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A cardinal $\kappa$ is huge iff there is $\lambda>\kappa$ and a $\kappa$-complete normal ultrafilter on $P_{\leq \kappa}(\lambda)$, or, equivalently, on the set of families of subsets of $\lambda$ of order-type $\kappa$. [1] I want to know whether one can further assume that $U$ is concentrated on something smaller? For example, can one further assume that $U$ is concentrated on $<\kappa$-directed families of sets of size $\kappa$? Only on those directed families that contain $P_{\kappa}(\lambda)$? Covering families as above ?

In notation: Let $U$ be such an ultrafilter. Let $D_\kappa=\{X\in P_{\leq\kappa}(\lambda): \forall S\subseteq X(|S|<\kappa\implies \cup S \in X\} $ be the subset of all $\kappa$-directed subsets of $P_{\leq \kappa}(\lambda)$. Is $D_\kappa\in U $ ? Let $D'_\kappa=\{X\in D_\kappa: P_{<\kappa}(\lambda)\subseteq X\}$ be the subset of all $\kappa$-directed subsets of $P_{\leq \kappa}(\lambda)$ containing all sets of size less than $\kappa$. Let $C'_\kappa=\{X\in D_\kappa: \forall S\in P_{\leq\kappa}(\lambda)\exists x\in X(S\subseteq x)\}$.
Is $D'_\kappa\in U$ ? Is $C'_\kappa\in U$? Can one find such an $U$ that the answers are positive?

I am also looking for any references describing in detail ultrafilter characterizations of large cardinals.

[1] The Ultrafilter Characterization of Huge Cardinals Robert J. Mignone Proceedings of the American Mathematical Society Vol. 90, No. 4 (Apr., 1984), pp. 585-590 http://www.ams.org/proc/1984-090-04/S0002-9939-1984-0733411-6/S0002-9939-1984-0733411-6.pdf

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1 Answer 1

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The hugeness of $\kappa$ is witnessed by an embedding $j:V\to M$ for which $M^\lambda\subset M$, where $\lambda=j(\kappa)$. In particular, for such an embedding we have $j''\lambda\in M$, and one may accordingly consider the induced measure on $P_{\leq\kappa}(\lambda)$ defined by $X\in U\iff j''\lambda\in j(X)$. The ultrapower by $U$ is a factor of the original embedding $j$ and also witnesses the hugeness of $\kappa$.

Since $j''\lambda$ is $\lt\lambda$-closed, in the sense that every subset of it of size less than $\lambda$ is bounded in it, then we see that $U$ concentrates on such kind of objects. This kind of reasoning is usually the key for understanding the essential nature of the sets on which a hugeness measure must concentrate.

But I am confused about your question concerning $\kappa$-directedness, since it seems to me that every subset $\sigma\subset\lambda$ with order type $\kappa$ will be $\kappa$-directed, since any small subset of $\sigma$ will be bounded in $\sigma$. Have you asked the question you intended to ask?

In your comment you state another property, which I would call something like $\lt\kappa$-closure rather than $\kappa$-directedness, since it refers to the least upper bound $\cup S$ rather than merely to an upper bound of $S$. In this case, $j''\lambda$ is not $\lt\lambda$-closed, since the union of the first $\kappa$ many elements of it is $\kappa$ itself, which is not an element of $j''\lambda$. So the usual hugeness measures will not concentrate on $D_\kappa$ defined that way.

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Thank you! I meant to ask: Is $D_\kappa=\{X\in P_{\leq\kappa}(\lambda): \forall S\subseteq X(|S|<\kappa\implies \cup S \in X\}$ in $U$ ? That is, I am looking at directed families of subsets, not families of directed subsets. –  mmm Jul 18 '13 at 15:53
    
Maybe I'm misunderstanding something, but it seems to me that, to see that $U$ concentrates on $<\kappa$-closed things, we need to use that $j''\lambda$ is $<j(\kappa)$-closed, i.e., $\lambda$-closed (which it is, of course), not merely that $j''\lambda$ is $<\kappa$-closed. –  Andreas Blass Jul 18 '13 at 17:56
    
Andreas, you are totally right. –  Joel David Hamkins Jul 18 '13 at 18:09
    
mmm, what you write is not $\kappa$-directedness, but some kind of $\kappa$-completeness, since you ask for $\cup S\in X$ rather than a cover of $\cup S$. Since $j''\lambda$ does not have $j$ of your property, the usual hugeness measures do not concentrate on $D_\kappa$. –  Joel David Hamkins Jul 18 '13 at 18:13
    
Thanks! And what about $\beta$-completeness? That is, $D''_\beta=\{X \in P_{\leq \beta}(\lambda): \forall S\subseteq X(|S|<\beta\implies\exists y\in X(\cup S\subseteq y))$, for some $\beta\leq\kappa$ ? –  mmm Jul 18 '13 at 18:19

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