# Real Numbers

# Why

We want a set which corresponds to our notion
of points on a line.

# Rational cuts

We call a subset $R$ of $\Q $ a
rational cut if (a) $R
\neq \varnothing$, (b) $R \neq \Q $, (c) for
all $q \in R$, $r \leq q \implies r \in R$,
and (d) $R$ has no greatest element.
Briefly, the intuition is that the point is
the set of all rationals to less than (or,
potentially, equal to) some particular rational
number.

# Definition

The set of real numbers
is the set of all rational cuts.
This set exists by an application of the
principle of selection (see Set Selection) to the power set (see Set Powers) of $\Q $.
We call an element of the set of real numbers
a real number or a
real.
We call the set of real numbers the
set of reals or
reals for short.
# Notation

We follow tradition and denote the set of real
numbers by $\R $, likely a mnemonic for “real.”

## Other terminology

Some authors call a real number a
quantity or a
continuous quantity.
The real numbers, then, are said to be
continuous.
When contrasting (using this terminology) a
finite set with the real numbers, one refers to
the finite set as
discrete.