John Barkley Rosser Sr. (December 6, 1907 – September 5, 1989) was an American logician, a student of Alonzo Church, and known for his part in the Church–Rosser theorem, in lambda calculus. He also developed what is now called the Rosser sieve, in number theory. He was later director of the Army Mathematics Research Center at the University of Wisconsin–Madison. Rosser wrote mathematical textbooks as well.
In 1936, he proved Rosser’s trick, a stronger version of Gödel’s first incompleteness theorem which shows that the requirement for ω-consistency may be weakened to consistency. Rather than using the liar paradox sentence equivalent to « I am not provable, » he used a sentence that stated « For every proof of me, there is a shorter proof of my negation ».
In prime number theory, he proved Rosser’s theorem.
The Kleene–Rosser paradox showed that the original lambda calculus was inconsistent.