Natural Numbers:Postulates

Formal definitions

Peano's Axioms

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

1. 0 is a natural number.
2. Every natural number has a successor which is also a natural number.
3. 0 is not the successor of any natural number.
4. If the successor of $x$ equals the successor of $y$, then $x$ equals $y$.
5. The axiom of induction: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.

These 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 due to John von Neumann 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