BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Martin Goldstern (Technical University Vienna\, Austria) DTSTART:20211207T200000Z DTEND:20211207T210000Z DTSTAMP:20260423T004133Z UID:PALS/20 DESCRIPTION:Title: Ca rdinals below the continuum\nby Martin Goldstern (Technical University Vienna\, Austria) as part of PALS Panglobal Algebra and Logic Seminar\n\n \nAbstract\nGeorg Cantor's "Continuum Hypothesis" (CH) postulates that eve ry\ninfinite set S of reals is either countable or equinumerous with\nthe set of all reals. Using the axiom of choice this means that\nthe "continu um" (the cardinality of the set of reals) is equal\nto aleph1\, the smalle st uncountable cardinal.\n\nDavid Hilbert's first problem asked if CH is t rue\; we know now that\nneither CH nor non-CH can be proved from the usual axioms of\nset theory (ZFC). Paul Cohen's method of forcing allows us\nt o build universes (structures satisfying ZFC) where the continuum\nis arbi trarily large. \n\nThere are many relatives of the continuum\, such as th e answers\nto these questions: How many nulls sets (Lebesgue measure zero) \ndo we need to cover the real line? How many points do we need\nto get a non-null set? How many sequences (or convergent series)\ndo we need to ev entually dominate all sequences (convergent series)?\netc.\nAll these card inals are located in the closed interval\nbetween aleph1 and the continuum .\n\nIn my talk I will present some of these cardinals and hint\nat the me thods used to construct universes where these cardinals\nhave prescribed v alues\, or satisfy strict inequalities.\n LOCATION:https://researchseminars.org/talk/PALS/20/ END:VEVENT END:VCALENDAR