Sunday, 5.5.13, 00:00-01:00, Hörsaal II, Albertstr. 23b
The pseudointersection number and the tower number
Wednesday, 8.5.13, 16:30-17:30, Raum 404, Eckerstr. 1
Wednesday, 15.5.13, 16:30-17:30, Raum 404, Eckerstr. 1
Unsound ordinals
Wednesday, 15.5.13, 16:30-17:30, Raum 404, Eckerstr. 1
An ordinal zeta is unsound if there are subsets An (n in omega) of it such that as b ranges through the subsets of omega, uncountably many ordertypes are realised by\nthe sets $\bbigcup{n \bin b} An\(.\n\nWoodin in 1982 raised the question whether unsound ordinals\nordinals exist; the answer I found then (to be found in a paper\npublished in the Mathematical Proceedings of the Cambridge Philosophical Society volume 96 (1984) pages 391--411) is this:\n\n\nAssume DC. Then the following are equivalent:\n\ni) the ordinal \)\bomega1^{\bomega + 2}$ (ordinal exponentiation) is unsound\n\nii) there is an uncountable well-ordered set of reals\n\nThat implies that if omega1 is regular and the ordinal mentioned in i) is sound, then omega1 is strongly inaccessible in the constructible universe. Under DC, every\nordinal strictly less than the ordinal mentioned in i) is sound.\n\n\nThere are many open questions in this area: in particular, in\nSolovay's famous model where all sets of reals are Lebesgue measurable,\nis every ordinal sound ? The question may be delicate, as Kechris and Woodin have shown that if the Axiom of Determinacy is true then there\nis an unsound ordinal less than omega_2.\n\n
Resurrecting Ramsey ultrafilters
Wednesday, 29.5.13, 16:30-17:30, Raum 404, Eckerstr. 1