Definable well-ordering, the GCH, and large cardinals
Andrew Brooke-Taylor
Kurt Goedel Research Center for Mathematical Logic,
The University of Vienna,
Waehringer Strasse 25
A-1090 Vienna Austria
andrewbt@logic.univie.ac.at
It is known that one may force a definable well-order of the universe to
exist, for example by McAloon's technique of coding into the continuum
function, or using Jensen coding to make the universe constructible from
a real. However, it would be nice to be able to force to obtain a
definably well-orderable universe where the GCH holds and very large
cardinals exist, a situation that cannot be achieved by either of those
methods. We show how one can achieve it, using a class length forcing
iteration which may be constructed so as to preserve a proper class of
such large cardinals as $n$-superstrong, $n$-huge, or
$\kappa^+$-supercompact $\kappa$.