Proof that a Program Could Reproduce Itself

In December 1949 mathematician John von Neumann delivered lectures at the University of Illinois at Urbana-Champaign on the  Theory of Self-Reproducing Automata. In these lectures, which built upon lectures that he gave in Pasadena in 1948, von Neumann showed that in theory a program could reproduce itself. He described what came to be known as the von Neumann Universal Constructor, a self-replicating machine in a cellular automata (CA) environment. After von Neumann's death the lectures were completed and edited by A. W. Burks and published by the University of Illinois Press in 1966. 

Years later one application of this plausibility result in computability theory was the development of what came to be known as malware.