Eventual failure detectors, such as Ω or ♦P, can make arbitrarily many mistakes before they start providing correct information. This paper shows that any detector implementable in a purely asynchronous system can be implemented as a function of only the order of most-recently heard-from processes. The finiteness of this representation means that eventual failure detectors can be enumerated and their relative strengths tested automatically. The results for systems with two and three processes are presented. Implementability can also be modelled as a game between Prover and Disprover. This approach not only speeds up automatic implementability testing, but also results in shorter and more intuitive proofs. I use this technique to identify the new weakest failure detector anti-Ω and prove its properties. Anti-Ω outputs process ids and, while not necessarily stabilizing, it ensures that some correct process is eventually never output.