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CATEGORIES:ok261's list
SUMMARY:Unsound ordinals - A.R.D. Mathias
DTSTART;TZID=Europe/London:20130513T110000
DTEND;TZID=Europe/London:20130513T120000
UID:TALK45335AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/45335
DESCRIPTION:An ordinal zeta is *unsound* if there are subsets
A_n (n in omega) of it such that as b ranges throu
gh the subsets of omega\, uncountably many orderty
pes are realised by the sets\n$\\bigcup_{n \\in b}
A_n$.\n\nWoodin in 1982 raised the question wheth
er unsound ordinals ordinals exist\; the answer I
found then (to be found in a paper published in 19
84 in Math Proc Cam Phil Soc) is this:\n\nAssume D
C. Then the following are equivalent:\n\ni) the or
dinal $\\omega_1^{\\omega + 2}$ (ordinal exponenti
ation) is unsound\n\nii) there is an uncountable w
ell-ordered set of reals\n\nThat implies that if o
mega_1 is regular and the ordinal mentioned in i)
is sound\, then omega_1 is strongly inaccessible i
n the constructible universe. Under DC\, every ord
inal strictly less than the ordinal mentioned in i
) is sound.\n\nThere are many open questions in th
is area: in particular\, in Solovay's famous model
where all sets of reals are Lebesgue measurable\,
is every ordinal sound ? The question may be del
icate\, as Kechris and Woodin have shown that if t
he Axiom of Determinacy is true then there\nis an
unsound ordinal less than omega_2.
LOCATION:MR13
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