It is easy to see that there is no largest natural number. Suppose
there was one, call it
. Now add one to it, forming
. We
know that
, contradicting our assertion that
was the
largest. This *lack* of a largest object, lack of a boundary, lack
of termination in series, is of enormous importance in mathematics and
physics. If there is no largest number, if there is no ``edge'' to
space or time, then it in some sense they run on *forever*, without
termination.

In spite of the fact that there is no *actual* largest natural
number, we have learned that it is highly advantageous in many context
to invent a *pretend* one. This pretend number doesn't actually
exist *as a number*, but rather stands for a certain *reasoning
process*.

In fact, there are a number of properties of numbers (and formulas, and
integrals) that we can only understand or evaluate if we *imagine* a
very large number used as a boundary or limit in some computation, and
then let that number mentally increase *without bound*. **Note
well** that this is a mental trick, encoding the observation that there
is no largest number and so we can increase a number parameter without
bound, no more. However, we use this mental trick all of the time - it
becomes a way for our finite minds to encompass the idea of
unboundedness. To facilitate this process, we invent a *symbol* for
this unreachable limit to the counting process and give it a name.

We call this unboundedness *infinity*^{4} - the
lack of a finite boundary - and give it the symbol
in
mathematics.

In *many contexts* we will treat
as a number in all of the
number systems mentioned below. We will talk blithely about ``infinite
numbers of digits'' in number representations, which means that the
digits simply keep on going without bound. However, it is *very
important* to bear in mind that
is *not* a number, it is a
*concept*. Or at the very least, it is a highly *special*
number, one that doesn't satisfy the axioms or participate in the usual
operations of ordinary arithmetic. For example:

(4) | |||

(5) | |||

(6) | |||

(7) |

These are certainly ``odd'' rules compared to ordinary arithmetic!

For a bit longer than a century now (since Cantor organized set theory
and discussed the various ways sets could become infinite and set theory
was subsequently axiomatized) there has been an *axiom of infinity*
in mathematics postulating its formal existence as a ``number'' with
these and other odd properties.

Our principal use for infinity will be as a limit in calculus and in
series expansions. We will use it to describe *both* the very large
(but never the largest) *and* reciprocally, the very small (but
never quite zero). We will use infinity to name the *process* of
taking a small quantity and making it ``infinitely small'' (but nonzero)
- the idea of the *infinitesimal*, or the complementary operation
of taking a large (finite) quantity (such as a limit in a finite sum)
and making it ``infinitely large''. These operations do not always make
arithmetical sense, but when they do they are *extremely valuable*.