The ultrafilter characterization of huge cardinals

Abstract. We define a filter on [\]K with properties similar to those of the closed unbounded filter in PK(X). This filter's behaviour depends on set theoretical hypotheses.The study of the combinatorial properties of the collection of subsets of uncountable cardinals has been a main line of research in the theory of large cardinals. For k < A regular uncountable cardinals, the space PK( X ) is the collection of subsets of A of cardinality smaller than k. This space was introduced in the investigation of strongly compact cardinals and of supercompact cardinals. In We recall the definition of huge cardinal. We say that k is huge with target A if there is an elementary embedding j': V -» M of the universe into a transitive model M containing all the ordinals such that k is the critical point of j, j(k) = X and XM C M. We denote this by k -» (A). (See [BDPT].) In this case the axiom of choice allows us to show that the set (A)K = {P C A | order type of P = k} belongs to the normal ultrafilter on [A]K, and thus we can characterize the fact that k -* (A) by the fact that there exists a normal, K-complete, fine ultrafilter on (A)\ Thus, under the axiom of choice the first characterization implies the second. This is not so in the absence of the axiom of choice; for instance, under the axiom of determinateness the implication fails, as shown by Mignone in [Mig].A natural problem is to find a filter on [A]K analogous to the closed unbounded filter for PK(X) constructed by Jech in [Jel]. The filter we construct is a K-complete, normal, fine, nontrivial filter, and, as shown by J. Baumgartner, it is the smallest filter on [A]K with these properties. Under the assumption that k is huge with target A, all elements of our filter have measure 1 with respect to the normal measure

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An extension of the closed unbounded filter

Mignone¹

1988

Proc. Amer. Math. Soc.

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ABSTRACT. A natural extension of the closed unbounded filter is introduced. This extension coincides with the closed unbounded filter on uncountable, regular cardinals re, but in general does not for PKA and [A]K.Henceforth, k will be a regular, uncountable cardinal, unless specified otherwise. A closed unbounded subset of k is a cofinal subset of k which contains the supremum of all increasing sequences from the subset of length less than k. The collection of all closed unbounded subsets of n generates a /c-complete, normal filter over k called the club filter. The notion of a closed unbounded subset of a cardinal k has been generalized to the set of all subsets of X of cardinality less than k, PkX (see [3]), and the set of all subsets of A of cardinality k, [X]k (see [2]). In both instances the collection of closed unbounded sets generate /c-complete, fine, normal filters, the club filters. This paper introduces a natural extension of the club filter. This extension coincides with the club filter on k, but in general does not for PKX and this property is satisfied by sequences (or chains) {pa : a < S} which satisfy w < U ip*i-A canonical example of a sequence which fails to satisfy this is {pa : a < 6} where pa = a and 6 -Wi. DEFINITION. A subset "o" of k is said to be L-closed if for any sequence {pa : a < S} Cb with S < k and |<5| < UQ<¿¡ \Pa\, then \Ja<6P* € °-

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The ultrafilter characterization of huge cardinals

Cited by 4 publications

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On the space (λ)κ

On the space (λ)κ

A filter on [𝜆]^{𝜅}

An extension of the closed unbounded filter

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