Kernel
Gödel's 1931 incompleteness theorems showed that any sufficiently expressive consistent mathematical system contains true statements it cannot prove. The result reset twentieth-century philosophy of mathematics and made the limits of formal reasoning a quantitative question rather than a metaphysical one.
Contribution
The two incompleteness theorems. The completeness theorem for first-order logic (1929). The consistency-of-the-continuum-hypothesis result (1940). Late-career platonist work on the nature of mathematical truth. The cosmological model with rotating dust that admits closed timelike curves (1949).
Civilization-scale significance
The figure through whom the question "what can be known?" was sharpened from a philosophical question into a theorem. The post-1931 understanding of mathematics, of computer science, and of the foundations of physics is all post-Gödelian.