Peano's Axioms Uniquely Define Natural Numbers
Theorem
Peano's Axioms uniquely define the set of natural numbers.
That is:
- not only do the natural numbers satisfy Peano's Axioms;
- but conversely, any set that satisfies Peano's Axioms also satisfies all the properties held by the set $\N$ of Natural Numbers.
Thus the structure of the set $\N$ of natural numbers is characterised completely by these axioms:
- P1: $\exists 0 \in \N$
- P2: $\forall n \in \N: \exists n' \in \N$
- P3: $\neg \left({\exists n \in \N: n' = 0}\right)$
- P4: $\forall m, n \in \N: n' = m' \implies n = m$
- P5: $\forall A \subseteq \N: \left({0 \in A \land \left({n \in A \implies n' \in A}\right)} \right) \implies A = \N$
These can be expressed in natural language as follows:
- P1: There exists a natural number $0$.
- P2: For every natural number $n$ there exists another, known as the successor of $n$.
- P3: No number has $0$ as its successor.
- P4: If two numbers have the same successor, they are the same number. Or: different numbers have different successors.
- P5: A subset of the natural number with $0$ in it, such that it has the successor of every number in it, is the same set as the natural numbers.
In this context, the element $n'$ denotes the (immediate) successor element of $n$, which (in the context of the natural numbers) is understood as meaning $n + 1$.
Proof
First we note from Equivalence of Peano Axiom Schemas that the axiom schema as defined above is logically equivalent to Peano's axioms.
We have that a Naturally Ordered Semigroup Satisfies Peano's Axioms.
So any algebraic structure that fulfils the requirements for being a naturally ordered semigroup also satisfies Peano's Axioms.
We also have that the Natural Numbers are a Naturally Ordered Semigroup
Thus the natural numbers satisfy Peano's Axioms.
We also have that Peano's Axioms define Naturally Ordered Semigroup.
Finally, we note that up to isomorphism, there is only one naturally ordered semigroup.
And, of course, the Natural Numbers are a Naturally Ordered Semigroup.
So, Peano's Axioms define a unique structure (up to isomorphism), that is the natural numbers.
Hence the result.
$\blacksquare$
Comment
It is apparent that Peano's Axioms are not truly "axiomatic" in the strict sense of the term, as they can be deduced from the construction of the natural numbers from different, more basic and powerful, axiom schemas. However, they are compact and comprehensible and are frequently used as a basis of understanding of the structure of the number systems.
