What is mathematical logic? What does mathematical logic mean?

In the book Analysis 1 by Terence Tao, it says:


  The purpose of this appendix is to give a quick introduction to
  mathematical logic, which is the language one uses to conduct
  rigourous mathematical proofs.


Checking Wikipedia:


  Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory.


This seems like a completely different definition. Why, for example, is set theory considered part of logic?
Mathematical logic is a strange beast.

It is a perfectly ordinary branch of mathematics whose goal is ... to study mathematics itself. 

Thus, the different branches of mathematical logic are devoted to the study of some basic building blocks of mathematical practice : language, model, proof, computation.
Modern mathematical logic is "mathematical" also in another sense: some branches of it are simply mathematical disciplines, like e.g. Computability theory and Constructive mathematics,