First, we will learn about Zeno’s Dichotomy Paradox: Imagine you are starting at one point in space trying to get to another point in space. In order to reach that point, you must first cover half of the distance towards your point. Then, you must cover half of the new distance from where you are now to the point. Then, you must cover half of the new distance once again, and so on and so on forever and ever. Since you are only covering half of the distance each time, theoretically you will never reach your destination, therefore motion is impossible.
Although Zeno’s theory about the impossibility of motion seems pretty absurd given our experiences in reality, his initial thinking is not too crazy. After all, how can an infinite number of segments add up to a finite number? This is where the limit comes into play and saves us from being unable to move anywhere. In the example of the Dichotomy Paradox, we are adding up segments that are ½ of the total remaining distance. So consider the example from the number 0 to number 1. First, we will cover ½ of the distance. Half of the remaining distance would be ¼, and half of that remaining distance would be ⅛, and so on, smaller and smaller pieces. This is known as an infinite series, and although we can keep adding smaller and smaller segments, no matter if we continue this pattern forever, we will never pass the number 1. The sum of the segments grows infinitely close to 1.
One of the key ideas for understanding infinity is the difference between potential and actual infinity. Potential infinity, as described by Aristotle, is a process that has no end. Using our example of the Dichotomy Paradox, Aristotle would have described this as a potential infinity because you can keep splitting the distance in half an infinite number of times. Aristotle himself wrote “This infinite is potential, never actual: … But this number is not separable from the process of bisection, and its infinity is not a permanent actuality but consists in a process of coming to be”. (Aristotle) The idea was that infinity could be approached, but never fully completed. Actual infinity is the idea of having a full picture of infinity. This was contradictory to the idea of infinity at the time, which is why Aristotle rejected this idea.
Another way to think about it is like this: Imagine you are counting up to as high as you can. No matter what number you stop at, you are always able to add 1 more, never reaching the “end” of infinity. This is an example of potential infinity. Now, imagine you have a magic box that contains all whole numbers. You know what numbers are in the box, and you can manipulate them mathematically, and work with them as a set of numbers. This is an example of actual infinity.
This “magical box” is exactly what Cantor was envisioning in set theory, where the magical box is an “infinite set”. Cantor described a set as “a Many which allows itself to be thought of as a One” (Jørgen Veisdal, 2018). In this example, the “Many” are all of the numbers, and the “One” is the box. This allowed Cantor to use actual infinity to expand our conceptual understanding of infinity. This led Cantor to determine that not all infinities are equal. This concept is not very intuitive at first, so let’s look at an example.
For example, let’s take our magical box from before. Remember, this box contains all whole numbers (1, 2, 3, 4, 5… and so on). Now, let’s compare it to another box that only contains even numbers (2, 4, 6, 8, 10… and so on). At first, it seems like the second box is smaller because it only includes every other number from the first box, however this is not exactly correct. We can see this when we match the items from each box. We see 1 match to 2, 2 to 4, 3 to 6, and so on. No matter what number we pick from box 1, there will be a match from box 2, which means that the ratio of numbers in each box is 1:1. This idea is called cardinality, and so these two boxes have the same cardinality.
Now let’s compare our first box to a box of all real numbers between 0 and 1. This box contains (0.1, 0.01, 0.001, 0.0001… and so on, and also includes 0.2 0.02, 0.002… and so on). No matter how you try to pair the numbers from the first box to the second box, there will always be an infinite amount of numbers left over in the second box. Our third box has a greater cardinality than our first or second box.