Infinity is a subject we have touched on before. However, it is not a subject we have covered in great depth. Today we are going to fix that and go into more depth. We will see why infinity is not a number, is not a quantity and how it refers to a kind of potential.
What is Infinity
If you look at Google, we find that it provides the following mathematical definition of infinity:
A number greater than any assignable quantity or countable number (symbol ∞).
Is infinity a number? What is a number and does infinity qualify as a number?
A number is an abstraction that refers to quantity. The concept of quantity counts something, measures relationships or measures some measurable aspect of something.
Counting something is identifying a kind of relationship and saying “there are these things that are related in as far as they are instances of the same class of object”. The “number” of such objects refers to a collection of instances of something. This one and this one and so forth until each of them has been identified. In the context of counting, the number of entities is simply a way of referring to all the entities in the collection.
Numbers are also used to measure relationships or properties of things. Note that such numbers are simply measurements that refer to relationships between entities or properties of entities. The numbers refer to some aspect of the relationship or attributes in question.
Note that in either case you are either referring directly to entities or measuring some aspect of something. You are identifying quantifiable relationships. That is the purpose of numbers, to measure such quantifiable relationships.
So, then, is infinity a number?
No, it is not a number. Why not? Because it does not refer to quantity. It does not refer to countable things or quantifiable and measurable attributes of entities. It refers to a potentiality. What kind of potentiality?
To understand this, we will need to return to a classic example of a series that is said to be infinite, the natural numbers.
The natural numbers, also called the counting numbers are all the numbers used for counting. So, it includes 1, 2, 3 … 100 … one hundred billion and so forth.
How many natural numbers are there? Does this sequence of numbers have a size? Are there only so many natural numbers and no more? No.
You could start listing natural numbers at any point. You can start with 1 or three hundred Googleplex (10(10^100) or one with ten to the power of one hundred zeroes). But at some point, you must stop counting them. You cannot keep listing these numbers indefinitely, you must stop at some point.
No matter where you eventually stop, there is always the potential to have progressed and to have counted out more natural numbers.
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