We study exchangeable, Markov survival processes – stochastic processes giving rise to infinitely exchangeable non-negative sequences $(T_{1},T_{2},\ldots)$. We show how these are determined by their characteristic index $\{\zeta_{n}\}_{n=1}^{\infty}$. We identify the harmonic process as the family of exchangeable, Markov survival processes that compose the natural set of statistical models for time-to-event data. In particular, this two-dimensional family comprises the set of exchangeable, Markov survival processes with weakly continuous predictive distributions. The harmonic process is easy to generate sequentially, and a simple expression exists for both the joint probability distribution and multivariate survivor function. We show a close connection with the Kaplan-Meier estimator of the survival distribution. Embedded within the process is an infinitely exchangeable ordered partition. Aspects of the process, such as the distribution of the number of blocks, are investigated.