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
(by lack of clear definitions, which would require a different start).
A usual and natural solution is to start with a set theory not fully formalized as an axiomatic
theory. This section (1.2) will intuitively introduce some first few concepts of set theory :
those of set, function and operation. But it will start by introducing some qualifications
of variables only meant as extrinsic qualities, namely to describe the status of a given variable
relatively to some kinds of contexts (viewpoints) which are not yet themselves introduced at
this stage. The reader is invited to not be stopped by this kind of unclarity
which is likely to get resolved along the specific uses in later sections.
Then 1.3 will start introducing model theory, by which any theory
(and thus any set theory) can be formalized. More subtleties (paradoxes..)
in the whole picture of the foundations of mathematics will be explained later.
Constants
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:
- The variable is called fixed when seen "from inside", which
means it has a given value, and is thus usable as a constant.
- It is called bound when seen from
the «outside» where the diversity of its possible values is
considered fully known, gathered and
processed as a whole.
- It is called free to describe a coexistence of both statuses (views over it):
a local view seeing it as fixed, and an external view giving the context of its variations.
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»...)
Functions
A function is an object f made of the following data:
- A set called the domain of f, denoted Dom
f
- For each element x of Dom f,
an object written f(x), called the image of x
by f or value of f at x.
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)
Operations
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.
Other languages:
FR :
1.2. Variables,
ensembles, fonctions et opérations