We can grasp the concept of infinity — that which increases with no boundary — yet we cannot completely comprehend its value. Infinity can be both enormously immeasurable or bewildering small. We sometimes apply it to a limit so extraordinary it might as well be limitless, such as the vast expanse of space.
Young children are introduced to the infinite when learning integers. By counting with their fingers, using their real world senses, they begin to build a comprehension of the abstract. It's soon after exploring the number units — one, tens, and hundreds — that children suddenly grasp the concept, shouting "infinity" when counting to large numbers. Buzz Lightyear perpetuated this notion, and then extended it ... "to infinity and beyond".
While we couldn't accomplish the task, we can understand how someone would go about counting the infinite set of integers. We can easily divide dessert portions into equal one-thirds to share with friends. However if we had to do this with a definite precision, we'd never make the cut, caught in an endless loop of mathematical division.
There is more than one infinity, in fact there are an infinite number of infinities. When we explore the integer set, between each integer, oft ignored, is an infinite amount of other numbers: real numbers (fractions and decimals). These are so numerous it is impossible to count them. Integers are a subset of real numbers, so the infinite set fits within the infinite set. Clearly all infinities are not the same.
We use Aleph numbers to help us gauge the depth of infinite sets, the cardinality of the set. The smallest infinity, known as aleph-zero, are the natural numbers: 1, 2, 3, 4, ... and so on. You can always add 1 to any natural number and get a larger number.
Aleph-zero is truly mind-boggling. If we create a new list of numbers, let's say all the even natural numbers, we think this must be smaller than the list of all natural numbers, in fact we'd expect it to be half as big. Yet it is still infinite, and still considered aleph-zero, because we can always add one more to the even number set. Infinity plus 1 is infinity, just as infinity divided by 2 is infinity, and infinity times infinity is still infinity.
"Infinity is not the biggest number, instead it is how many numbers there are."
- Michael Stevens of Vsauce
And as Buzz said, there is something beyond aleph-zero. Rather than growing larger to create the next number, we consider the order of the set instead. For example, if you ran a race and finished after the aleph-zero runner, this would mean you finished after an infinite number of people finished the race. This is called omega, it designates the order of things, not necessarily the quantity of things.
It turns out that the order of things matters greatly. While we often focus on measuring the quantity of results, the order of those results describes our journey to get there. There is no bounds to our level of curiosity, we cannot measure the value of finding inspiration. Instead it's the omega, the sequence and steps we follow that create these moments that lead to enlightenment, that spark innovation, that allow us to create something new, something the universe has never seen before, even in all it's infinite time.
Learn more about Infinity as well as Hilbert's hotel paradox, the Banach-Tarski paradox, and what the Hyperwebster is, by watching this excellent video from Michael Stevens at Vsauce.
The Paradox Pairs series is an exploration of the contradictory forces that surround us. A deeper study finds that these forces often complement each other if we can learn to tap into the strength of each. See the entire series by using the Paradox Pairs Index.