What Is Finite?

Finite is the opposite of infinite; something is finite if it's not infinite. But there are many different notions of infinite; there are several for the idea of finite.

In geometry, a set may or may not be contained in a finite portion of the plane (or space). A portion of the plane is finite, if it's contained in a ball - however big. In other words, a portion of the plane is finite if the set of all distances from its points to a fixed point (say, origin) is bounded. A figure lying in a finite portion of the plane is said to be bounded. A segment is a bounded - also and frequently finite - portion of an infinite line. This is so, even if a segment contains an infinite number of points.

There are also Finite Geometries that contain a finite number of points and lines.

The set of rational numbers between 0 and 1 belongs to a finite segment but, in itself, is infinite.

Among numbers, the notion of finiteness is an outgrowth of our ability to count. Roughly speaking, a set of objects is finite if it can be counted.

The numbers 1, 2, 3, ... are known as "counting" just because this is what we do while counting: we call the names of those numbers one at a time while pointing (even if mentally) to members of a set. The last number to be called is the cardinality of the set. A set of cardinality N is in a 1-1 correspondence with the set {1, 2, ..., N}.

Thus a set is finite if its cardinality is an integer. A set is infinite if it is not finite.

We may establish another definition of infinitude as a consequence of what has been said so far

Theorem

A set is infinite if and only if it contains a proper subset of the same cardinality.

(The empty set Ø is considered finite as well - it is certainly does not appear infinite.)

Proof

The proof will emerge in a succession of Lemmas.

Adding an element to a finite set leaves a finite set.

Let there be set A of cardinality N: |A| = N. The elements of A are in a 1-1 correspondence with the integers {1, 2, ..., N}. Adding an element to A increases its cardinality by 1, for the 1-1 correspondence between A and {1, 2, ..., N} is naturally expanded to augmented set and {1, 2, ..., N, N+1}. Just assign N+1 to the new element.

But N + 1 is a counting number as much as N itself. Therefore, the augmented set is also of finite cardinality, i.e., is finite.

Removing and element from an infinite set leaves an infinite set.

Indeed, if we remove an element from an infinite set, the remaining set is bound to be infinite; for, otherwise, putting that element back in we would get a finite set.

The set N = {1, 2, 3, ...} of natural numbers is infinite.

Assuming that |N| = k, for an integer k, leads to a contradiction because already |{1, 2, ..., k, k+1}| = k+1 so that counting elements of N we can reach k+1 which is greater than k.

A set whose cardinality equals that of N is said to be countable and is called a sequence.

Every infinite set contains a sequence.

Just keep removing one element at a time.

A sequence contains a subset of the same cardinality.

Indeed, this is true of N which is equivalent to the set of odd numbers, the set of even numbers, the set of squares - and what not. For what follows one example is especially handy. The 1-1 correspondence n→n+1 shows that N and {2, 3, ...} have the same cardinality.

Any infinite set contains a set of the same cardinality.

Let set A contains a sequence S. If A = S, we are finished. Otherwise, let T = A - S, and s ∈ S. There exist a 1-1 correspondence between S and S-{s}. This correspondence is expanded to a correspondence between A and A-{s} by assigning elements from T to themselves.

Finally,

Any subset of a finite set is finite.

and

A superset of an infinite set is infinite.

References

R. Smullyan, Satan, Cantor, and Infinity, Dover Publications (March 26, 2009)

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