Peano Axioms 


In essence, the axioms define a zero and the concept of a "next number." It then defines what we mean when we say two numbers are equal, and what it means to add and multiply. All arithmetic on the natural numbers can be reduced to working with the axioms, and although it can be instructive to do so, it can also be incredibly tedious.
The arithmetic in this system is called Peano Arithmetic.
Being a formal system we can start to talk about proofs in a formal way. In 1977 Paris and Harrington gave a "natural" example of a statement which is true for the integers but unprovable in Peano arithmetic. This was the first case of an example of Gödel's first incompleteness theorem that wasn't explicitly constructed as part of the proof.
Further reading: