Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Among the most basic mathematical concepts are: number, shape, set, function, algorithm, mathematical axiom, mathematical definition, and mathematical proof. Typical questions in foundations of mathematics include: What is a number? What is a shape? What is a set? What is a function? What is an algorithm? What is a mathematical axiom? What is a mathematical definition? What is a mathematical proof? What are the most basic concepts of mathematics? What is the logical structure of mathematics? What are the appropriate axioms for numbers? What are the appropriate axioms for shapes? What are the appropriate axioms for sets? What are the appropriate axioms for functions?

Obviously, foundations of mathematics is a subject of the greatest mathematical and philosophical importance. Beyond this, foundations of mathematics is a rich subject with a long history, going back to Aristotle and Euclid and continuing in the hands of outstanding modern figures such as Descartes, Cauchy, Weierstraβ, Dedekind, Peano, Frege, Russell, Cantor, Hilbert, Brouwer, Weyl, von Neumann, Skolem, Tarski, Heyting, and Gödel. An excellent reference for the modern era in foundations of mathematics is van Heijenoort [272].

In the late 19th and early 20th centuries, virtually all leading mathematicians were intensely interested in foundations of mathematics and spoke and wrote extensively on this subject. Today that is no longer the case. Regrettably, foundations of mathematics is now out of fashion.

The Main Question

The purpose of this book is to use the tools of mathematical logic to study certain problems in foundations of mathematics. We are especially interested in the question of which set existence axioms are needed to prove the known theorems of mathematics.

The scope of this initial question is very broad, but we can narrow it down somewhat by dividing mathematics into two parts. On the one hand there is set-theoretic mathematics, and on the other hand there is what we call “non-set-theoretic” or “ordinary” mathematics. By set-theoretic mathematics we mean those branches of mathematics that were created by the set-theoretic revolution which took place approximately a century ago. We have in mind such branches as general topology, abstract functional analysis, the study of uncountable discrete algebraic structures, and of course abstract set theory itself.

We identify as ordinary or non-set-theoretic that body of mathematics which is prior to or independent of the introduction of abstract set-theoretic concepts. We have in mind such branches as geometry, number theory, calculus, differential equations, real and complex analysis, countable algebra, the topology of complete separable metric spaces, mathematical logic, and computability theory.

The distinction between settheoretic and ordinary mathematics corresponds roughly to the distinction between “uncountable mathematics” and “countable mathematics”. This formulation is valid if we stipulate that “countable mathematics” includes the study of possibly uncountable complete separable metric spaces.