The Peano axioms, formulated by Giuseppe Peano in 1889, provide a rigorous foundation for the natural numbers. These elegant axioms define the natural numbers using just a few simple principles:
- 0 is a natural number
- Every natural number has a successor
- 0 is not the successor of any natural number
- Different natural numbers have different successors
- If a property holds for 0 and holds for the successor of every number that has it, then it holds for all natural numbers
From these simple axioms, we can build all of arithmetic! Addition, multiplication, and even more complex operations can be defined using just these fundamental principles.