Peano Axioms Explained: Natural Numbers Built from Set Theory

The Peano axioms are a set of three rules that define the natural numbers using only a starting element and a “successor” function. This article presents two equivalent versions of the axioms (starting at 1 or at 0), builds the natural numbers from pure set theory, verifies that this set-theoretic construction satisfies the axioms, and examines whether the axioms are independent, categorical, and consistent.

Key Takeaways

• A Peano system requires a base element, an injective successor function $s$, and an induction principle; no other structure is assumed.
• The natural numbers can be built entirely from set theory by defining $0=\emptyset$ and $n+1 = n\cup\{n\}$, with $\mathbb{N}$ as the smallest inductive set.
• The three Peano axioms are logically independent: dropping any one of them while keeping the other two allows structures that violate the third.
• Any two systems satisfying the Peano axioms are isomorphic, meaning the axioms are categorical.
• Gödel’s incompleteness results mean the Peano axioms cannot be proven consistent from within themselves; their consistency can only be shown relative to set theory.

What Are the Peano Axioms?

The Peano system in Bloch’s presentation uses a distinguished element $1\in \mathbb{N}$, corresponding to the intuitive set $\mathbb{N} = \{1,2,3,…\}$. An equally valid version starts the count at zero instead, giving $\mathbb{N} = \{0,1,2,3,…\}$.

Under the zero-indexed version, a Peano system is a set $\mathbb{N}$ with a distinct element $0\in \mathbb{N}$ and a function $s:\mathbb{N}\rightarrow \mathbb{N}$ satisfying three conditions:

• There is no $n\in \mathbb{N}$ such that $s(n) = 0$.
• The function $s$ is injective.
• If $G$ is a set with $0\in G$, and $g\in G$ implies $s(g)\in G$, then $G=\mathbb{N}$.

Under this indexing, addition is defined recursively as:

• $n+0 = n$
• $n+s(m) = s(n+m)$

Multiplication is also defined by recursion:

• $n\cdot 0 = 0$
• $n\cdot s(m) = n\cdot m + m$

How Are the Natural Numbers Constructed from Set Theory?

A Peano system can be built using nothing but sets, taking $0$ as the “first element.” The idea is:

$$0 = \emptyset,~1 = \{\emptyset\},~2 = \{\emptyset,\{\emptyset\}\},…$$

In general, $0=\emptyset$, and once $n$ is defined, its successor is $n+1 = n\cup \{n\}$.

More formally, a set $X$ is called inductive if $\emptyset \in X$ and, for every $x\in X$, $x\cup\{x\}\in X$. Set theory takes it as an axiom that an inductive set exists. The natural numbers are then defined as the smallest inductive set:

$$\mathbb{N}=\bigcap\{X~\vert~X~\text{is inductive}\}.$$

This set $\mathbb{N}$ satisfies the Peano axioms with $0 = \emptyset$ and $s(x) = x\cup \{x\}$.

Proof: No Element Maps to Zero

There is no $n\in \mathbb{N}$ such that $s(n) = 0$. If such an $n$ existed, then $n\cup \{n\} = \emptyset$, which would force $n\in \emptyset$. That is impossible, since the empty set has no elements.

Proof: The Successor Function Is Injective

First, every $n\in \mathbb{N}$ is transitive: if $m\in n$, then $m\subseteq n$. Let $X\subseteq \mathbb{N}$ be the set of all transitive elements of $\mathbb{N}$. Showing $X$ is inductive proves $X=\mathbb{N}$.

The empty set $\emptyset$ is transitive, so $\emptyset \in X$. Assume $n\in X$. For any $m\in n\cup\{n\}$, either $m=n$, giving $m\subseteq n\cup\{n\}$, or $m\in n$, and transitivity of $n$ gives $m\subseteq n$. So $X$ is inductive.

Now suppose $s(n) = s(m)$, meaning $n\cup \{n\} = m\cup \{m\}$. If $n\neq m$, then $n\in m$ and $m\in n$. Transitivity gives $m\subseteq n$, and $n\in m$ would then force $n\in n$, which is impossible for any natural number, as shown next.

Let $Y$ be the set of all $n\in \mathbb{N}$ such that $n\notin n$. The empty set belongs to $Y$. Assume $n\in Y$. If $n\cup\{n\}\in n\cup\{n\}$, then either $n\cup\{n\}\in n$ or $n\cup\{n\} = n$, and both lead to contradictions:

• If $n\cup\{n\}\in n$, transitivity of $n$ gives $n\cup\{n\}\subseteq n$, so $n\in n$, contradicting the assumption.
• If $n\cup\{n\} = n$, then $n\in n$, again a contradiction.

So $n\cup\{n\}\in Y$, making $Y$ inductive and therefore $Y=\mathbb{N}$. This proves that $s$ is injective.

Proof: Induction Holds

If $G\subseteq \mathbb{N}$ contains $0$ and is closed under the successor function, then $G=\mathbb{N}$. This follows immediately because $G$ is inductive and $\mathbb{N}$ is defined as the smallest inductive set.

What Is a Natural Number, Philosophically?

Two definitions of natural numbers have now been given: one as any set satisfying the Peano axioms, and one as an explicit set-theoretic construction where, for example, $1 = \{\emptyset\}$. Whether the number 1 “is” literally that set is a philosophical question rather than a mathematical one.

Mathematicians generally care about the properties of the natural numbers rather than their metaphysical nature. Any structure satisfying the Peano axioms is treated as a valid model of the natural numbers. Constructing such a model inside set theory means no assumptions beyond set theory are needed to discuss natural numbers.

How Is Exponentiation Defined for Natural Numbers?

Exponentiation of natural numbers can be defined recursively:

• $m^1 = m$
• $m^{n+1} = m^n \cdot m$

From this definition, the basic exponentiation laws can be proven as an exercise:

• $a^{m+n} = a^m a^n$
• $a^{mn} = (a^m)^n = (a^n)^m$
• $(ab)^m = a^m b^m$

Are the Three Peano Axioms Independent?

The three Peano axioms are logically independent, meaning that assuming any two of them does not let you prove the third. Three examples demonstrate this:

Dropping Induction

Assume the first two axioms hold but not induction. The structure $[1,+\infty)$ with $s(x)=x+1$ has an injective successor function and no element mapping to 1, but induction fails: the subset $\{1,2,3,\dots\}$ satisfies the base and successor conditions without covering the whole structure.

Dropping Injectivity

Assume the first axiom and induction hold but not injectivity. Take $X=\{1,2,\dots,n\}$ with $s(1)=2, s(2)=3,\dots, s(n-1)=n$ and additionally $s(n)=2$. The first axiom holds and induction can be arranged, but $s$ is not injective since $s(n)=s(1)=2$.

Dropping the “No Predecessor for the Base Element” Axiom

Assume injectivity and induction hold but not the first axiom. Take $X=\{1,\dots,n\}$ with $s(1)=2, s(2)=3,\dots, s(n-1)=n$, and $s(n)=1$, forming a cyclic structure. Many theorems about natural numbers carry over to such cyclic contexts, producing examples like cyclic groups and rings.

Is the Peano System Categorical?

A system of axioms is categorical if it determines a unique structure up to isomorphism. The Peano axioms are categorical: every two systems satisfying them are isomorphic, meaning they are equivalent up to renaming elements.

Are the Peano Axioms Consistent?

An axiomatic system is consistent if it cannot be used to prove a contradiction, that is, a statement that is both provably true and provably false. For example, the two statements “$X$ is nonempty” and “$X$ is empty” together form an inconsistent system, since they directly contradict each other.

Kurt Gödel showed that the consistency of a sufficiently strong axiomatic system cannot be proven from within that same system. As a result, the consistency of the Peano axioms can only be demonstrated relative to another system. Constructing the natural numbers inside set theory shows that if set theory is consistent, then the Peano axioms are consistent as well.

The Peano system formalizes the everyday counting numbers, but it also asserts that infinitely many numbers exist. That claim seems obvious for small numbers, yet the existence of extremely large numbers such as $10^{10^{10}}$ is not something anyone can verify physically. Anyone who doubts the existence of such infinite collections must also examine whether the axioms describing them are consistent. The set-theoretic construction shows that, within set theory, a coherent model of the Peano axioms does exist.

Glossary of Terms

• Peano system: a set with a base element and a successor function satisfying three specific axioms, used to formally define the natural numbers.
• Successor function ($s$): a function mapping each natural number to the “next” one.
• Inductive set: a set containing the empty set and closed under the operation $x\mapsto x\cup\{x\}$.
• Injective function: a function that never maps two different inputs to the same output.
• Categorical axiom system: an axiom system whose models are all isomorphic to one another.
• Consistency: the property that an axiom system cannot be used to derive a contradiction.

Frequently Asked Questions

What is the difference between starting the natural numbers at 0 versus 1?

Both choices produce equivalent Peano systems; only the labeling of the base element changes. Starting at 0 requires slightly different recursive definitions for addition and multiplication, using $n+0=n$ and $n\cdot 0=0$ as base cases, compared to starting at 1.

Why construct the natural numbers using sets like $\emptyset$ and $\{\emptyset\}$?

This construction shows that the natural numbers can be built entirely within set theory, without assuming anything extra. It demonstrates that a model of the Peano axioms exists as long as set theory itself is consistent.

Does “$1 = \{\emptyset\}$” mean the number 1 really is that set?

Not necessarily. Mathematicians are primarily concerned with the properties the natural numbers satisfy, namely the Peano axioms, rather than their underlying metaphysical identity. The set-theoretic model is one valid representation among possibly many.

Are the three Peano axioms independent of each other?

Yes. Each axiom can fail while the other two hold, as shown by specific counterexample structures: an infinite half-line without induction, a finite structure with a repeated successor value, and a cyclic structure lacking a true base element.

Can the Peano axioms be proven consistent?

Not from within the Peano system itself. Gödel’s incompleteness results show that a sufficiently strong system cannot prove its own consistency. Consistency can only be shown relative to another system, such as set theory.

What does it mean for the Peano axioms to be categorical?

It means every structure satisfying the Peano axioms is isomorphic to every other such structure. In other words, there is essentially only one “shape” of natural numbers, up to renaming its elements.