Foundations of mathematics
A study of the foundation of mathematics is essentially the study of logic and set theory. Logic is the language of mathematics; all mathematical statements can ultimately be decomposed into statements of pure logic. In some ways, logic is analogous to natural languages, such as English. The English language consists of words (a lexicon) which may be combined according to a set of rules (a grammar) to construct sentences. The infinite ways in which these words and sentences can be rearranged makes English a rich tool that can describe the world as we perceive it (such as a history book), other worlds of the imagination (such as a fantasy novel) and everything in between.
Similarly, logic is also a language with a lexicon and grammatical structure. It allows us to model the world we see around us by allowing us to define abstract mathematical objects. For example, a bullet exiting a gun can be modelled using numbers and vectors which describe its position in space, its mass, and its velocity. Numbers and vectors are examples of mathematical objects that can be constructed with logical statements. But logic, much like English, is not confined to modelling the real world. It can be used to create exotic mathematical worlds that don't seem to have any counterpart in the real world. For example, while complex numbers are not a part of our daily perception, they are very useful as a tool for solving problems in many areas of physics and engineering.
However, that is where the similarities end. Logic can only make statements about truth. This restriction is necessary as mathematics is dependent on demonstrating that a mathematical argument is valid. Mathematics is very black and white. A statement is either right or wrong, there is no grey area. Logic is also precise and, unlike English, purposely avoids ambiguous constructs. Logic also comes with a set of rules (known as rules of inference), that allow us to derive new truths from existing truths. This is a very important aspect of logic, as proofs of mathematical theorems are simply applications of the logical rules of inference.
The foundation of any branch of mathematics begins by constructing a theory. A theory is simply a collection of logical statements, known as axioms, which we assert to be true without question. Each axiom is written within the language of logic. Axioms are necessary because we cannot derive new truths without having some truths to start with. Peano arithmetic is an example of a theory which formally describes the properties of the natural numbers, as well as the operations of addition and multiplication. Using the rules of inference, new truths (known as theorems) can be derived from the axioms. From these theorems, other theorems can be derived, and so on. The discovery or proof of new theorems is the day job of the working mathematician.
One of the most remarkable achievements of the 20th century is the creation of set theory. Set theory is constructed from a very small (less than ten) number of axioms which together define the properties of objects called sets. Intuitively, sets can be understood as containers that hold an ordered collection of objects. In set theory, the only objects that exist are sets; therefore, the objects that sets contain are also sets themselves. As basic and uninteresting as this sounds, incredibly all objects in mathematics (including numbers, vectors, and geometrical objects such as lines and planes) can be formalized in terms of sets only. In a sense, sets can be considered the "atoms" that build the mathematical universe.
This book can be roughly divided into three sections. The first is a complete examination of a system of logic known as natural deduction. Second, the most widely used theory of sets, known as Zermelo-Fraenkel set theory, or ZFC, is introduced as the foundational theory from which mathematics is constructed. Finally, a complete construction of the real numbers and its operations (via the natural, integers and rational numbers) is demonstrated within ZFC.