  We give a fully constructive proof that there is a proper cartesian
$\omega$-combinatorial model structure on the category of simplicial sets,
whose generating cofibrations and trivial cofibrations are the usual boundary
inclusion and horn inclusion. The main difference with classical mathematics is
that constructively not all monomorphisms are cofibrations (only those
satisfying some decidability conditions) and not every object is cofibrant. The
proof relies on three main ingredients: First, our construction of a weak model
categories on simplicial sets, then the interplay with the semi-simplicial
versions of this weak model structure and finally, the use of Kan
Ex$^{\infty}$-functor, and more precisely of S.Moss' direct proof that the
natural map $X \rightarrow $ Ex$^{\infty} X$ is an anodyne morphism, which we
show is constructive when $X$ is cofibrant.
