The answer is no ; for more information, see the section describing the reasons for this .

In other words, does there exist any number system which, as well as including the familiar numbers we are used to, also includes an "infinity" concept?

Now the question is, does infinity exist in the same way that these concepts (negative numbers, fractions, etc.) do?

Number systems come in many sizes. There is the "natural number system", which is just the set of numbers used in counting: 1, 2, 3, 4, and so on. Or, one can expand this number system to include additional concepts, such as negative numbers, fractions, even the so-called "imaginary" numbers (which are not really imaginary at all). Each of these concepts exists provided we look for it in the context of a large enough number system.

A number system is any collection of objects that has the basic properties (like addition, multiplication, and so on) we normally associate with numbers. More information is available on this.

You've probably never heard the term "topological space" before; it occurs in an advanced branch of mathematics. Don't worry; we don't need to get into that advanced area. All we need is the following rough idea:

Roughly speaking, a topological space is any collection of objects for which there is a definition of which sequences of objects converge to other objects, and which sequences don't.

The real number system is a topological space: there's a definition of what it means for a sequence of numbers to converge. For instance, the sequence 1.1, 1.01, 1.001, 1.0001, etc. converges to the number 1, while the sequence 1, 2, 1, 2, 1, 2, 1, 2, etc. does not converge to anything.

In areas such as calculus, one often speaks of a sequence like 1, 2, 3, 4, . . . as "converging to infinity". Is this just a convenient phrase, or can there actually exist an object "infinity" that this sequence is converging to?

In other words, the question is: does there exist some topological space (that is, a set of objects plus a definition of what convergence means) which, as well as including the familiar real numbers we are used to, also includes an "infinity" concept to which some sequences of real numbers converge?

The answer is yes; for more information, see the section describing the reasons for this.

It is important to realize, though, that this topological space is not a number system. Although it includes an additional object called "infinity" as well as the familiar real numbers, you cannot add, subtract, multiply, or divide this additional object the way you can numbers.

Sometimes you will see a statement like " ". This does not really mean what it seems to say. You are not really dividing 1 by infinity. Instead, it is a statement about sequences. What it means is that if a sequence , , , . . . converges to infinity, then the sequence of reciprocals , , , . . . converges to zero.

(Similar looking expressions like " " don't make any sense, for just because two sequences , , , . . . and , , , . . . each converge to infinity tells you nothing about what the sequence , , , . . . does.)