The revolution in mathematics is that of set theory, which responds both to the problem of a universal and rigorous conceptual language and to that of a single foundation for all mathematical disciplines. Although set theory was Cantor’s original work, the limitations of the initial theory led to a sophisticated, axiomatic version, written in the formal logic of the early 20th century. We are thus accustomed to thinking that the ultimate foundation of mathematics lies in a formal axiomatic theory that distances us from the intuitive simplicity of the concept of set and from everyday mathematical practice. This is in fact not the case, and by learning from axiomatic set theory and simply analysing its own logical presuppositions, it is possible to reconstruct an intuitive, natural and rigorous foundation for the whole of mathematics.
The concepts discussed in this article are taken from the scientific publication Natural Set Theory: An Ultimate Foundation for Mathematics.
1 The advent of set theory
1.1 Axiomatic set theory
Axiomatic set theory is one of the major mathematical and scientific achievements of the 20th century, and of the history of science in general. Since the advent of this very particular branch of mathematical logic, which developed in close relation to it, set theories such as ZF (‘Zermelo-Fraenkel’, named after the mathematicians Enrst Zermelo and Abraham A. Fraenkel) or ZFC (‘ZF’ with the added axiom of choice), with possible additions or modifications, are today considered by a vast majority of mathematicians as a satisfactory ultimate foundation for all mathematics, classical and modern.
1.2 The limits of Cantor’s intuitive theory
Now, these axiomatic theories are refinements of Georg Cantor‘s set theory, the father of the discipline and the inventor of a revolutionary intuitive theory that nevertheless suffered from a number of fatal flaws. Among these, certain paradoxes linked to ‘self-reference’ were brought to light, first by Cantor himself and then by other mathematicians. These include Russell’s paradox (there cannot be a set of all sets), Burali-Forti’s paradox (there cannot be a set of all ordinal numbers) and Cantor’s paradox (there cannot be a set of all cardinal numbers). Moreover, an intuitive theory of sets does not allow the question of their definition to be dealt with mathematically.
1.3 The contribution of formal mathematical logic
So it was thanks to predicate calculus, a kind of ABC of nascent mathematical logic, that it was possible to build a set theory free of these restrictions. This ‘first-order logic’ provides a rigorous mathematical description of both simple mathematical ‘structures’ and the ‘language’ used to describe them. Like polynomial theory, of which it is an analogue, it provides a representation of the linguistic objects used in mathematics and their relationship to the mathematical objects they denote. It is therefore possible, thanks to formal logic, to take Cantor’s ‘naive’ – i.e. intuitive – theory and treat it as a mathematical object in its own right, whose syntactic formations and meanings we control.
2 The problem of the foundations of mathematical science
2.1 The relationship between formal theories and an intuitive basis
Yet predicate calculus itself is a mathematical theory, based on the very notion of set, and must therefore be founded on an intuitive set theory. It might seem that there is a certain circularity here; however, the sets of the axiomatic theories written in first-order logic are not the sets of the intuition on which the latter is based: they are elements of certain ‘models’ of these theories. Thus, formal axiomatic theories such as ZF or ZFC, and all those of this type, cannot in themselves represent an ultimate mathematical foundation, i.e. one beyond which it is not possible to go back, because they rest in the final analysis on an intuitive basis which they cannot replace.
2.2 The metamathematical interest of formal theories
That being said, axiomatic set theory has a number of advantages. On the one hand, it has greater expressive power than naive set theory. For example, it can be used to define all the ordinal and cardinal numbers in each ‘universe’, as well as a cumulative hierarchy of sets, and to formalise mathematical definitions in general, so that their relationship with the sets they describe can be conceptualised. On the other hand, the mathematical treatment of axioms and theories themselves makes it possible to study in a scientific way questions of independence of certain axioms (whether or not they are determined by the rest of the theory) or questions of relative consistency (is the addition of one or more axioms to a theory – logically – consistent with it?). These are sometimes referred to as ‘metamathematical’ questions, which are difficult or even impossible to tackle using an intuitive approach.
2.3 The limits of formal theories for foundational questions
However, the absolute logical consistency of these theories cannot be demonstrated in general. In other words, it is never possible to establish rigorously that they describe a genuine universe of sets, and therefore that they correspond to any kind of reality! This means that this consistency must be accepted, and that logical formalism provides no additional justification for these theories in relation to their natural intuitive foundation: the mathematical edifice is always based on an act of faith in its first principles… Thus, in terms of logical consistency, the formal axiomatic method offers no advantage over naive set theory. Furthermore, the formalism of mathematical logic – not to be confused with the usual mathematical symbolism! – does not correspond to current mathematical practice, which consists of working with the sets of intuition, and the symbolism applied to the logic of natural language.
3 Principles of a natural set theory
3.1 Towards a higher intuitive theory with no formal logic
Because of the limitations of both naive set theory and formal axiomatic set theory, it is necessary to look for an alternative way of founding mathematics. It seems natural to demand that such a solution have both the same intuitive character as the original theory – in order to reflect mathematical practice – and the same power as the formal theories; but does such a solution exist? Paradoxically, it is essentially an idea of Kurt Gödel’s to distinguish between sets and classes in the formal axiomatic theory NBG (‘von Neumann-Bernays-Gödel’), which makes it possible for a theory of this kind to be constructed, without logical formalism. Gödel simply defines a set as a class that is an element of another class, and by transposing his idea to natural logic, we can free ourselves from the formalism of predicate calculus by reintroducing the notion of object, which still appears in Zermelo’s work. It is on this basis that we proposed a refoundation of mathematics in a ‘natural set theory’ in Natural set theory: an ultimate foundation for mathematics.
3.2 Extensional logic and class theory
As summarised in Figure 1, all set theory is based on extensional logic. In natural logic, certain concepts give rise to extensions, i.e. the multiplicity of objects possessing the properties that define a concept. For example, the concept ‘Parisienne’ has as its extension the class of all the women living in Paris – if these ladies will forgive us for considering them, on a purely philosophical basis, as ‘objects’. Thus, the intuitive notion of ‘class’ corresponds to that of ‘concept’, and it is on this basis that we intuitively understand the notion of ‘set’. It is therefore from natural logic that we propose to start, with the following definition:
Definition 1
A class is a multiplicity of objects which we call its elements.
All the terms in this definition are understood in an intuitive sense! We are returning to Cantor’s approach, which defined sets without any formal logic as ‘collections of objects that we assemble into a whole’, but this poses no problem insofar as it is the distinction between objects and classes, and their relationship, that are essential on the conceptual level. If $a$ is an object and $C$ is a class, we note as usual ‘$a\in C$’ the property ‘$a$ is an element of $C$’. In principle, all objects can be assembled into a single class; at least this is our intuition, and since it is not obvious that any concept can give rise to a class, we will admit this as the first axiom of the theory:
Axiom of Reality (1)
There exists a class containing all objects as elements.
We will call $\mathbb U$ the meta-universe, in other words this class of all objects: it is the universe of mathematical discourse. This class is well defined, in other words it is unique, by the principle of extensionality, which fully describes the equality of classes according to their elements:
Principle (Extensionality)
Two classes $C$ and $D$ are equal if and only if they have the same elements.
3.3 The definition of sets… and Russell’s ‘paradox’
Any serious student of modern mathematical logic would chuckle to hear of a definition of the notion of set: formal mathematical logic now defines sets as ‘points’ of a ‘model’ of a theory such as ZF or ZFC. We think we were that serious student, but having become suspicious of academic seriousness we have found it possible to define sets again. For Gödel, in NBG a set is ‘a class which is an element of another class’; this is the definition we could adopt, and which turns out to be equivalent to the following solution thanks to the axiom of reality :
Definition 2
i) A set is a class which is also an object (or an object which is also a class).
ii) A proper class is a class which is not a set (i.e. which is not an object).
iii) A urelement is an object of the meta-universe which is not a set (i.e. a primitive element).
Obviously, since objects are supposed to be the elements of the classes, a class that is an element of another class is a set. Reciprocally, a set is an object, and therefore an element of $\mathbb U$: we recover the Gödelian definition at the level of natural intuition, by reintroducing the concept of object! In natural logic, the notion of a set can therefore be seen as an ambivalent concept: as a class, the set (possibly) has elements, and as an object, it can itself be an element.
The distinction between sets and classes appears classically in Russell’s paradox: there can be no set of all sets. This is one of the reasons why we distinguish them here, and we can then form the class of all sets, which we will denote $\mathbb S$, and which no longer poses a problem since we did not admit that any class could be an element of another class; only, the class $\mathbb S$ is now a proper class, just like the meta-universe $\mathbb U$… So we now have a ‘universe’ containing all sets, with no formal logic and no paradoxes.
3.4 Subclasses and Boolean operations
If the classes of the theory are the abstract counterparts of the concept extensions of natural logic, the logical relations and operations must be represented in them, starting with the inclusion between two classes:
Definition 3
We say that a class $C$ is a subclass of a class $D$ (or that $C$ is included in $D$), if every element of $C$ is an element of $D$, which is denoted by $C\subset D$.
Thus, by definition, every class is a subclass of the meta-universe $\mathbb U$! We then adopt the operations of natural logic for all classes:
Definition 4
Let $C$ and $D$ be two classes.
i) The intersection of $C$ and $D$ is the class noted $C\cap D$ and which elements are all the objects which are at the same time elements of $C$ and elements of $D$.
ii) The union of $C$ and $D$ is the class noted $C\cup D$ and which elements are all the objects which are elements of $C$ or elements of $D$.
iii) The difference of $C$ and $D$ is the class noted $C-D$ and which elements are all the objects which are elements of $C$ but not elements of $D$.
iv) In particular, if $D$ is a subclass of $C$, the difference $C-D$ is a subclass of $C$, called the complement of $D$ in $C$.
Inclusion and Boolean operations are obviously valid for sets as particular classes, but also in a specific way, which is an opportunity to get to the heart of the theory by tackling the first axioms concerning the class $\mathbb S$.
4 The basic axioms of the theory
Having set out the essential definitions and principles of the class theory in which our natural set theory is embedded, we must now introduce the elementary axioms of the theory, i.e. the unprovable assertions that we accept about the class $\mathbb S$ of all sets.
4.1 The comprehension and power axioms
Firstly, if $E$ is a set and $C$ is a subclass of $E$, is $C$ a set? If intuition suggests that proper classes are too ‘big’ to be sets, then subclasses of sets should be sets… However, to ensure this, we need to introduce a specific axiom, known as the comprehension axiom, the second axiom of the theory:
Axiom of Comprehension (2)
If $C$ is a class and $E$ is a set, then the class $C\cap E$ is a set.
By considering classes as extensions of properties, this axiom implicitly states that ‘the class of elements of a set having a certain property is itself a set’. It is equivalent to the following property:
Proposition 1
Every subclass of a set $E$ is a set, called a subset of $E$.
Since any subset $S$ of $E$ is a set we can then define the class $\mathcal P(E)$ of all subsets of $E$, as a subclass of $\mathbb S$. But again there is no guarantee that this class is itself a set, which requires a third axiom:
Power Set Axiom (3)
If $E$ is a set, then the class $\mathcal P(E)$ of all subsets of $E$ is a set.
Thus, the subclasses of a set $E$ are sets, which form a set $\mathcal P(E)$ (the ‘power set of $E$’): we would say in a more ‘algebraic’ way that the class $\mathbb S$ of all sets is ‘stable’ under subclasses and under power set formation.
4.2 The pairing axiom and Cartesian products
The set-theoretic representation of mathematical concepts is essentially based on the notions of set, relation and function. To extend the concept of set to relations and functions, we need to be able to produce products of any two sets. In fact, the product of any two classes always exists, but must be defined on the basis of (ordered) pairs of objects, which brings us back to the following concept
Definition 5
If $a$ and $b$ are two objects, the pair $a,b$ is the class, denoted $\{a,b\}$, which has $a$ and $b$ as its only elements.
The pair $\{a,b\}$ always exists as an explicitly described subclass of the meta-universe $\mathbb U$. This is intuitively a finite class, and intuition also suggests that these should be sets. At this stage however we cannot prove this, and to define products in general we need to adopt a new axiom:
Pairing Axiom (4)
If $a$ and $b$ are two objects, then the pair $\{a,b\}$ is a set.
When $a=b$, we denote $\{a\}$ the pair $\{a,b\}$: it is the ‘singleton $a$’, a set the only element of which is $a$. From pairs of objects, we can define ordered pairs, and from there unambiguously the Cartesian product of two classes:
Definition 6
i) If $a$ and $b$ are two objects, the ordered pair $a,b$ is the pair $\{\{a\},\{a,b\}\}$, denoted $(a,b)$.
ii) If $C$ and $D$ are two classes, the (Cartesian) product of $C$ and $D$ is the class denoted $C\times D$ and consisting of all ordered pairs $(a,b)$ for which $a\in C$ and $b\in D$.
We define the Cartesian product from ordered pairs rather than pairs, because the definition of pairs is insufficient: while two ordered pairs $(a,b)$ and $(a‘,b’)$ are equal if and only if $a=a‘$ and $b=b’$, this is not true for the pairs $\{a,b\}$ and $\{a’,b’\}$.
Note that we can form the product $\mathbb U\times \mathbb U$, which is as such a subclass of $\mathbb U$, and even of $\mathbb S$: the idea seems counter-intuitive, but corresponds well to the intuition underlying the properties of infinite sets $E$, where $E\times E$ is always represented as a subset of $E$.
4.3 The axiom of union
If the class of subsets of a set form a set, then conversely we can consider the class formed from the elements of a set of sets, a version of what is generally called a ‘family of sets’. In general, if $C$ is a class whose elements are all sets, we can generalise the notion of the union of two sets to $C$ :
Definition 7
If $C\subset \mathbb S$ is a class of sets, the union of $C$ is by definition the class denoted $\bigcup C$ and which elements are the elements of the elements of $C$.
Symbolically, we write $\bigcup C=\{x\in \mathbb U : \exists S\in C,\ x\in S\}$, and this class is well defined as a subclass of $\mathbb U$ (it can, for example, be ‘constructed’ from elementary operations). However, obviously in general the union of $C$ is not a set:
Example 1
The union $\bigcup \mathbb S$ of the class $\mathbb S$ of all sets is none other than the meta-universe $\mathbb U$. Indeed, if $S$ is a set and $a\in S$, then $a\in \mathbb U$ by definition, whereas if $a\in \mathbb U$, then $a\in \{a\}$ by the Pairing Axiom, thus $a\in \bigcup \mathbb S$.
Thus, we must introduce an additional axiom for reunions of sets of sets:
Axiom of Union (5)
If $E\subset \mathbb S$ is a set of sets, then its reunion $\bigcup E$ is a set.
In particular, the union of two sets $E$ and $F$, defined as a class, is a set, since it is the reunion of the pair $\{E,F\}$! We can deduce the same thing about Cartesian products:
Proposition 2
If $E$ and $F$ are two sets, then the Cartesian product $E\times F$ is a set.
Indeed, by definition the elements of the class $E\times F$ are the pairs $(a,b)$ such that $a\in E$ and $b\in F$ : we have $a,b\in E\cup F$, a set by the Axiom of Reunion. By the Power Set Axiom, we can therefore form the set $\mathcal P(E\cup F)$ of its subsets, and again the set $\mathcal P(\mathcal P(E\cup F))$ of its subsets. We then have ${a},{a,b}}in \mathcal P(E\cup F), $ so $(a,b)={a},{a,b}}in \mathcal P(\mathcal P(E\cup F)). Therefore, $E\times F$ is a subclass of $\mathcal P(\mathcal P(E\cup F)),$ and therefore a set by the Axiom of Comprehension!
The dual operation of the union is the intersection: one can define in general the intersection noted $\bigcap C$ of a class $C$ of sets, i.e. of a subclass $C$ of $\mathbb S,$ by putting $\bigcap C:={x\in \mathbb U : \forall E\in C,\ x\in E},$ i.e. the class of all the objects of the meta-universe which belong to all the elements of $C.$.
When the class $C$ is empty, the condition which defines its intersection is also empty, so that all the objects are concerned and we then have $\bigcap \emptyset=\mathbb U.$ Conversely, the meeting of the empty class contains no element, so it is $\bigcup \emptyset=\emptyset.$ In the opposite case, we have the :
Proposition 3
If $C$neq \emptyset,$ the intersection $\bigcap C$ is a set.
Indeed, if $C$ is not empty, it contains an element $E,$ which is a set: by definition, we then have $\bigcap C\subset E,$ a set by the axiom of comprehension.
4.4 Relations and functions
The plasticity of natural set theory ensures that the notions of (binary) relations and functions can be very simply combined in the same concept:
Definition 8
Let $C$ and $D$ be two classes.
i) A relation between $C$ and $D$ is a subclass $R$ of the product $C\times D$. The set of all $x\in C$ such that there exists $y\in D$ for which $(x,y)\in R$ is the domain or field of $R$, the set of all $y\in D$ such that there exists $x\in C$ for which $(x,y)\in R$ is the co-domain or image of $R$.
ii) A relation $R$ of $C$ in $D$ is said to be functional if any element of its domain has at most one correspondent in its image, in other words if for all $x\in C$ and $y,z\in D$ such that $(x,y),(x,z)\in R$ we have $y=z$.
iii) A function from a class $C$ into a class $D$ is a functional relation $F$ between $C$ and $D$ such that every element of $C$ has a corresponding element in $D$, in other words such that for every $x\in C$, there exists $y\in D$ (necessarily unique) such that $(x,y)\in F$.
We denote $F:C\to D$ a function from $C$ into $D$ to emphasise its operational nature, and if $x\in C$, we denote $F(x)$ the unique object $y\in D$ such that $(x,y)\in F$.
4.5 Family reunions and the replacement axiom
This general notion of a function makes it possible to properly define families of sets:
Definition 9
A set family is a function $F:I\to \mathbb S$, where $I$ is a set (the “indices” of the family).
In this case, an “element” of the family is one of the values of $F$, i.e. one of the sets $F(i)$, for $i\in I$. The notion of a family of sets, universally used in mathematics, gives rise to another presentation of the axiom of reunion, if we consider families of sets rather than sets of sets. In the first place, since the image $C$ of $F$ is a subclass of $\mathbb S,$ we can take its union:
Definition 10
The union of a family of sets $F:I\to\mathbb S$ is the class $\bigcup F,$ union of the image $C$ of $F.$.
The union of $F$ is therefore the class of objects $x$ of the meta universe, for which there exists an index $i\in I$ such that $x\in F(i)$. Now, $I$ being a set, the union $\bigcup F$ of $F$ should be a set, which however does not derive from the axioms already introduced. In order to guarantee this, and to be able to form the Cartesian product of the family $F$ in the class $\mathbb S$, we introduce the last basic axiom of natural set theory:
Axiom of Replacement (6)
If $F:C\to D$ is a function and $E\subset C$ is a set, then the image of $E$ by $F$ is a set.
So, in the case of a family of sets $F:I\to \mathbb S$, the image $C$ of $F$ is a set, and therefore so is its union $\bigcup F$ by the axiom of union (5). We can therefore state the axiom of union in the following form:
Proposition 4
If $F:I\to \mathbb S$ is a family of sets, then its union $\bigcup F$ is a set.
The Cartesian product $\prod F$ of the family of sets $F$, often denoted $\prod_{i\in I} F(i)$, can then be unambiguously defined as the set of functions $f:I\to \bigcup F$ such that $f(i)\in F(i)$ for all $i\in I$, thanks to all the axioms introduced so far. Such a function is in fact a subset of the Cartesian product $I\times \bigcup F$ (a set by proposition 2), so that $\prod F$ is a subclass of $\mathcal P(I\times \bigcup F),$ and therefore a set by the parts axiom (3) and the comprehension axiom (2).
Conclusion
It is possible to found mathematics rigorously on intuition, without using formal logic, and without sacrificing the essential contributions of axiomatic set theory. This requires us to learn the lesson taught by Russell’s paradox, to distinguish fundamentally between classes, of a logical nature, and sets, of a mathematical nature, and to reintroduce the objects of natural logic to even define sets. This approach is ‘logicist’ – it aims to ground mathematics in logic – without being ‘formalist’ – it does not locate the essence of mathematics in logical formalism. On the contrary, by seeking the logical foundation of mathematics in intuition, natural set theory fully embraces current mathematical practice without abandoning rigour. From now on, it will be possible to reconstruct the whole of mathematics from within this natural set theory on the basis of the other axioms, which will have to be introduced in another chapter of this philosophico-mathematical narrative.