Francisco Kibedi: Maximal Saturated Linear Orders

Place: Bahen Centre  (located right beside the Fields institute), Room 1220.

Date and time: Friday 22 May (13:30-15:00)

Speaker: Francisco Kibedi

Title: Maximal Saturated Linear Orders
Abstract:
In his 1907 paper about pantachies (maximal linearly ordered subsets of the space of real-valued sequences partially ordered by eventual domination), Felix Hausdorff poses several questions that he was unable to answer, including a question he labels $(\alpha)$: Is there a pantachie with no $(\omega_1, \omega_1)$-gaps?
Hausdorff knew that CH implies the answer is no; in other words, under CH, a pantachie must have $(\omega_1, \omega_1)$-gaps. However, Hausdorff’s question turns out to be independent of ZFC. We answer question $(\alpha)$ by proving something a bit stronger, namely, Con(ZFC + $\lnot$CH + $\exists$ a maximal saturated linear order of size continuum in the space of real-valued sequences partially ordered by eventual domination). We then extend this result to include Martin’s Axiom — i.e., we prove Con(ZFC + MA + $\lnot$CH + $\exists$  a maximal saturated linear order of size continuum in the space of real-valued sequences partially ordered by eventual domination).

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