The full basis theorem does not imply analytic wellordering

Abstract

We make use of a finite support product of ${\omega }_1$ clones of the Jensen minimal ${\Pi }_2^1$ singleton forcing to obtain a model of ZFC in which every non-empty lightface analytically definable set of reals contains a lightface analytically definable real (the full basis theorem), but there is no analytically definable wellordering of the continuum.