paying attention to the language used in mathematics, to the ways abstract
objects are defined, and to the laws of logic which govern us as we reason
about these objects. The logician undertakes this study with the hope of
understanding the phenomena of mathematical experience and eventually
contributing to mathematics, both in terms of important results that arise
out of the subject itself (Godel’s Second Incompleteness Theorem is the
most famous example) and in terms of applications to other branches of
from Handbook of mathematical logic, Part A - Model theory, An Introduction to First Order Logic, 1. Foreword, page 6.
The book we have now here https://www.academia.edu/30987187/_Jon_Barwise_Handbook_of_Mathematical_Logic_Stud_BookZZ_org_ if you wish.