In mathematical logic, he was noted especially for his internal set theory, and views on ultrafinitism and the consistency of arithmetic.

His father, who spoke fluent Russian, stayed in St. Petersburg in connection with issues related to prisoners of war.

After the war, his family returned to Italy and he attended the Liceo Scientifico Giovanni Verga in Rome.

He held a position at Princeton University starting in 1959, attaining the rank of professor there in 1964 and retiring in 2013.

In 1950, Nelson formulated a popular variant of the four color problem: What is the chromatic number, denoted

One of his goals was to extend IST (Internal Set Theory—a version of a portion of Abraham Robinson's non-standard analysis) in a natural manner that includes external functions and sets, in a way that provides an external function with specified properties unless there is a finitary obstacle to its existence.