Mathematical Analysis I, 2004a

§§1–4. Axioms and Basic Definitions 25
Note 1. The uniqueness assertions in Axioms IV and V are actually redundant
since they can be deduced from other axioms. We shall not dwell on
this.
Note 2. Zero has no reciprocal; i.e., for no x is 0x =1. Infact,0x =0.
For, by Axioms VI and IV,
0x +0x =(0+0)x =0x =0x +0.
Cancelling 0x (i.e., adding −0x on both sides), we obtain 0x = 0, by Axioms III
and V(a).
Note 3. Due to Axioms VII and VIII, real numbers may be regarded as
giveninacertainorder under which smaller numbers precede the larger ones.
(This is why we speak of “axioms of order.”) The ordering of real numbers can
be visualized by “plotting” them as points on a directed line (“the real axis”)
in a well-known manner. Therefore, E 1 is also often called “the real axis,” and
real numbers are called “points”; we say “the point x” instead of “the number
x.”
Observe that the axioms only state certain properties of real numbers without
specifying what these numbers are. Thuswemaytreattherealsasjustany
mathematical objects satisfying our axioms, but otherwise arbitrary. Indeed,
our theory also applies to any other set of objects (numbers or not), provided
they satisfy our axioms with respect to a certain relation of order (0ornegative
if xy”meansthesameas“y