[FOM] "Mathematician in the street" on AC
tomek.bartoszynski at gmail.com
Sun Aug 16 13:29:05 EDT 2009
> That you need AC to get non-measurable sets "everyone
> knows", so I think that the existence of non-measurable
> sets is too surprising a property to get a clear "yes,
> of course that's true" answer.
This indeed seems to be a common perception. However it is false.
Theorem(Shelah) In ZF+DC
If there is an uncountable well ordered set of reals then there is a
Lebesgue nonmeasurable set.
If aleph_1 from the universe is not an inaccessible cardinal in the
constructible universe then we have such a set, and the nonmeasurable
set is in fact definable (possibly with parameters).
This has been proved 25 years ago in
ISRAEL JOURNAL OF MATHEMATICS. Vol. 48. No. 1. 1984. CAN YOU TAKE.
SOLOVAY'S INACCESSIBLE AWAY? . BY. SAHARON SHELAH.
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