We explore the foundation of natural arithmetic starting from Peano’s axioms within set theory, revealing an innovative approach to conceptualizing natural numbers. We question the traditional use of ordinals and propose an alternative formulation of the axiom of infinity, which relies on the elementary notion of hereditary sets rather than complex ordinal structures.
This reformulation allows us to establish an essential logical equivalence: the axiom of infinity in its formulation involving the class of finite ordinals, and the existence of a hereditary set, a set-theoretic version of the axiom that avoids ordinal theory and does not rely on the axiom of choice.
By reconstructing Peano arithmetic in this way, we move toward a more intuitive and less constrained understanding of mathematical foundations, aligning with the visions of Georg Cantor and Gottlob Frege on the unification of mathematics through logic and set theory.

1. Founding Natural Arithmetic in Set Theory

If set theory is to serve as the foundation for all mathematics, then it must be possible to encompass within it the first and most elementary of mathematical disciplines: arithmetic, or the theory of numbers. However, like all other sciences, including mathematics itself, arithmetic can only be grounded on indemonstrable principles.

The principles that history has retained for their simplicity and power are the three so-called Peano axioms, which we will present here in a particular form. When we constructed natural arithmetic from these axioms, we implicitly assumed the existence of a set $\mathbb{N}$ of all natural numbers and stated the fundamental postulates regarding the successor function $s: \mathbb{N} \to \mathbb{N}$, which associates to any integer $n$ its successor $s(n) = n + 1$.

In the context of set theory, however, the very existence of this set $\mathbb{N}$, equipped with the function $s$ satisfying these properties, is problematic. We therefore reintroduce this axiomatic framework in a more radical manner, as follows:

Definition 1
A realization of Peano arithmetic is a triple $(E,*,f)$ consisting of a set $E$, an element $ *$ of $E$, and a function $f: E \to E$, satisfying the following properties:
i) $*$ is not in the image of $f$
ii) $f$ is injective (if $x, y \in E$ and $f(x) = f(y)$, then $x = y$)
iii) $f$ is “inductive”: for any subset $S$ of $E$ such that $*\in S$ and $f(x) \in S$ whenever $x \in S$, then $S = E$.

This definition, where the Peano axioms can be recognized for $E = \mathbb{N}$, $* = 0$, and $f = s$, implicitly considers that these axiomatic properties—and thus natural arithmetic and all of classical mathematics—do not depend on the particular elements of the set $\mathbb{N}$ that correspond to the natural numbers of intuition.

In other words, it is the “external structure” of such a realization that matters here. Since the formation of the set $\mathbb{N}$ is not self-evident in set theory, reconstructing natural arithmetic as the cornerstone of the mathematical edifice requires assuming that Peano arithmetic is consistent, i.e., that there exists a realization, a triple $(E, *, f)$ as described in Definition 1.

2. The Realization of Peano Arithmetic through Finite Ordinals

That being said, in set theory, we have a “standard representation” of natural numbers as finite ordinals. These ordinals, which are sets of the form $\mathbf{0} = \emptyset$, $\mathbf{1} = \{\emptyset\} = \{\mathbf{0}\}$, $\mathbf{2} = \{\emptyset, \{\emptyset\}\} = \{\mathbf{0}, \mathbf{1}\}$, and so on, form a class denoted by $\omega$ (“omega”).

A natural “successor” function $s$ on $\omega$ is defined as $s(\mathbf{n}) := \mathbf{n} \cup \{\mathbf{n}\}$, such that for any finite ordinal $\mathbf{n}$, we always have $\mathbf{n+1} := s(\mathbf{n}) = \{\mathbf{0}, \mathbf{1}, \ldots, \mathbf{n}\}$ (a finite ordinal is always the set of its “predecessors”).

It can be shown in an elementary way that the function $s: \omega \to \omega$ satisfies the properties stated as Peano axioms in Definition 1, with $* = \emptyset$. In other words, finite ordinals provide an almost-complete realization of Peano arithmetic, except for the fact that we do not know if the class $\omega$ is a set. This is why natural arithmetic is classically reconstructed in set theory on the basis of the axiom that explicitly states this property, namely the axiom of infinity, according to which there exists an infinite ordinal, and which we present here in a form suitable for natural set theory:

Axiom of Infinity (Ordinal Version)
The class $\omega$ of finite ordinals is a set.

Thus, in set theory, arithmetic is reconstructed through the set $\omega$, which can be identified with the set $\mathbb{N}$ of intuitive integers. From the other axioms of the theory, all known mathematics can also be derived, since the fundamental sets of numbers are reconstructed from the set $\mathbb{N}$, and the theory contains all the processes needed to develop mathematical science from these.

Moreover, it can be shown that the axiom of infinity is equivalent to the consistency of Peano arithmetic: the class $\omega$ is a set if and only if there exists a realization $(E, *, f)$ of arithmetic in the sense of Definition 1. In other words, reconstructing mathematics on this version of the axiom of infinity is merely an option, albeit a natural and convenient one, as it relies on a class whose elements are already well-understood.

3. The Axiom of Choice and the Limits of the Ordinal Approach

If the axiom of infinity is usually formulated in terms of finite ordinals, it is because ordinal theory is perfectly grounded in set theory and does not require invoking mysterious external objects such as the intuitive natural numbers.
However, this theory requires significant elaboration and thus appears to impose an excessive constraint for founding natural arithmetic, which does not deal with ordinals. After all, we only need integers and sets, and operations on both as they are commonly used.

An alternative formulation of the axiom of infinity, closer to mereological intuition—that is, concerning the relationships between wholes and their parts—is possible thanks to the axiom of choice, if we adopt Dedekind‘s characterization of infinite sets: a set $E$ is infinite if and only if there exists a bijection between $E$ and a subset $S$ of $E$ different from $E$.

Proposition 1
Under the axiom of choice, the axiom of infinity is equivalent to the existence of an infinite set.

It is thus possible to obtain an alternative, non-ordinal formulation of the axiom of infinity by adopting the axiom of choice. In principle, ordinal theory is therefore not necessary for the foundation of natural arithmetic: to reconstruct it, it suffices to admit the axiom of choice and the existence of an infinite set!

However, in practice, this abstract version of the axiom of infinity can only be exploited using finite ordinals… We go in circles, and it is therefore necessary to find another way to build arithmetic simply on set theory.

4. Toward an Alternative Formulation of the Axiom of Infinity

If we wish to avoid ordinal theory, it nonetheless provides an example of a “formal” representation (that is, through sets) of natural numbers and the successor function, which can be useful for achieving an abstract version of Peano arithmetic.

Indeed, the class $\omega$ of finite ordinals is characterized by the following two properties:
i) The empty set $\emptyset$ is an element of $\omega$;
ii) For any ordinal $\mathbf{n} \in \omega$, the successor ordinal $\mathbf{n+1} = \mathbf{n} \cup \{\mathbf{n}\}$ is an element of $\omega$.

Now, since ordinals are sets, intuition suggests that the class $\omega$ is the “smallest class” possessing these two properties, which leads us to consider the following sets:

Definition 2
A set $S$ is said to be hereditary if $\emptyset \in S$ and $X \cup \{X\} \in S$ for all $X \in S$.

From this perspective, under the classical form of the axiom of infinity, the class $\omega$ is a hereditary set, and it can be shown that it is included in every hereditary set: it is thus the “smallest” in terms of set inclusion.

In fact, it can be reciprocally demonstrated by the principle of induction that if a hereditary set $S$ exists, then there exists an injective function from $\omega$ into $S$ (so that $\omega$ is a set by the axioms of Replacement and Comprehension).

5. A Natural Realization of Arithmetic

Thus, using the elementary concept of a hereditary set, which does not explicitly involve the notion of ordinals, we arrive at the logical equivalence of the following two properties, without invoking the axiom of choice:

Proposition 2
The class $\omega$ of finite ordinals is a set if and only if there exists a hereditary set.

Since the first clause is precisely the axiom of infinity in its ordinal version, we can now disregard ordinals and adopt the following alternative version of this axiom, formulated within set theory without any reference to a set of intuitive natural numbers:

Axiom of Infinity (Set-Theoretic Version)
There exists a hereditary set, i.e., a set $S$ such that $\emptyset \in S$ and $X \cup \{X\} \in S$ for all $X \in S$.

However, even in this elegantly simple form, this version of the axiom of infinity does not immediately provide a realization of Peano arithmetic as defined in Definition 1. For example, the property of inductiveness (the third property in the list) may fail in a hereditary set $S$.

We must therefore reconstruct a realization of Peano arithmetic based on this new version of the axiom, which we achieve by rebuilding the class $\omega$ of finite ordinals as sets—but this time without knowing what an ordinal is!

Indeed, since under the axiom of infinity $\omega$ is the smallest hereditary set, we simply define it as a formal representation of the set of natural numbers as follows:

Definition 3
The set of (set-theoretic) natural numbers is the intersection, denoted $\mathbb{N} := \bigcap C$, of the class $C$ of all hereditary sets.

This definition assumes, of course, the set-theoretic version of the axiom of infinity, which ensures that the intersection $\bigcap C$, i.e., the class $\{x \in \mathbb{U} : \forall S \in C, \ x \in S\}$ of objects $x$ in the meta-universe such that $x \in S$ for all $S \in C$, is a set (Natural Set Theory, Proposition 3).

Naturally, the set $\mathbb{N}$ is then the set $\omega$, which is itself an element of $C$. However, with this approach, we do not need to know that its elements are finite ordinals to define it, and we can easily prove that $\mathbb{N}$ is a “standard” realization of Peano arithmetic, as stated in the following theorem, which is the true starting point of all mathematics:

Theorem 1
The function $s: \mathbb{N} \to \mathbb{N}$, which associates to a natural number $n$ the natural number $n + 1 := s(n) := n \cup \{n\}$, has the following properties:
i) $0 := \emptyset \in \mathbb{N}$;
ii) If two natural numbers $n$ and $m$ have the same successor ($s(n) = s(m)$), then they are equal ($n = m$);
iii) If $S$ is a subset of $\mathbb{N}$ such that $0 \in S$ and $n + 1 \in S$ whenever $n \in S$, then $S = \mathbb{N}$.

Conclusion

Finally, starting from the elementary axioms of natural set theory and the set-theoretic version of the axiom of infinity, we can now consider the existence of the set $\mathbb{N}$ and the axiomatic properties of the successor function $s: \mathbb{N} \to \mathbb{N}$ as formulated by Peano, as theorems. From these, we can reconstruct all of natural arithmetic, and from there, all of mathematics.

In this way, we fulfill the program of Georg Cantor, to ground the mathematics of infinity in an intuitive set theory encompassing all finite and transfinite (i.e., infinite) numbers, as well as that of his contemporary Gottlob Frege, to ground arithmetic in logic, if we consider that natural set theory is as much a logical theory as it is a mathematical one.