  A precise definition of a black hole has long been absent in causal set
theory. I first show that the local finiteness of the theory cannot be
interpreted as a complete discretization condition, and the theory still admits
continua, which I call singular anti-chains. After providing a couple of
suggestive calculations from general relativity to support the argument that a
singular anti-chain behaves like a causal singularity in a space-time, I
propose a definition of a causal set black hole based on a singular anti-chain.
The paper concludes with brief discussions of questions that causal set black
holes and singularities may give rise to.
