1.2. Variables, sets, functions and operations

Starting mathematics is a matter of introducing some simple concepts from the founding cycle, which may seem as self-sufficient as possible (while they cannot be absolutely so). A usual and natural solution is to start with a set theory not fully formalized as an axiomatic theory. This will be briefly done in 1.2, intuitively explaining the concepts of set and function. Then 1.3 will start introducing model theory, by which any theory including set theory can be formalized, and later sections will explain the main subtleties (paradoxes) in the resulting picture of mathematical foundations.

The below tries to introduce some concepts without clear definitions, especially some qualifications of variables which are not meant as intrinsic qualities but extrinsic ones, that is, only holding relatively to some kinds of contexts which are not yet themselves introduced at this stage, and therefore may look obscure. The reader is invited to continue with next sections despite possible feelings of unclarity with the current section, feelings which are then likely to dissipate on the way.


A constant symbol is a symbol seen as denoting a unique object, called its value. Examples: 3, ⌀, ℕ. Those of English language usually take the form of proper names and names with «the» (singular without complement).

Free and bound variables

A variable symbol (or a variable), is a symbol which, instead having an a priori definite value, comes with the concept of its possible values, or possible interpretations, each of which give it a particular value. Each possibility gives it a role of constant. There may be any number of these possible values, including infinitely many, only one or even none.
It can be understood as limited by a box, whose inside has multiple versions in parallel, articulating different viewpoints over it: More precisely with respect to given theories, fixing a variable means taking a free variable in a theory and more lengthily ignoring its variability, therefore simulating the use of the other theory obtained by holding this symbol as a constant.
The diverse «internal viewpoints», corresponding to each possible value seen as fixed, may be thought of as abstract «locations» in the mathematical universe, while the succession of views over a symbol (qualifying it as a constant, a free variable or a bound variable), can be seen as a first expression of the flow of time in mathematics: a variable is bound when all the diverse "parallel locations inside the box" (possible values) are past. All these places and times are themselves purely abstract, mathematical entities.

Ranges and sets

The range of a variable, is the meaning it takes when seen as bound: it is the «knowledge» of the considered totality of its possible or authorized values (seen in bulk: unordered, ignoring their context), that are called the elements of this range. This «knowledge» is an abstract entity that, depending on context, may be able to actually process (encompass) infinities of objects (unlike human thought). Any range of a variable is called a set.
A variable has a range when it can be bound, i.e. when an encompassing view over all its possible values is given. Not all variables of set theory will have a range. A variable without a range can still be free, which is no more an intermediate status between fixed and bound, but means it can take some values or some other values with no claim of exhausitivity.

Cantor defined a set as a «gathering M of definite and separate objects of our intuition or our thought (which are called the "elements" of M) into a whole». He explained to Dedekind : «If the totality of elements of a multiplicity can be thought of... as "existing together", so that they can be gathered into "one thing", I call it a consistent multiplicity or a "set".» (We expressed this "multiplicity" as that of values of a variable).
He described the opposite case as an «inconsistent multiplicity» where «admitting a coexistence of all its elements leads to a contradiction». But non-contradiction cannot suffice to generally define sets: the consistency of a statement does not imply its truth (i.e. its negation may be true but unprovable); facts of non-contradiction are often themselves unprovable (incompleteness theorem); and two separately consistent coexistences might contradict each other (Irresistible force paradox / Omnipotence paradox).

A variable is said to range over a set, when it is bound with this set as its range. Any number of variables can be introduced ranging over a given set, independently of each other and of other variables.
Systematically renaming a bound variable in all its box, into another symbol not used in the same context (same box), with the same range, does not change the meaning of the whole. In practice, the same letter can represent several separate bound variables (with separate boxes), that can take different values without conflict, as no two of them are anywhere free together to compare their values. The common language does this continuously, using very few variable symbols («he», «she», «it»...)


A function is an object f made of the following data: In other words, it is an entity behaving as a variable whose value is determined by that of another variable called its argument with range Dom f : whenever its argument is fixed (gets a name, here "x", and a value in Dom f), f becomes also fixed, written f(x). This amounts to conceiving a variable f where the "possible views" on it as fixed, are treated as objects x conceptually distinct from the resulting values of f. As we shall see later, such an entity (dependent variable) f would not be (viewable as) a definite object of set theory if its argument had no range, i.e. could not be bound (it would only be a meta-object, or object of model theory, that we shall call a functor in 1.4)


The notion of operation generalizes that of function, by admitting a finite list of arguments (variables with given respective ranges) instead of one. So, an operation gives a result (a value) when all its arguments are fixed. The number n of arguments of an operation is called its arity ; the operation is called n-ary. It is called unary if n=1 (it is a function), binary if n=2, ternary if n=3...
The concept of nullary operation (n=0) is superfluous, as their role is already played by their unique value; 2.3 will show how to construct operations with arity > 1 by means of functions.

Like for functions, the arguments of operations are basically denoted not by symbols but by places around the operation symbol, to be filled by any expression giving them desired values. Diverse display conventions may be used (1.5). For instance, using the left and right spaces in parenthesis after the symbol f, we denote f(x,y) the value of a binary operation f on its fixed arguments named x and y (i.e. its value when its arguments are assigned the fixed values of x and y).

An urelement (pure element) is an object not playing any other role than that of element: it is neither a set nor a function nor an operation.

Set theory and Foundations of mathematics
1. First foundations of mathematics
1.1. Introduction to the foundations of mathematics
1.2. Variables, sets, functions and operations
1.3. Form of theories
1.4. Structures of mathematical systems
1.5. Expressions and definable structures
1.6. Logical connectives
1.7. Classes in set theory
1.8. Binders in set theory
1.9. Axioms and proofs
1.10. Quantifiers
1.11. Second-order quantifiers
Time in model theory
Truth undefinability
Introduction to incompleteness
Set theory as unified framework
2. Set theory - 3. Algebra - 4. Arithmetic - 5. Second-order foundations
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FR : 1.2. Variables, ensembles, fonctions et opérations