## Orders of ∞

The idea of infinity is easy to look over upon first glance. It can simply be defined as the idea that numbers go on forever and it is easy to end there. However, there are very developed and well-defined notions of infinity suggest that there are different orders and types of infinity which come with various properties. The consideration of all these are vital to our understanding of infinite quantities and especially in set theory. The exploration of these transfinite numbers  can be split up into two very related categories: ordinals and cardinals.

Ordinals help define order and can even extend into transfinite regimes. The following example should provide some method that hints at the existence of these numbers.

Assume a point moves from $\inline&space;0$ to $\inline&space;\frac{1}{2}$. From there, it moves to $\inline&space;\frac{3}{4}$ and then $\inline&space;\frac{7}{8}$ and this process continues as it cuts in half the distance between it and 1 with each step. It would take an infinite amount of steps to describe this process. One could number these steps with the rule that $\inline&space;0\rightarrow&space;1&space;=&space;S_0$ and $\inline&space;1-\left(\frac{1}{2}&space;\right&space;)^n\rightarrow&space;1-\left(\frac{1}{2}&space;\right&space;)^{n+1}=S_n$. However, assume this process it repeated from $\inline&space;1$ to $\inline&space;2$. This second steps would not have a proper ordering. One could use all the even for the first set and the odds for the second but this would be ill-defined as we want a chronological ordering. This requires an infinite ordinal. Call it $\inline&space;\omega$. This means we count all the steps up to $\inline&space;1$ and the next step is the $\inline&space;\omega$ step. The next steps would simply be numbered $\inline&space;\omega+1,\omega+2,...$ until we reach $\inline&space;\omega&space;\cdot2$. $\inline&space;2\omega$ is slightly different and the distinction is important##. This makes it clear how to order these new numbers. One would number the second set of steps the same way but add $\inline&space;\omega$. In fact, assume this process was repeated an infinite amount of times. Then the move $\inline&space;a-\left(\frac{1}{2}&space;\right&space;)^n\rightarrow&space;a-\left(\frac{1}{2}&space;\right&space;)^{n+1}$ would be called $\inline&space;S_{\omega\cdot(a-1)+n}$. From this, we have created a well-defined ordering of the process. These ordinals can keep continuing $\inline&space;\omega\cdot3,\omega\cdot4,...\omega\cdot\omega,\omega\cdot\omega\cdot\omega,...\omega^\omega,\omega^{\omega^{\omega}},...$. Of course, this goes on forever and it is easy to list out the notation for higher ordinals but this is not really vital to understanding what these numbers really do.

Cardinals help define the size of numbers in some sense and also extend into transfinite regimes. To understand this idea, one must grasp the idea of cardinality. Two sets $\inline&space;A$ and $\inline&space;B$ have the same cardinality if there exists a one-to-one and onto (bijective) mapping $\inline&space;\phi:A\rightarrow&space;B$ .For example, the following sets have the same cardinality as shown.

This idea helps create a well-defined notion for the size of infinities. Consider all the natural numbers and all the even numbers. A simply mapping between the two would be $\inline&space;\phi(x)=2x$. Because there exists a bijective mapping, this means there are just as many even numbers as natural numbers. This is counterintuitive but is simply a result of the fact that infinity does not work under the same algebra that we are familiar with. It is a bizarre creature.

Consider another and even more counterintuitive case. All the rational numbers $\inline&space;\mathbb{Q}$ can be arranged into a table with numerators increasing along one axis and the denominators across the other as shown below.

This begs the question if there are even exists a set that could have a cardinality greater than countable infinity. In fact, there is and the most known example is the real numbers. The real numbers cannot be listed in any order. This fact is made clear with Cantor’s diagonal argument. Assume there was a list of all the real numbers laid out. Then one could always construct another real number that is not in the list by taking one decimal spot from each number and changing it. The image below visualizes this process.From this one can create a winding pattern along the diagonals. In this way, you are covering the fractions in a sequential manner and by ignoring the repeats along this path, one can create another bijective function from the naturals to the rationals. This is an amazing fact because it means that there are as many rationals as there are naturals. In fact, there are so many sets with the same cardinality as the naturals that there is a name for it called countable infinity or $\inline&space;\aleph_0$ because you can literally count in order every term in the set in a well-defined manner. In fact, all the ordinals shown above are countable infinite which I will leave without explanation as an exercise. However, there do exist uncountable cardinals.

By definition, that last number is different from every other number on the list by at least one digit. Because this can always be done, no complete list can ever be created and in some sense this represents a higher level of infinity called uncountable infinity or $\inline&space;\aleph_1$.

From here, one can discuss various topics of transfinite set theory like power sets, transfinite induction, the continuum hypothesis, Beth numbers, Reinhardt cardinals, and so much more. The idea of this post was simply to establish that there is a lot more complexity to infinity than what is usually assumed and it is worth studying it because it can lead to some very interesting results.

I did not really learn this from a singular source but you can start getting an idea of these concepts of infinity from this video by Vsauce.