Understanding Infinity

A Guide to Abstract Thinking in Mathematics

Photo by freddie marriage on Unsplash

We constantly use the word infinity in both our conversational language and to describe scientific concepts. However, infinity as a physical concept is impossible to visualize. One example is the statement ‘the Universe is infinite’. It’s a fine statement to make, but no human being can visualize an infinite universe. You can articulate that the universe is infinite, but you cannot appreciate what that actually means in terms of physical appearance.

I can actually prove this, under the assumption that our lives are finite. Here’s my proof. Imagine that you are visualizing the universe. You start at a point in your visualization and you move forward. One of two things happen. Either you stop visualizing and make a comment like ‘and so on forever’. Thereby you have made your visualization finite. Or you never stop moving forward in your visualization until you die, and then you have stopped and your visualization is finite.

So, in order to understand and work with infinite quantity, we need to think about it abstractly. We need to develop a way of understanding infinity that does not require diagrams or visuals. Luckily, that is what mathematicians are very, very good at.

How do mathematicians think about infinity?

To mathematicians, there are two types of set. There are finite sets — meaning that you can count the elements of the set and you can be sure that your counting will stop at a certain point.

Or there are non-finite sets. Non-finite sets are sets where there is no way of counting the elements in a way that your count will terminate. There are two important consequences of this:

• We cannot think of the size of a non-finite set in the way that we think about the size of finite sets. With finite sets we can use simple arithmetical logic — for example. the union of two disjoint sets of size 3 and 8 is a set of size 11, or if set A has 5 elements and set B has 5 elements we can say that sets A and B are the same size.
• We need to have a new way to compare non-finite sets to determine if they are the same size or not.