Transfinite numbers

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In my 2023-11-18 post, Cantor's Paradise: From Empty to Infinity and Beyond, I introduced the idea of multiple sizes of infinity and Cantor's diagonalization proof. Let's quantify those sizes of infinity. Once you get into the infinite, or what Cantor called the transfinite, you need two different notions of number: cardinal and ordinal.

Cardinal numbers measure the size of the set, whereas ordinal numbers measure the position of in an ordered transfinite sequence. Let's start by naming some of these transfinite cardinals. Consider the set ℤ of integers. It is countably infinite, whereas the set ℝ of real numbers is uncountably infinite. The symbols used to name these first two transfinite numbers are:

$$\begin{align*} |\mathbb{Z}| &= \aleph_0 \\ |\mathbb{R}| &= c \\ \end{align*}$$

The |S| notation means "size of S", and the \( \aleph_0 \) notation means "aleph null". Aleph is the first letter of the Hebrew alphabet because the mathematicians ran out of Greek letters.

Ordinal numbers will immediately be familiar to anyone who remembers being a kid and playing the "who can name the bigger number" game. Your friend will say infinity, and you, being a clever genius, will say infinity + 1. And the game will continue until one or both of you gives up or gets creative. Infinity times infinity!

Well, dear friend, I'm here to validate your childish but brilliant scheme. Adding 1 to a transfinite ordinal is totally legitimate. Ordinal numbers are defined in terms of well-ordered sets. A set is well-ordered if every subset has a least element in it. So ℤ is not well-ordered, because the set of negative integers, \(\{...,-4,-3,-2,-1\}\), does not have a least element. But the set ℕ of natural numbers is well-ordered. In fact, ℕ is the prototypical well-ordered set.

Ordinal numbers are typically defined as sequences of natural numbers like this:

$$\begin{align*} 0 &= \{\} \\ 1 &= \{0\} \\ 2 &= \{0, 1\} \\ 3 &= \{0,1,2\} \\ &\vdots \\ \omega &= \mathbb{N} \end{align*}$$

Remember that ℕ is the set \(\{0, 1, 2, 3, ...\}\) of all non-negative integers. Also ω is the Greek letter omega. ω is the smallest countable ordinal. Here is where we can distinguish the meaning of cardinals and ordinals.

Using the recursive method suggested above, where we define the next ordinal in the sequence as the set containing all the previous ordinals, we can define "infinity + 1" like this:

ω+1 = {0,1,2,3,...,ω}

From this, we can use set equality to show that ω ≠ ω+1, so infinity plus one is not equal to infinity, so long as we interpret "infinity" as meaning "some transfinite ordinal".

$$\omega + 1 \neq \omega$$

For cardinal numbers like \( \aleph_0 \), countable infinity, we define two cardinals as being the same size whenever we can put them into a one-to-one correspondence. A good example of a one-to-one correspondence is the proof that there as just as many even integers as there are integers. That is: you can take a countably infinite set like ℤ, delete all the odd numbers, and the set is still the same size:

$$\begin{align*} \{...,-3,& -2,& -1,& 0,& 1,& 2,& 3,& ... \} \\ \{...,-6,& -4,& -2,& 0,& 2,& 4,& 6,& ... \}  \end{align*}$$

You can see from above that you can delete the odd numbers, then "pull in" all the even numbers and perfectly pair them up with the integers, so by definition, the two sets are the same size.

This means that for transfinite cardinal numbers like \( \aleph_0 \), adding 1 to the set will not increase it:

$$ \aleph_0 + 1 = \aleph_0 $$

So now that we have introduced cardinals and ordinals, and some transfinite numbers, we will pick up in the near future to talk about the hyperreal numbers, surreal numbers, and infinitesimals. The hyperreals contain the real numbers, the infinitesimals (which Newton used to invent calculus), and the transfinite ordinal numbers. The surreal numbers even let you do things like this: 1/ω = ε, where ε is the largest infinitesimal number. The fact that there is a consistent algebraic number field that lets you divide and multiply infinities and infinitesimals is mind blowing. I'll work up to it, but for now it's late, and I have to sleep.

#math #infinity