1.1. Introduction to the foundations of mathematics
Mathematics and theories
Mathematics is the study of systems of elementary objects, whose only 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, that may be formalized as (axiomatic) theories.
Each theory is the study of a supposedly fixed system (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
Each theory starts with a foundation, that is the data of a list of pieces of
description specifying what it knows or assumes of its model(s) (its kind or shape).
This includes a list of formulas (statements) called axioms, expressing the
required properties of models, i.e.
selecting its accepted models as the systems where the axioms are true,
from the whole range of possible systems where they can be interpreted.
Then, the study of a theory progresses by choosing some of its possible developments :
new concepts and information about its models, resulting from its given foundation,
and that we can add to it to form its next foundation.
In particular, a theorem of a theory, is a formula deduced from its axioms,
so that it is known as true in all its models. Theorems can be added to the list of
axioms of a theory without modifying its meaning.
Other possible developments (not yet chosen) can still be operated later, as the part of
the foundation that could generate them is preserved. 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
(before the Completeness theorem
will finally show how the range of possible theorems precisely reflects
the more interesting reality of the diversity of possible models).
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 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 unlimited diversity of possible variants
(not always equivalent to each other).

Model theory is the general theory of theories (describing their formalisms
as systems of symbols), and their possible models.
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, especially for this following last piece:

Proof theory completes model theory by describing a possible formal
system of rules of proofs giving the theorems of any theory.
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.
Model theory and proof theory are essentially unique, giving a clear natural
meaning to the concepts of theory, theorems and consistency of each theory.
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FR :
Introduction au fondement des mathématiques