Draft:Finiteness
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Finiteness is the state of being limited or completed. For example, if notches in a bone are used to count the number of years in the life of an animal, the notches are finite in that there are a limited number of visibly distinct notches which can be made in the bone. The notches are also finite in that they are limited by the life of the animal. During the life of the animal, the notches are not finite in that they are not yet completed.
The terms finite and infinite are often used to describe abstract concepts. The number 10 is a finite number, not infinite. The rotation of the earth around the sun can be approximated using an ideal model in which the sun never dies and the earth never stops rotating around it. In this ideal model, the rotation of the earth around the sun is infinite because the rotation is never completed and is not limited by some amount of time or number of rotations.
Definitions are used to communicate abstract concepts. The definition of the ideal model above is finite because the definition has been completed. The finite definition describes an infinite rotation. The number 10 also has a finite definition. The number of unique ways in which the number 10 could be defined is, of course, not finite (5 + 5, 11 - 1, ...).
One way to define an abstract concept is to use recursion. A recursive definition is one in which concepts are defined in terms of other previously defined concepts. Following is a recursive definition of the term natural number:
- The number one is a natural number.
- If a number x is a natural number, then x plus one is also a natural number.
The recursion begins by defining the number one as a natural number. It then introduces the variable x, which can be defined to be equal to one. If x does equal one, then the number two is also defined to be a natural number. This pattern continues by redefining x to be equal to the number two.
The definition of the recursive process is complete, which defines the term natural number. The application of the recursive process is not. There is no limit to the number of natural numbers which can be defined using this recursion. There is no limit to the number of times the variable x can be redefined. There is no point in the recursion which has been identified as an endpoint marking completion.
Because it is a theoretical process, the recursion can define an arbitrarily large number of natural numbers in an arbitrarily short period of time. The only theoretical limit on the shortness of the period of time is that it be greater than zero, since x can only be equal to one natural number at a time.
The number which can be represented by a one followed by ten billion zeros is an example of an arbitrarily large number. It would take a lot of time and a very large bone for a human to make that many notches in the bone. Theoretically, the number can be completely defined in only one breath. It may seem infinitely large from one's limited human perspective, but it is limited to only ten billion zeros, no more.
In mathematics, the term set is used to refer to a collection of some number of elements, which number may be limited or not limited. A set in the state of finiteness is one that is limited, or completed, or both.