Giuseppe Peano proposed the following five axioms for the natural numbers; they have come to be known as the Peano axioms or Peano postulates. They state:
- There is a natural number 0.
- Every natural number a has a successor, denoted by a + 1.
- There is no natural number whose successor is 0.
- Distinct natural numbers have distinct successors: if a ≠ b, then a + 1 ≠ b + 1.
- If a property is possessed by 0 and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers.
These axioms are sometimes paraphrased differently, starting at 1 instead of 0. These axioms are given here in a second-order predicate calculus form. See first-order predicate calculus for a way to rephrase these axioms to be first-order.
Dedekind proved, in his 1888 book Was sind und was sollen die Zahlen, that any model of the second order Peano axioms is isomorphic to the natural numbers. On the other hand, the last axiom listed above, the mathematical induction axiom, is not itself expressible in the first order language of arithmetic.
If one replaces the last axiom with the schema:
for each first order property P(x) (an infinite number of axioms) then although natural numbers satisfy these axioms, there are other, nonstandard models of arbitrary large cardinality - by the compactness theorem the existence of infinite natural numbers cannot be excluded in any axiomatization; by an "upward Löwenheim-Skolem theorem", there exist models of all cardinalities.
- If P(0) is true and for all x, P(x) implies P(x + 1)
When the axioms were first proposed, people such as Bertrand Russell agreed these axioms implicitly defined what we mean by a "natural number". Henri Poincaré was more cautious, saying they only defined natural numbers if they were consistent. If a proof can exist that starts from just these axioms, and derives a contradiction such as P AND NOT P, then the axioms are inconsistent, and don't really define anything. David Hilbert posed a problem of proving consistency using only finite logic as one of the problems on his famous list. The motivation was to eliminate infinity, by justifying it with this proof, and show that it brings nothing new.
But in 1931, Kurt Gödel in his celebrated second incompleteness theorem showed such a proof cannot exist. It is even impossible to prove consistency of Peano arithmetic while assuming the axioms themselves. Furthermore, we can never prove that any axiom system is consistent within the system itself, if it is at least as strong as Peano's axioms. In 1936, Gerhard Gentzen proved the consistency of Peano's axioms, using transfinite induction.
All mathematicians assume that Peano arithmetic is consistent, although this relies on intuition only. However, early forms of naive set theory also intuitively looked consistent, before the inconsistencies were discovered. This has been a source of confusion for a number of people, especially nonmathematicians.
The point is that we do have to rely on our intuition, and that it brings something new. Roger Penrose has argued that this intuition is what differentiates men from machines, but his arguments are dubious. The modern set theory often considers axioms postulating existence of large cardinals - none of them can be proved within set theory, nor is it possible to prove consistency of these axioms. But mathematicians generally do not exclude the possibility that some of these axiom systems are inconsistent. The intuition here is much less clear than in the case of natural numbers. Some people argue that even Peano arithmetic could be inconsistent - since intuition is not really a reliable source of truth. This argument can be extended and make us doubt even finite logic itself - these questions go back to Kant and his famous Critique of Pure Reason.
Founding a mathematical system upon axioms, such as the above Peano axioms for natural numbers or axiomatic set theory or Euclidean geometry is sometimes labeled the axiomatic method or axiomatics. In Euclidean geometry, axiomatics allows such results as the Banach-Tarski paradox. This states that a ball can be dissected into five parts and re-assembled by Euclidean transformations into two balls of size equal to the original. A simpler approach to a mathematical system is found in Generating arithmetic.