Logic and Set Theory
Mathematical logic is a branch of mathematics which is mainly concerned with the relationship between “semantic” concepts (i.e., mathematical objects) and “syntactic” concepts (such as formal languages, formal deductions and proofs, and computability). A distinctive feature is its role in the foundations of mathematics, especially concerning whether the truth value of mathematical statements can be obtained algorithmically within various axiom systems.
A closely related branch is set theory, which provides a simple, uniform background in which to do virtually all mainstream mathematics. A set theorist explores what is provable—and what is not provable—when one is allowed to use all of the usual tools of mathematical constructions and proofs (tools such as creating function spaces, completions of various kinds of objects, taking quotients, forming products, induction and recursion, etc.).
Together, mathematical logic and set theory have produced some of the most beautiful theorems in all of mathematics; for example:
- Gödel’s incompleteness theorems, which prove that in any sufficiently useful axiomatic system there must be statements which can neither be proved nor disproved
- Independence results, such as the Cohen/Gödel theorems that both the continuum hypothesis and the axiom of choice are undecidable from the standard axioms of mathematics