Original: One of the most significant logicians of all time , Gödel made an immense impact upon scientific and philosophical thinking in the 20th century , a time when many , such as Bertrand Russell , A. N. Whitehead and David Hilbert , were pioneering the use of logic and set theory to understand the foundations of mathematics .
Simplified: The more famous incompleteness theorem says that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers ( Peano arithmetic ) , there are true propositions about the naturals that can not be proved from the axioms .