1. First foundations of mathematics

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: 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 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. 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).

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. Algebra - 4. Arithmetic 5. Second-order foundations
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FR : Introduction au fondement des mathématiques