1.1. Introduction to the foundations of mathematics
Mathematics and theories
Mathematics is the study of systems of elementary objects, whose only considered
nature is to be exact, unambiguous (two objects are equal or different, related or not;
an operation gives an exact result...). Such systems are conceived independently
of our usual world, even if many of them can resemble (thus be used to describe)
diverse aspects of it. Mathematics as a whole can be seen as «the science
of all possible worlds» of this kind (of exact objects).
Mathematics is split into diverse branches, implicit or explicit frameworks of any
mathematical work, which may be formalized as (axiomatic) theories. Each
theory is the study of a supposedly fixed system that is its world of
objects, called its model. But each model of a theory may be just one of its
possible interpretations, among other equally legitimate models. For example,
roughly speaking, all sheets of paper are systems of material points, models
of the same theory of Euclidean plane geometry, but independent of each other.
Foundations and developments
The content of a theory, describing its model(s), is made of
components which are pieces of description (concepts and information,
described in 1.3). A theory starts with a choice of foundation made of a
logical framework and an initial version of its content (hopefully rather small, or at
least simply describable). The components of this initial version are qualified as
primitive.
The study of the theory progresses by choosing some of its possible
developments : new components resulting from its current content, and
that can be added to it to form its next content. These different contents, having
the same meaning (describing the essentially same models), play the role of
different presentations of the same theory.
Any other possible development (not yet chosen) can still be added later, as the
part of the foundation that could generate it remains. Thus, the totality of
possible developments of a theory, independent of the order chosen to process
them, already forms a kind of «reality» that these developments explore.
To express the properties of its models, the content of a theory includes a list of
statements, which are formulas meant as true when interpreted in any model.
Primitive statements are called axioms. Further statements called theorems
are added by development to the content, under the condition that they are proven
(deduced) from previous ones : this ensures them to be true in all models,
provided that previous ones were. Theorems can then be used in further
developments in the same way as axioms. Other kinds of developments
(definitions and constructions) which add other components beyond statements,
will be described in 4.8 and 4.9.
A theory is consistent if its theorems will never contradict each other.
Inconsistent theories cannot have any model, as the same statement cannot be
true and false on the same system.
The Completeness
theorem (4.6) will show that the
range of all possible theorems precisely reflects the more interesting reality
of the diversity of models, which indeed exist for any consistent theory.
There are possible hierarchies between theories, where some can
play a foundational role for others. For instance, the foundations of
several theories may have a common part forming a simpler theory,
whose developments are applicable to all.
A fundamental work
is to develop, from a simple initial basis, a convenient body of knowledge
to serve as a more complete "foundation", endowed with efficient tools
opening more direct ways to further interesting developments.
The cycle of foundations
Despite the simplicity of nature of mathematical objects, the general
foundation of all mathematics turns out to be quite complex
(though not as bad as a physics theory of everything). Indeed,
it is itself a mathematical study, thus a branch of mathematics,
called mathematical logic.
Like any other branch, it is made of definitions and theorems
about systems of objects. But as its object is the general form
of theories and systems they may describe, it provides the general
framework of all branches of mathematics... including itself.
And to provide the framework or foundation of each considered
foundation (unlike ordinary mathematical works that go forward from
an assumed foundation), it does not look like a precise starting point,
but a sort of wide cycle composed of easier and harder steps. Still this
cycle of foundations truly plays a foundational role for mathematics, providing
rigorous frameworks and many useful concepts to diverse branches of
mathematics (tools, inspirations and answers to diverse philosophical questions).
(This is similar to dictionaries defining each word by other words, or to
another science of finite systems: computer programming. Indeed computers
can be simply used, knowing what you do but not why it works; their working
is based on software that was written in some language, then compiled by
other software, and on the hardware and processor whose design and production
were computer assisted. And this is much better than at the birth of this field.)
It is dominated by two theories:

Set theory describes the universe of «all mathematical
objects», from the simplest to the most complex such as infinite
systems (in a finite language). It can roughly be seen as one theory,
but in details it will have an endless diversity of possible variants
(indeed differing from each other).

Model theory is the study of theories (their formalisms as systems
of symbols), and systems (possible models of theories). Proof theory
completes this by describing formal systems of rules of proofs. While these
are usually meant as general topics (admitting variants of concepts), the
combination of both can be specified into precise versions (mathematical
theories) called logical frameworks, each giving a precise format of
expression for a wide range of possible theories, and a format in which all
proofs in any of these theories can in principle be expressed. There is an
essentially unique preferable logical framework called firstorder logic,
by which the concepts of theory, theorem (as provable statement) and
consistency of each theory, find their natural mathematical definitions.
Each one is the natural framework to formalize the other: each set theory is formalized
as a theory described by model theory; the latter better comes as a development from
set theory (defining theories and systems as complex objects) than directly as a theory.
Both connections must be considered separately: both roles of set theory, as a
basis and an object of study for model theory, must be distinguished. But these
formalizations will take a long work to complete.
Starting mathematics is a matter of introducing some simple concepts from
the founding cycle, which may seem as selfsufficient 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. 1.2 will do this very shortly,
intuitively explaining the concepts of set and function. Then
1.3 will start introducing the main picture of foundations (model theory) by
which set theory can be formalized, with its main subtleties (paradoxes).
Other languages:
FR :
Introduction au fondement
des mathématiques