Based on the notions of object and class derived from natural logic, we have redefined the concept of set in an intuitive way, thus establishing a natural set theory without resorting to formal logic. This approach is based on six axioms that form the basis of this fundamental theory. Progressing towards a complete reconstruction of the mathematical universe and a refoundation of our science of infinity, we introduce the four higher, properly mathematical axioms: infinity, choice, foundation and universe. Anchored in an intuitive logicist perspective, they dispense with the need for formal mathematical logic and also allow us to approach natural cumulative hierarchies. These organise the sets above a given base set and, through inaccessible cardinalities, offer concrete examples of mathematical universes. These structures form the basis of our handling of category theory, which describes the large-scale structure of the meta-universe.

The concepts discussed in this article are taken from the scientific publication Natural Set Theory: An Ultimate Foundation for Mathematics.

1 Planting a tree in the universe

1.1 Integrating mathematics into set theory

In laying the foundations of a natural set theory, based on the notions of object and class and without any formal logic, we have adopted an intuitive approach based on the natural language. The whole point of such a theory is to be able to incorporate the whole of mathematics, like an infinite tree planted in the fertile soil of the meta-universe. The 6 axioms of Reality, Pair, Comprehension, Power, Union and Replacement ensure that within the class $\mathbb S$ of all sets, all the usual mathematical definitions and constructions can be unrolled. For the scientific interest of set theory is precisely to be able to describe all mathematical objects, and to make mathematics a science through their rigorous representation by means of structures.

1.2 Reconstructing natural sets using a structural approach

This is why mathematical science only really begins with the reconstruction of natural sets and their arithmetical structure, in particular the set $\mathbb N$ of natural numbers, the set $\mathbb Z$ of integers, the set $\mathbb Q$ of rational numbers and the set $\mathbb R$ of real numbers. The existence of these sets is indeed axiomatic in a way, and their emergence within set theory depends on the adoption of at least one additional hypothesis, for example that of the existence of the first of them, if we want to found arithmetic itself as a science. From there, we can define, i.e. construct, all the others, which has become the traditional approach, and extend the natural arithmetical structure to recover the whole mathematical universe from set-theoretic axioms and operations. In this sense, the axiom of infinity is the seed that enables us to plant the mathematical tree in natural set theory.

2 The axiom of infinity and the natural numbers

2.1 The infinite set of natural numbers

The existence of the set $\mathbb N$ of natural numbers is not self-evident. The intuitive notion of a natural number, although rigorously definable from a philosophical point of view, does not immediately provide a definition or construction of the set $\mathbb N$. We must therefore either admit its existence, with its axiomatic properties, or adopt a representation of the natural numbers in set theory, such as finite ordinals. Knowing that ordinals are sets, and that the notion of a finite set is usually defined in terms of the natural numbers, we end up in a circularity from which we can, however, extricate ourselves using the following original definition:

Definition 1
A natural number is an ordinal for which every finite non-empty subset has a greatest element.

The concept is now perfectly defined mathematically, and we can then define the notion of a finite set on the basis of natural intuition, remembering that for any natural number $n$, the integer interval $[[1,n]]$ is the set of natural numbers $i$ between $1$ and $n$ :

Definition 2
A set $E$ is said to be finite if there exists a natural number $n$ and a bijection $f:E\cong [[1,n]]$.

We then show that a natural number is exactly a finite ordinal as a set, and we can form the class, denoted $\omega$, of natural numbers, an infinite subclass of the class $Ord$ of ordinals. However, there is no a priori guarantee that this class is a set, which is why the 7th axiom, known as the infinity axiom, is introduced here in the following form:

Axiom of Infinity (7)
The class $\omega$ of finite ordinals is a set.

In general, the axiom of infinity is expressed as follows: ‘There exists an infinite ordinal’, but since the existence of $\omega$ is assured as a class, the present formulation fits in better with the dialectic between sets and classes that is the hallmark of this natural theory of sets.
In fact, it is even possible to introduce this axiom without explicitly using the notion of ordinal, in the following form: there exists a set $E$ such that $\emptyset\in E$ and $S\cup{S}\in E$ for all $S\in E$. We recognise here a form of the induction principle of natural arithmetic, and under this hypothesis, the intersection of all sets $E$ having this property is precisely the class $\omega.$

2.2 Infinity and arithmetic

Under the axiom of infinity, the set $\omega$ of finite ordinals is the first explicit example of an infinite set and their archetype, since it serves as a representation of the set $\mathbb N$ of natural numbers. Now, we know how to define the successor of an ordinal $\alpha$ as the ordinal $\alpha+1:=\alpha\cup{\alpha}$, so that the function $s:\omega\to\omega$ which associates to a finite ordinal $n$ the finite ordinal $n+1$ is also a representation of the successor function of the natural numbers. In fact, using set theory alone and without the axiom of infinity, it can be shown that the class $\omega$ of finite ordinals and the function $s$ possess the following axiomatic properties, which serve as the basis of arithmetic itself:

Proposition 1
The function $s:\omega\to\omega$ has the following properties:
i) $0:=\emptyset$ is not the successor of any finite ordinal
ii) If two finite ordinals $n,m$ have the same successor – i.e. if $s(m)=s(n)$ – then $m=n$.
iii) If $C$ is a subclass of $\omega$ which contains $0=\emptyset$ and which contains $s(n)$ as soon as $n\in C$, then $C=\omega$.

Here we recognise Peano’s three axioms, which uniquely determine the arithmetic structure of the set $\mathbb N$, assuming it exists. In fact, it can also be shown, using the recursion theorem adapted to the class $\omega$, that in natural set theory the infinity axiom (7) is equivalent to the logical consistency of Peano’s arithmetic, i.e. essentially to the existence of a representation of the set $\mathbb N$ of natural numbers. In this sense, natural arithmetic as the ‘pillar’ of mathematical science is based in set theory on the axiom of infinity, which is therefore the first properly mathematical axiom of the theory.

3 The axiom of choice and the infinite sets

3.1 The axiom of choice and the characterisation of mathematical infinity

The axiom of infinity states the existence of a certain infinite set, but this set comes with a ‘structure’, essentially the successor function of finite ordinals. If we were to adopt an abstract version of it, i.e. admit the existence of an infinite set, in order to deduce from it the axiom of infinity – or the consistency of arithmetic – we would have to introduce a new axiom, which formulation and applications are always confusing, and which is the following principle of natural set theory:

Axiom of Choice (8)
If $C$ is a class of non-empty sets, then there exists a function $f:C\to \bigcup C$ such that $f(S)\in S$ for all $S\in C$.

Let us rephrase this axiom in order to understand it, noting that the use of the union $\bigcup C$ only comes into play to properly define the function $f$, which values are elements of the elements of $S$. The axiom states that if $C$ is a collection of non-empty sets, then an element of $S$ can be ‘chosen’ from each element $S$ of $C$. This is what is meant by the existence of the function $f$, called a ‘choice function’, because it associates with each set $S\in C$ an element $f(S)$ of $S$. The axiom of choice makes it possible to define many mathematical objects without having to give an explicit construction. In particular, in conjunction with the previous axiom, it enables infinite sets to be characterised both from the set $\omega$ and intrinsically (remember that a subset $S$ of a set $E$ is said to be proper if $S\neq E$):

Theorem 1
i) A set $E$ is infinite if and only if there exists an injective function $f:\omega\hookrightarrow E$.
ii) A set $E$ is infinite if and only if there exists a bijection from $E$ onto an proper subset $S$ of $E$ (Dedekind property).

Dedekind’s characterisation thus allows us to recover in natural set theory the notion of infinity as an intrinsic and positive property, in contrast to the notion of finiteness, which is always either extrinsic (relating to a possible enumeration, Definition 2) or negative (as a negation of Dedekind’s property).

3.2 Enumeration and cardinality

The mathematical notion of infinity must in any case be that of ‘non-infinity’, i.e. the negation of Definition 2: a set is infinite when it is impossible to enumerate it by a natural number. Cantor’s theorem shows that infinite sets do not all have ‘the same number of elements’: there is no bijection between a set $E$ and the set $\mathcal P(E)$ of all subsets of $E$; for example, there are strictly more real numbers than rational numbers. But if it is impossible to ‘count’ infinite sets in the usual sense, how can we distinguish their quantities? Here, the axiom of choice makes it possible to extend enumeration to infinite sets, i.e. to count their elements. This is possible thanks to a recursion principle known as ‘transfinite’, i.e. an extension of the recursion theorem to infinite ordinals, based on the following famous corollary:

Proposition 2 (Zermelo’s theorem)
If $E$ is any set, then there exists a bijection between $E$ and an ordinal $\alpha$.

That being said, the enumeration of all sets using ordinals reveals something new: while all possible ways of counting the elements of a finite set give the same result (the cardinal of the set), the infinite ordinal number associated with the enumeration of an infinite set $E$ depends on this enumeration! This is how the ‘arbitrary’ nature of the axiom of choice becomes apparent in the implementation of Zermelo’s theorem; and this is why the quantity associated with an infinite set $E$ must be chosen from all the ordinals that enable it to be enumerated, as the smallest of them, which is called the cardinal of $E$.

4 The foundation axiom and cumulative hierarchies

With the axioms of infinity and choice, we have everything we need to construct the objects of modern mathematics and state their properties. Having said that, there are still two axioms that are essential to a natural set theory, if it is to be well founded in the logical sense and if it is to be able to integrate category theory, which describes the ‘large-scale structure’ of the mathematical universe.

4.1 Cumulative hierarchies

We have implicitly used the hypothesis that a set cannot be an element of itself, a possibility that arises artificially in Russell’s paradox. More generally, we want to avoid in set theory the existence of membership cycles, i.e. finite sequences of sets $E_1,E_2,\ldots,E_n$ for which $E_1\in E_2\in \ldots E_n\in E_1$, which for transitive sets lead us back to $E_1\in E_1$. Intuitively, eliminating these cycles allows us to consider that the ‘construction’ of sets from basic objects (empty sets or urelements) is done in a hierarchical manner, with clearly identified levels. For example, if $a,b$ are two objects, the sets ${a}$ and ${a,b}$ are at a higher level, and the pair $(a,b)={{a},{a,b}}$ at the next level; similarly, if $E$ is a set, the set $\mathcal P(E)$ of its subsets should be ‘more elaborate’ and therefore live at a higher level. In general, a ‘hierarchy’ of sets associated with a base set (which elements are the elementary constituents) can be defined as follows:

Definition 3
The cumulative hierarchy of sets above a set $E$ is the sequence $(V_\alpha(E))$ of sets, indexed by ordinals, and defined by :
i) $V_0(E)=\emptyset$
ii) $V_\alpha(E)=\mathcal P(E\cup V_\beta(E))$ for any successor ordinal $\alpha=\beta+1$
iii) $V_\alpha(E)=\bigcup{V_\beta(E) : \beta<\alpha}$ for any limit ordinal $\alpha$.

4.2 The axiom of foundation

The cumulative hierarchy of sets above the sets $E$ then forms a class $V(E)=\bigcup{V_\alpha(E):\alpha\in Ord}$ made up of all the sets that the theory allows us to ‘construct’ recursively from $E$ and the axioms of the class $\mathbb S$. Now, given a set $S$, we can ‘decompose’ its structure using its transitive closure, i.e. the smallest transitive set $cl(S)$ containing $S$, which leads us to the following notion:

Definition 4
A set $S$ is said to be constructed over a set $E$ if $cl(S)\subset E\cup \mathbb S$.

This property essentially means that the set $S$ consists only of elements of $E$ and of sets formed from $E$: thus, the elements of $V(E)$ are obviously ‘constructed over $E$’. The converse should be true: if a set is constructed over $E$, we should be able to ‘rank’ it in the class $V(E)$. However, for this to be true, we need to add a new axiom, which guarantees both this possibility and the non-existence of membership cycles:

Axiom of Foundation (9)
For any non-empty set of sets $S$, there exists $x\in S$ such that $x\cap S=\emptyset$.

The foundation axiom implies that there are no infinite descending chains of sets $E_1\ni E_2\ni E_\nildots \ni E_n\ni \ldots$ elements of each other (this is in fact an equivalent statement), and therefore no cycles, and we deduce the following equivalent formulation in terms of cumulative hierarchies:

Theorem 2
If $E$ is a transitive set, then any set $S$ constructed above $E$ is an element of $V(E)$.

5 The universe axiom and the categories

All the axioms introduced so far form a ‘complete’ natural set theory, not in the logical sense that any statement in it would be decidable, but in the sense that we have introduced everything necessary for an integral recapitulation of mathematics as a science of infinity in a class ontology rooted in natural logic. That said, category theory, which revolutionised mathematics for a second time in the 20th century, takes the structuralist approach to the extreme, providing the means to study mathematical objects at a higher level. Categories are perfectly well definable as sets in natural set theory, but the modular nature of the constructions they allow leads us to extend these foundations to a tenth axiom, which allows us to embrace them in their algebraic and topological sophistication.

5.1 Grothendieck’s universes

The meta-universe $\mathbb U$ of all objects falls under the following concept, which generalises in our context – where we now accept the axiom of foundation – the notion of (Grothendieck) universe:

Definition 5
A Grothendieck class is a transitive class $U$ satisfying the following properties:
i) $\emptyset\in U$.
ii) For all objects $a,b\in U$, we have ${a,b}\in U$.
iii) For any family $f:I\to U\cap \mathbb S$ of sets of $U$ indexed by $I\in U\cap \mathbb S$, the set $\bigcup f$ is an element of $U$.
iv) For any set $S$ of $U\cap \mathbb S$, the set $\mathcal P(S)$ of the parts of $S$ is an element of $U$.
The class $U$ is called a (Grothendieck) universe if it is a set.

We have omitted the foundation property, which is redundant with the axiom. A Grothendieck class is therefore essentially a class in which the basic operations of set theory are permitted. If the meta-universe is the ultimate natural example, there are a priori many other distinct Grothendieck classes, the typical example of which is given by cumulative hierarchies:

Proposition 3
For any transitive set $M$, the class $V(M)\cup M$ is a Grothendieck class.

We distinguish in particular the case where $M=\emptyset$, with $V(\emptyset)\cup \emptyset=V$, the ‘standard’ cumulative hierarchy of all ‘pure’ sets, i.e. which transitive closure contains no urelement (a primitive object of the meta-universe – see Natural Set Theory, Definition 2).

5.2 The universe axiom

From the point of view of category theory, the most interesting situation is when we can work with Grothendieck universes: we can then define categories of structures associated with a given universe, and therefore associated categories of functors, higher category structures, etc. The typical situation is where the objects we are working with are all included in the same set $E$, which we want to incorporate into a universe. To do this, we need to introduce our tenth and final axiom (for now):

Axiom of the Universe (10)
For any set $E$, there exists a Grothendieck universe $U$ such that $E\in U$.

In this situation, can we explicitly ‘construct’ a Grothendieck universe from a given set, as we did with cumulative hierarchies? The answer is yes, but only if we accept the existence of a certain type of cardinal number, which corresponds to the following concept:

Definition 6
A cardinal number $\kappa$ (‘kappa’) is said to be (strongly) inaccessible if:
i) $\kappa>\omega$
ii) For any family $f:I\to \mathbb S$ of sets such that the cardinal of $I$ and the cardinal of $f(i)$ for any $i$ are $<\kappa$, the cardinal of the union $\bigcup f$ is $<\kappa$ (we say that $\kappa$ is regular)
iii) For any cardinal $\lambda<\kappa$, we have $2^\lambda<\kappa$ (we say that $\kappa$ is (strongly) limit) – here $2^\lambda$ is the cardinal of $\mathcal P(\lambda)$.

Thus, if $\kappa$ is an inaccessible cardinal, then it can be shown that for any transitive set $M$ of cardinal $<\kappa$, the set $V_\kappa(M)\cup M$ is a Grothendieck universe. Given any set $E$, if there exists an inaccessible cardinal $\kappa$ strictly greater than that of $E$, and therefore than that of its transitive closure $M=cl(E)$, we can explicitly construct a Grothendieck universe $U=V_\kappa(M)\cup M$ which contains $E$! But the existence of such cardinal numbers is not guaranteed by the first nine axioms of the theory, and is in fact equivalent to the axiom of the universe:

Proposition 4
The axiom of the universe is equivalent to the following property: for any cardinal number $\lambda$, there exists an inaccessible cardinal $\kappa$ such that $\lambda < \kappa$.

In the end, Number, to which Magnitude and Form are reduced, continues to reign over mathematics even in the abstraction of set theory: as with the axiom of infinity, a property of higher structure like the axiom of the universe consists essentially in the existence of certain numbers.