1. An infinity is a number which is larger than any natural number

This definition is still surprisingly popular, even (I regret to say) among mathematicians, although it was exploded by Cantor more than a century ago.

It is easy to prove that an infinity is larger than some particular natural number; it does not follow that it must, in principle, be larger than ANY natural number, because the natural numbers are themselves extendable to infinity. Indeed, the set of the natural numbers is the definitive example of what mathematicians call a 'countable' or 'aleph-null' infinity. In an aleph-null infinity, every member can be counted off with a natural number; therefore, there is necessarily some natural number which is equal to the infinity, though of course we can never know what it is, because (paradoxically) we can never reach it by counting.

It is possible to prove that an infinity may be smaller than some natural number. (For the sake of brevity, in what follows, I do not distinguish between a set and its cardinality).

(1) Consider the infinite set of the natural numbers, beginning from 0. We will call this set A. Now consider the infinite set of the natural numbers which are greater than or equal to 10. Call this set B.

(2) In the arithmetic of the countable infinities, A = B; but: A - B = 10.

(3) In other words, if we pair the elements of infinite set A one-to-one with those of infinite set B, we have 10 numbers left over.

(4) Therefore, an infinity may be smaller than some natural number.

We can never assign a specific value to "some natural number"; but every aleph-null infinity is equal to or smaller than some natural number. There other kinds of infinity which are definitely larger, but that is another subject.

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2. An infinity is any non-finite number

A finite number is a number whose value can be expressed with perfect exactitude. It is finished, complete, precise, and nothing more can be added to it.

If we can identify one non-finite number which is not an infinity, this definition will fail. Is there such a number? As it happens, there are infinitely many. The infinitesimal, as defined by the symbol 'h', is not a finite number. In contexts where 'h' is important, this may imply that 0 also is not a finite number; however, considered as a natural number, 0 is finite. Any number which entails an infinite series of non-trivial decimal places, like pi or any of the irrationals, is a non-finite number; it is not bounded and its true or final value is unknowable. Infinitely many fractions between 0 and 1 are non-finite in this sense.

The definition, then, is incorrect and seriously misleading. All infinities are non-finite, but only some non-finite numbers are infinities. The confusion arises from the careless assumption that "infinite" and "non-finite" are synonymous. If they were really synonymous, the definition would be vacuous and non-informative instead of merely incorrect.


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Correct Definition:

3. An infinity is the cardinality of a set which contains one or more proper subsets equinumerous with itself

A 'proper subset' is a collection of some of the elements of a set, but not all of them, and not including anything taken from any other set; it's really just a part of a set. "Equinumerous" means, having the same cardinality or numeric value. So the definition says that if a part of the set turns out to have the same number of members as the whole set, even though it is only supposed to include some of them, then the parent set must be an infinity.

In classical arithmetic, the part is always less than the whole. If we cut a cake into portions, every portion is necessarily something smaller than a whole cake. In the arithmetic of infinity, however, at least one portion ('proper subset') always persists in being just as large as the original cake, no matter where or how often we cut it. (Sounds almost like a children's fairy tale, doesn't it!)

Suppose we have N quantity of coins in a jar, and we take out five coins. Then we check the number of coins remaining in the jar. If there are N-5 coins, we will know that N is a finite number and, if we keep taking coins out, eventually the jar will be empty. On the other hand, if we find that the jar still contains N coins, and never contains less than N coins, however often we take some out, then N must be an infinity.

Set theory is not as clearly defined as some other branches of mathematics, so there is some room for dispute, but this essentially defines infinity in terms of what it IS, rather than what it isn't, and for that reason, it succeeds where the other two popular definitions fail. Infinity can only be correctly defined in terms of this simple, special logical property, which is unique to itself. The definition makes no stipulations about finitude or absolute magnitude relative to any other kind of number; the criteria to which the other two definitions appeal are thus seen to be irrelevant and useless, and have no place in mathematics.

Moral of the story: believe only 33.3% of what you read in Wikipedia!