20,527 research outputs found

    Measurable cardinals and the cardinality of Lindel\"of spaces

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    If it is consistent that there is a measurable cardinal, then it is consistent that all points g-delta Rothberger spaces have "small" cardinality.Comment: 9 pag

    Rothberger gaps in fragmented ideals

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    The~\emph{Rothberger number} b(I)\mathfrak{b} (\mathcal{I}) of a definable ideal I\mathcal{I} on ω\omega is the least cardinal κ\kappa such that there exists a Rothberger gap of type (ω,κ)(\omega,\kappa) in the quotient algebra P(ω)/I\mathcal{P} (\omega) / \mathcal{I}. We investigate b(I)\mathfrak{b} (\mathcal{I}) for a subclass of the FσF_\sigma ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is ℵ1\aleph_1 while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consistent that there are infinitely many (even continuum many) different Rothberger numbers associated with fragmented ideals.Comment: 28 page

    A hierarchy of Ramsey-like cardinals

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    We introduce a hierarchy of large cardinals between weakly compact and measurable cardinals, that is closely related to the Ramsey-like cardinals introduced by Victoria Gitman, and is based on certain infinite filter games, however also has a range of equivalent characterizations in terms of elementary embeddings. The aim of this paper is to locate the Ramsey-like cardinals studied by Gitman, and other well-known large cardinal notions, in this hierarchy
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