Alonzo Church, building on the work of Kurt Godel, showed that no algorithm can decide in general whether statements in number theory are true or false. Formally, we write (N,+,X) to be the model whose universe is the natural numbers with the usual + and X relations. Church showed that TH(N,+,X), the theory of this model, is undecidable. (Note: Th(N,+) is decidable)

This has great importance philosophically because it demonstrates that mathematics cannot be mechanized.

 (Theory of Computation, Michael Sipser)