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This study is concerned with the question if existence is evidence of eternal recurrence, that a current observer is within a cyclic world, if the past is infinite. Michael Huemer proposed a proof of existence being evidence of immortality using a Bayesian approach, which is discussed, as well as various counter arguments. This study then uses transition systems, a non-Bayesian approach, to prove various results about worlds that can be described by them. It is proved that in transition systems with an infinite past, where time can be discretely subdivided, eternal recurrence is the case for every observer in a world described by such a system. Finally, the reasoning, potential and actual counter arguments, consequences, and future research are considered.