Different Sizes of Infinity

Before you read this make sure you have a basic knowledge of sets.

The "cardinality" of something is the size of a set. For example ­the cardinality of the set {3, 2, 17, -9} is 4, and the cardinality of the set {2, 6, -7} is 3. But when thinking it terms of infinity things become more confusing, I mean how can you categorise different sizes of infinity, infinity surely must just be infinity, right?

Well unfortunately, no. Using the ideas of sets we can find that there are actually different cardinalities of infinity. But to find this we need a more general definition of what being the "same sized set" is. A set {1, 2, 3, ..., x-1, x} has a cardinality of x and the set {1, 2, 3, ..., y-1, y} has a cardinality of y, the sets will be the same size if x = y. This then leads to the definition that being the "same size" is if the sets have the same amount of elements.

When we check if a set is the same size as another set, we simply count the number of elements in both sets and compare. But with infinite sets this is (obviously), impossible. But if we break the task down a bit we find ways of managing this. Take the elements from each set and pair them up, if they do match up perfectly with no doubling up or ignoring any elements then the sets are the same size.

Now if we pair the natural numbers, N {1, 2, 3, ...}, and the integers, Z {-2, -1, 0, 1, 2, ...}, we will be able to see which set is larger. We pair the numbers up like so:

And you can see, that despite your better knowledge, the numbers are pairing up and will continue to do perfectly infinitely. This then means that the sets of both the natural numbers and integers are the same size and thus have the same cardinality. This cardinality is referred to as: ℵ₀ (read as "aleph null" or "aleph zero"). The aleph, "ℵ", character is used as it is the first letter in the Hebrew word 'infinity'. ℵ₀ refers to 'countable infinity', or in simpler terms the lowest possible form of infinity.

When we continue to pair up the numbers we discover that the rational numbers and the natural numbers are too the exact same size and thus the cardinality of the rational numbers is also ℵ₀. In fact the prime numbers are also deemed to have the same cardinality as the natural numbers, integers and rational numbers. I know it seems insane, but it is true.

And going back to my earlier point there are larger cardinalities of infinity. As you would think the next cardinality of infinity is ℵ₁. The size of the real numbers in fact are larger that the natural numbers, integers, etc., this is because the real numbers (in now way at all) can be paired to the natural numbers. Because, between 0 and 1 there an infinite number of real values that are there, in fact the same can be said between 0 and 0.000000001. Thinking of it in this way helps to visualise the fact that the set of real numbers must be larger than the set of natural numbers.

As far as ℵ_x goes it will exist as long as x is a natural number, but as x goes past 1 it becomes much harder to visualise how this infinity is larger than the last. But there can not be ℵ_1.5 for example, this is the continuum hypothesis.