Natural Numbers:Postulates

Formal definitions

Many properties of the natural numbers can be derived from the five Peano axioms:

Peano's Axioms for the Natural Numbers

1. 1 is a natural number.
2. For every natural number $n$, the successor to $n$, ($n+1$), is also a natural number.
3. 1 is not a successor to any natural number.
4. If two numbers $n_{1}$ and $n_{2}$ have the same successor, then $n_{1}=n_{2}$.
5. If a set $S$ contains 1, and also contains the successor of every element $n$ in $S$, then every natural number is in $S$.

These axioms are used to build the set of natural numbers $\mathbb {N} =\{1,2,3,...,n,n+1,...\}$. They are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of $x$ is $x+1$. Replacing axiom 5 by an axiom schema, one obtains a (weaker) first-order theory called Peano arithmetic.

Constructions based on set theory

Von Neumann ordinals

In the area of mathematics called set theory, a specific construction defines the natural numbers as follows:

• Set $\emptyset =\{\}$, the empty set,
• Define $S(a)=a\cup \{a\}$ for every set $a$. $S(a)$ is the successor of $a$, and $S$ is called the successor function.
• By the axiom of infinity, there exists a set which contains 0 and is closed under the successor function. Such sets are said to be inductive. The intersection of all such inductive sets is defined to be the set of natural numbers. It can be checked that the set of natural numbers satisfies the Peano axioms.
• It follows that each natural number is equal to the set of all natural numbers less than it:

$0=\{\}$,

$1=0\cup \{0\}=\{0\}=\{\{\}\}$,

$2=1\cup \{1\}=\{0,1\}=\{\{\},\{\{\}\}\}$,

$3=2\cup \{2\}=\{0,1,2\}=\{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}$,

$n=n-1\cup \{n-1\}=\{0,1,...,n-1\}=\{\{\},\{\{\}\},...,\{\{\},\{\{\}\},...\}\}$, etc.

With this definition, a natural number $n$ is a particular set with $n$ elements, and $n\leq m$ if and only if $n$ is a subset of $m$. The standard definition, now called definition of von Neumann ordinals, is: "each ordinal is the well-ordered set of all smaller ordinals."

Also, with this definition, different possible interpretations of notations like ℝⁿ (n-tuples versus mappings of n into ℝ) coincide.

Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.

Zermelo ordinals

Although the standard construction is useful, it is not the only possible construction. Ernst Zermelo's construction goes as follows:

• Set $0=\{\}$
• Define $S(a)=\{a\}$,
• It then follows that

• $0=\{\}$,
• $1=\{0\}=\{\{\}\}$,
• $2=\{1\}=\{\{\{\}\}\}$,
• $n=\{n-1\}=\{\{\{...\}\}\}$, etc.

Each natural number is then equal to the set containing just the natural number preceding it. This is the definition of Zermelo ordinals. Unlike von Neumann's construction, the Zermelo ordinals do not account for infinite ordinals.

Resources

• Course Textbook, pages 196-201