This chapter will explain how to write definitions and statements with the
Nuprl
system, and in doing so it will introduce
a minimal set of system commands
to the beginning user
and explain how the type theory introduced in chapter 2 can be
used to express mathematical propositions. The first four sections of the
chapter concentrate on discussing the system commands and some relevant
syntactic features of the Nuprl
logic. The rest of the chapter describes
the conceptual issues involved in using type theory as a general--purpose
language for expressing mathematics. These conceptual issues are discussed
via examples drawn from four areas of mathematics: logic
(both constructive and classical), number theory, algebra and
set theory. (Chapter 11 contains
a summary of definitions and theorems from other
areas of mathematics.) It is intended that this chapter be read
while experimenting with the system. Throughout this chapter the
typewriter font represents text that is actually used in the
Nuprl
system, while the mathematical 
is used in general discussions.