1.2. Variables, sets, functions and operations

Let us start mathematics by to introducing some simple concepts from the founding cycle, which may seem self-sufficient. It is natural to start with a set theory not fully formalized as an axiomatic theory.
Let us first explain what is a set, then we will complete the picture with more concepts and explanations on the context of foundations (model theory) and its main subtleties (paradoxes).


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

Free and bound variables

A variable symbol (or a variable), is a symbol without a fixed value. Each possible interpretation gives it a particular value and thus sees it as a constant.
It can be understood as limited by a box, whose inside has multiple versions in parallel. The diverse «internal viewpoints», corresponding to the possible fixed values, may be thought of as abstract «locations» in the mathematical universe, while the succession of statuses of a symbol (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 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 can encompass infinities of objects, unlike human thought. A variable has a range when it can be bound, i.e. when an encompassing view over all its possible values is given.
Any range of a variable is called a set.

Cantor defined a set as «a grouping into a whole of distinct objects of our intuition or our thought». He explained to Dedekind : «If the totality of elements of a multiplicity can be thought of as «simultaneously existing», so that it can be conceived as a «single object» (or «completed object»), 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 non-contradiction of a statement does not imply its truth (the opposite statement may be true but unprovable); facts of non-contradiction can be 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 any object f behaving as a variable whose value is determined by that of another variable called its argument; the argument has a range called the domain of f and denoted Dom f. Whenever its argument is fixed (gets a name, say x, and a value in Dom f), f becomes a constant (written f(x)).
In other words, f is made of the following data:


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 nullary if n=0 (it is a constant), unary if n=1 (it is a function), binary if n=2, ternary if n=3...
Nullary operations are useless as their role is played by their unique value; we shall see how to construct those with arity > 1 by means of functions.

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 values of x and y), is denoted f(x,y). Generally, instead of symbols, the arguments are represented by the left and right spaces in parenthesis, to be filled by any expression giving them desired values.

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: notions, objects, meta-objects
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. Quantifiers
1.10. Formalization of set theory
1.11. Set generation principle
Philosophical aspects
Time in model theory
Time in set theory
Interpretation of classes
Concepts of truth in mathematics
2. Set theory (continued) 3. Model theory
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FR : 1.2. Variables, ensembles, fonctions et opérations