Infinity, an Infinite Number with Infinite Questions

Infinity has a way of twisting your brain. There is “countable” infinity, which, of course, could never be counted in anyone’s lifetime, along with “uncountable” infinity, which is even bigger than the “countable” one. There are even infinities that are defined as infinite powers of infinity. Another way to say it is, infinity comes in different sizes; some infinities can be counted step by step, others are so vast they cannot be counted at all, and mathematicians even describe infinities that are powers of other infinities.

To help us clear the confusion around this matter, we interviewed Dr. Karagila, an Academic Fellow at the University of Leeds in the UK. He is very knowledgeable in set theory, has a PhD in mathematics, and was interviewed in a Numberphile video one year ago.

We sent him some questions, and he responded with a loaded email – way too much to put into this article. Therefore, we are going to summarize the most important and interesting parts of his responses.

Before we start, we will discuss the notation of these large infinities. The first infinity, represented by ℵ0 (pronounced “aleph nought” or “aleph null”), is the size of the set containing all the natural numbers. This means that if you have the list 1,2,3,..., then ℵ0 is the size of that list. Another property of aleph numbers is that for any α, where α is an integer, ℵα+1 is the number of types of well-orderings there are of ℵα. A well-ordering is an order with a clear starting point and a clear way to continue the sequence. For example, 1,3,5,...,2,4,6,... is a well-ordering, since given any number in this series, you could say what is next.

The first question we asked was “What does 2^ℵ0 even mean? Is it the same as 3^ℵ0? Is it the same as ℵ0^ℵ0?”. This was his response: “The meaning of exponentiation x^y, for cardinal numbers (aka natural numbers), is ‘How many functions exist between sets X and Y, where X has cardinality x and Y has cardinality y’. Here, a function is defined on the entire set of X, but a function here is not just an equation or something like that, but rather the general and abstract mathematical concept.” This seems to be a pretty complicated response to a hard question, since we are dealing with infinities. Simply put,  x^y means the number of possible functions from a set of size x to a set of size y. For example, 2^2 could be represented as the number of functions between {1,2} and {3,4}. Dr. Karagila also noted that changing the value of x or y doesn’t change the result; therefore, 2^ℵ0 and 3^ℵ0 are the same. This is true for ℵ0^ℵ0 also.

In the Numberphile video, Dr. Karagila talked about the value of 2^ℵ0. He said that this value was greater than ℵ0, but they could not determine which infinity it was. We asked Dr. Karagila why this was, and he responded that we can think of this value in two different ways. It can either be the size of the real numbers or we could ask the question: “What value of solves the equation 2^ℵ0 = ℵα?” Set theory cannot solve this. In fact, it has been proven that this is unsolvable. He also mentioned that from a philosophical stance, the value of 2^ℵ0 could be assigned to different ℵα in different mathematical universes.

While this email only covers the basics of set theory, feel free to do some research of your own. To view the full contents of the emails, go here. Good luck on your set theory journey!