An Introduction to Infinities
How much do you know about infinities? You probably have heard of a number that is the biggest number possible, or impossibly large.
In order to explain infinities, I first need to define some terminology.
Integers are numbers used for counting. The set of integers is defined as all numbers without fractional components. Some examples are: -12, 0, 1, 3, 42, 17.
Infinity is the concept of an object beyond the reach of the natural numbers. It was first conceptualized by a Russian Mathematician named Georg Cantor, who not only introduced infinity, but also revealed that there were multiple infinities that existed.
Cantor provided a controversial proof of infinities, which said that some infinities were larger than others. At first, this may seem impossible—how can an object be larger than another object that is infinite?
Cantor based his proof on a branch of Mathematics that is seemingly useless: Set Theory. A set is a collection of objects—for example, we can have a set containing 1, 2 and 3. These objects in the set are called elements.
In mathematical notation, this would look like:
The number of elements in this set, or the Cardinality of the set, is 3.
This is denoted as:
If we now have another set, B, containing objects, such that
How do we know that the sets have the same size?
One way we can see this is by counting the number of elements in the second set. We can clearly see that this set also has 3 elements in it, and so we know that they have the same size.
Another way we can do this is by comparing our second set, B, to our first set, A by mapping.
We can map the first element from set A , 1, to the first element of set B, chair.
We can map the second element from set A, 2, to the second element of set B, table.
We can map the third element from set A, 3, to the third element of set B, hat.
Since every element from A is mapped on to exactly 1 element from B, the sets have equal size.
Let’s take a look at infinite sets now. The set of integers, which was mentioned at the beginning, is an infinite set—there are an infinite number of integers. The set of integers is denoted by the symbol
The set of even numbers, which we can denote as
is defined as
Intuitively, it may seem like
as there are 2 integers for every even integer: 1 odd integer and 1 even integer. Let’s verify this.
Let’s try comparing the sets by mapping elements from 1 set to the other.
We can map 1 to 2, 2 to 4, 3 to 6, and so on.
If this goes on throughout the sets, we can see that every elements from the integer set maps on to exactly 1 element from the even integer set.
Thus, though it may seem counter-intuitive, the size of the sets are equal.
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This is just the start of what Cantor discovered about infinities. We can use similar methods to compare other infinite sets with each other, such as the set of real numbers, and the set of natural numbers. If you want to know more about infinities, please do comment below to let us know.
