  The problem of a rigorous theory of singularities in space-times with torsion
is addressed. We define geodesics as curves whose tangent vector moves by
parallel transport. This is different from what other authors have done,
because their definition of geodesics only involves the Christoffel connection,
though studying theories with torsion. We propose a preliminary definition of
singularities which is based on timelike or null geodesic incompleteness, even
though for theories with torsion the paths of particles are not geodesics. The
study of the geodesic equation for cosmological models with torsion shows that
the definition has a physical relevance. It can also be motivated, as done in
the literature, remarking that the causal structure of a space-time with
torsion does not get changed with respect to general relativity. We then prove
how to extend Hawking's singularity theorem without causality assumptions to
the space-time of the ECSK theory. This is achieved studying the generalized
Raychaudhuri equation in the ECSK theory, the conditions for the existence of
conjugate points and properties of maximal timelike geodesics. Hawking's
theorem can be generalized, provided the torsion tensor obeys some conditions.
Thus our result can also be interpreted as a no-singularity theorem if these
additional conditions are not satisfied. In other words, it turns out that the
occurrence of singularities in closed cosmological models based on the ECSK
theory is less generic than in general relativity. Our work is to be compared
with previous papers in the literature. There are some relevant differences,
because we rely on a different definition of geodesics, we keep the field
equations of the ECSK theory in their original form rather than casting them in
a form similar to general relativity with a modified energy momentum tensor,
