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 $\kappa$.