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 infinity4 - 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:
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.