  We offer further evidence that discreteness of the sort inherent in a causal
set cannot, in and of itself, serve to break Poincar{\'e} invariance. In
particular we prove that a Poisson sprinkling of Minkowski spacetime cannot
endow spacetime with a distinguished spatial or temporal orientation, or with a
distinguished lattice of spacetime points, or with a distinguished lattice of
timelike directions (corresponding respectively to breakings of
reflection-invariance, translation-invariance, and Lorentz invariance). Along
the way we provide a proof from first principles of the zero-one law on which
our new arguments are based.
