The field of logic in mathematics deals with questions regarding the nature of formalized mathematical statements, proofs of statements, the structure of objects that are described by these statements, as well as questions regarding computability.
More precisely, mathematical logic studies questions that ask:
- How can we represent mathematical statements formally? How expressive must the language used be? Can certain properties, theorems, etc. be formulated in more restricted settings than others?
- What, exactly, is a proof? What do valid conclusions in proofs look like? For a given language, can we find an effective/algorithmic proof generation method?
- What can be said about the objects that are described by a set of formal statements? For example, can we assume that the objects will be unique? Will they have a fixed cardinality? What will be the common properties of all such objects?
A related subject is computability: Given a mathematically precise problem description, is there an algorithm that solves this problem? If not, why not?
What this tag does not include are basic questions about informal uses of logic - using mathematical arguments to prove some statement should be labeled with the proof-writing tag instead.