Abstrakt
We show first that it is consistent that $\kappa$ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of $H(\kappa^+)$. Then with further forcing we show that it is consistent that GCH fails at $\aleph_\omega$, $\aleph_\omega$ strong limit, while there is a lightface definable wellorder of $H(\aleph_{\omega+1})$ (''definable failure'' of the singular cardinal hypothesis at $\aleph_\omega)$.
The large cardinal hypothesis used is the existence of a $\kappa^{++}$-strong cardinal, where $\kappa$ is $\kappa^{++}$-strong if there is an embedding $j:V \to M$ with critical point $\kappa$ such that $H(\kappa^{++}) \sub M$. By work of M.~Gitik and W.~J.~Mitchell \cite{GITIKo2}, \cite{MITcoreI}, our large cardinal assumption is almost optimal.
The techniques of proof include the ''tuning-fork'' method of \cite{FRIEDMANperfect} and \cite{FRDOBtree}, a generalisation to large cardinals of the stationary-coding of \cite{FRFprojective} and a new ''definable-collapse'' coding based on mutual stationarity. The fine structure of the canonical inner model $L[E]$ for a $\kappa^{++}$-strong cardinal is used throughout.