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 do just that.
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 Google “infinity”, you will 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.
The purpose of numbers is to measure 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 look at an 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? If you kept listing them, would you eventually run out? No.
You could start listing natural numbers at any point. You can start with 1 or three hundred Googleplex (a Googleplex is 10(10^100) or one with ten to the power of one hundred zeroes).
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.
At no point will you have “run out of natural numbers”.
Had you have kept going, you always could have counted out more numbers in the sequence.
Or suppose you start with one and form a sequence that starts with one and then halves the previous number each time. So, you start the sequence like this:
1, 1/2, 1/4, 1/8 .... 1/256 ... 1/1,048,576 ....
Is there any point at which you can stop halving the previous number? There is not. There is always the potential to continue halving the previous number to get an even smaller number.
Eventually, you might start making up names for the extremely small numbers, but if you were to keep going, you could write down these extremely small numbers.
Or, suppose you start listing the digits of the number pi. You list one digit, and then you list another and another and you find that you can keep listing digits of pi for as long as you want, you can always find further digits of pi.
The sequence of digits in pi has no hard limit, you can keep listing them for as long as you want. But at some point you will have to stop, you cannot do for it forever.
This is what infinity refers to.
It refers to the fact that when progressing in certain mathematical sequences, you must stop at some point. But no matter where you do stop, there is always the potential to have progressed in the sequence.
That is, no matter where you stop in such a sequence, if you were to have kept going, you would never run out of terms in the sequence. The sequence will never run out of terms.
The size of such mathematical sequences is not quantifiable. It does not make sense to say that such sequences have a size. Yes, you can count how many terms you identify within a sequence, but the sequence itself does not have a size.
Such a sequence has terms and no matter how many of them you identify, there is the potential to have continued. But there is not a specific number of elements in the sequence, it does not have a size. There is not a quantifiable number of elements in the sequence. The concept of size simply does not apply to such sequences.
In this sense, the natural numbers are an infinite sequence. This simply means that no matter how many of them you find, there is always the potential to find more of them.
It is important to note that you cannot ever identify an infinite number of terms in such sequences. You can only identify so many terms before you must stop.
No identification of terms can go on forever. Such processes go for a finite period, they must stop at some point, as must all processes.
Infinity refers to a potentiality to continue in a sequence. Infinities are potentials but not actualities. There are no actual infinities.