem and it's proof?

Link to the article: https://arxiv.org/abs/2002.07299

Thank you.

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1 Answer 1

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Sorry I didn't give a reference. The theorem first appeared in spirit in Ketonen's paper "Strong compactness and other cardinal sins," Theorem 1.3.

The precise result I needed is proved in my thesis as Theorem 7.2.12, which states that if $j : V \to M$ is an elementary embedding, $\lambda$ is a regular uncountable cardinal, and $\delta$ is an $M$-cardinal, then $\text{cf}^M(\sup j[\lambda])\leq \delta$ if and only if $j$ is "$(\lambda,\delta)$-tight," which just means $j[\lambda]$ is contained in a set in $M$ of $M$-cardinality at most $\delta$. Lemma 7.2.7 states that an ultrapower embedding $j : V\to M$ is $(\lambda,\delta)$-tight if and only if $M$ has the "$({\leq}\lambda,{\leq}\delta)$-cover property," which just means that every subset of $M$ of cardinality at most $\lambda$ is contained in an element of $M$ that has $M$-cardinality at most $\delta$.

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    $\begingroup$ Excuse me for further asking, but is not needed for $\lambda$ to be weakly inaccessible in theorem 2.3 in the paper? $\endgroup$
    – D. Hershko
    Jul 20, 2020 at 19:46
  • $\begingroup$ No, just regular. I don't know why I wrote weakly inaccessible... $\endgroup$ Jul 20, 2020 at 19:58

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