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SUMMARY:Unsound ordinals - A.R.D. Mathias
DTSTART:20130513T100000Z
DTEND:20130513T110000Z
UID:TALK45335@talks.cam.ac.uk
CONTACT:31089
DESCRIPTION:An ordinal zeta is *unsound* if there are subsets A_n (n in om
 ega) of it such that as b ranges through the subsets of omega\, uncountabl
 y many ordertypes are realised by the sets\n$\\bigcup_{n \\in b} A_n$.\n\n
 Woodin in 1982 raised the question whether unsound ordinals ordinals exist
 \; the answer I found then (to be found in a paper published in 1984 in Ma
 th Proc Cam Phil Soc) is this:\n\nAssume DC. Then the following are equiva
 lent:\n\ni) the ordinal $\\omega_1^{\\omega + 2}$ (ordinal exponentiation)
  is unsound\n\nii) there is an uncountable well-ordered set of reals\n\nTh
 at implies that if omega_1 is regular and the ordinal mentioned in i) is s
 ound\, then omega_1 is strongly inaccessible in the constructible universe
 . Under DC\, every ordinal strictly less than the ordinal mentioned in i) 
 is sound.\n\nThere are many open questions in this area: in particular\, i
 n Solovay's famous model where all sets of reals are Lebesgue measurable\,
  is every ordinal sound ?  The question may be delicate\, as Kechris and W
 oodin have shown that if the Axiom of Determinacy is true then there\nis a
 n unsound ordinal less than omega_2.
LOCATION:MR13
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