  We study the low-regularity (in-)extendibility of spacetimes within the
synthetic-geometric framework of Lorentzian length spaces developed in [KS:17].
To this end, we introduce appropriate notions of geodesics and timelike
geodesic completeness and prove a general inextendibility result. Our results
shed new light on recent analytic work in this direction and, for the first
time, relate low-regularity inextendibility to (synthetic) curvature blow-up.
