Counting, uncounting, and dust, part seven

The word “dust” has appeared in the title of every post in this series, and we will finally begin to see why. Many counterintuitive marvels remain to be unmasked in the counterintuitive worlds of cardinality and measure theory. Georg Ferdinand Ludwig Philipp Cantor, the mathematician who laid the foundations for the modern understanding of infinity, was responsible for the discovery of some of these truly unbelievable marvels.

There seems to be a whole sub-genre of math blog posts that focus on defining or presenting the Cantor set at different levels of rigor! For our purposes, I will offer two descriptions (without proof). 

• The Cantor set can be described quite “simply” as the set of all real numbers between 0 and 1 whose ternary representation contains only 0 and 2. (We use the decimal place value system in everyday life, while computers use the binary, using only 0 and 1.) The ternary place value system uses 0, 1, and 2 as the only digits. So the number 0.1 in ternary represents the fraction \( \frac{1}{3} \), and 0.22 the fraction \( \frac{8}{9} \), and so on. It is also worth noting that just as we take the decimal 0.999… to be equal to the number 1, we must also take the ternary number 0.0222… to be equal to the ternary number 0.1, and so on.
• The Cantor set can also be seen as the limit of an infinite process of cutting up the closed interval \( [0, 1] \). The rule is simple: at every stage, delete the middle third (excluding the endpoints) of every remaining interval. Repeat ad infinitum. The first few stages of such a process are shown in this picture.

How does the Cantor set measure up (pun intended) using our four different definitions of size?

Counting, uncounting, and dust, part six

Let’s quickly recap of some of the interesting things we have seen so far.

• The real numbers are uncountable, and in fact every interval of the real numbers is uncountable. In other words, there are more real numbers between 0 and 1 than there are integers from negative infinity to positive infinity. 
• However, this alone does not really give us a sense of just how much bigger uncountable is than countable. For instance, we can see that the sequence of fractional numbers \( \{ 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \} \) is also contained within the interval \( (0, 1) \). It is clear, though, that this sequence has the cardinality of \( \mathbb{N} \). It is thus not really clear how it is so remarkable that there are more real numbers in \( (0, 1) \) than there are natural numbers anywhere at all.
• The Lebesgue measure of an isolated point is zero. Consequently, the Lebesgue measure of a countable number of isolated points is also zero. In particular, the Lebesgue measure of \( \mathbb{N} \), the natural numbers, is zero.
• But what if we take an uncountable number of isolated points? Wait, is that even possible?
• And what if we take “non-isolated” points, whatever that means?

Counting, uncounting, and dust, part five

Alright, so now we finally know that there isn’t one infinity any more. In fact, there is an infinite number of infinities, each infinitely larger than the one smaller than it, and so on upwards. Excelsior is a good motto for the infinitude of infinities (as it is for the Coat of Arms of the State of New York).

But we haven’t dealt with another problem: this way of measuring the size of a set just isn’t good enough for our purposes. We need something more refined, something that does distinguish between countable and uncountable while also capturing common-sense notions of the “length” of a line, the “area” of a sheet of paper, the “volume” of a hill, and so on. This is exactly what is studied in the branch of mathematics known as measure theory.

Counting, uncounting, and dust, part four

I’d been completely side-tracked from this set of posts, and I only just realized I had forgotten to finish it. Last time we checked, we had finally figured out that there are different levels of infinity. We had, however, ended up with a rather strange idea: the idea that the set of all real numbers between 0 and 1 is somehow as big as the set of all real numbers in its entirely. In effect, we have shown that there is no difference between a mountain and a molehill!


We thus have two different lines of inquiry to pursue:

1. Are there levels of infinity beyond the infinity of the real numbers?
2. Are there ways to measure the size of a set that can distinguish mountains from molehills?

Levels of infinity
We begin with the idea of a power set: the power set of a set is defined as the set of all subsets of the set. Huh? An example should make things clearer: if \( A = \{ a, b, c \} \), then the power set of \( A \), sometimes written as \( \mathcal{P}(A) \) and sometimes \( 2^{A} \) (we’ll see why), is the set \( \mathcal{P}(A) = \{ \{ a \}, \{ b \}, \{ c \}, \{ a, b \}, \{ b, c \}, \{ a, c \}, \{ a, b, c \}, \{ \}  \} \). All those braces are necessary, because every element of \( \mathcal{P}(A) \) is a subset (and not an element) of \( A \) itself. We note that \( A \) itself is an element of \( \mathcal{P}(A) \), as is the empty set \( \{ \} \).

Counting, uncounting, and dust, part three

Sequences and convergence
We saw earlier that the rational numbers Q are much more versatile than the natural numbers N for doing a wide variety of mathematical operations. However, there is still one thing the rational numbers cannot do adequately: it is possible to construct sequences of rational numbers that do not converge to a rational number. In formal terms, the rational numbers are not complete.

Why does this seemingly obscure mathematical quirk matter? Because this is the foundation of calculus. The ideas of sequences and limits are fundamental to calculus, and if such limits don't exist meaningfully, then neither does calculus. But there is another, intuitive reason for why completeness matters.

Counting, uncounting, and dust, part two

Extending countability
Having looked at the definition of countability and the natural numbers N, it is but natural (no pun intended) to ask how far this concept can be extended. The natural numbers are pretty awesome, after all, but there are a huge number of things you can't do with natural numbers, including something as basic as subtraction.

Integers
The only operations under which N is closed are addition (that is, you can add two natural numbers and the result will be a natural number) and multiplication, but the latter is pretty limited in the sense that it's basically just repeated addition. So all we can do with natural numbers is "go forth and multiply"; there is no way to reduce them. In other words, there is no "inverse"operation to addition under which N is closed.

Counting, uncounting, and dust, part one

I've wanted to write about this for a while but never really got the chance. One of the coolest things about modern mathematics is infinity: we have found all sorts of ways to study and classify different notions of "size" and "number" that can result in the most counterintuitive results imaginable. I want to look at a few examples, starting today.

Counting
Let's begin with the natural numbers, the set N = {1, 2, 3, ... }. This is "clearly" an "infinite" set, as indicated by the ellipsis. What does "infinite" mean in this case? Now we know that we can add natural numbers together to produce natural numbers; more simply, we can add the number 1 to a natural number n to produce a new natural number n + 1. N is infinite in the sense that there is simply no "end" to it, at least in one direction (because there is such a thing as the smallest natural number, which is 1). To rephrase that, there is simply no way to list every single element of N, whereas for any finite set, no matter how large it may be, you can eventually list out every element.

Before we go any further, let's actually get a handle of what counting means. In real life, when we have to count out a bunch of things, we essentially point at each item once (and only once) and call out the natural numbers in order. We are in effect uniquely indexing every item with one (and only one) of the natural numbers. There is nothing twisted or weird about this; this is the most "natural" (pardon the pun) way of counting things.

What this definition of counting allows us to do is to compare the size of different sets: we can know that a sack of three potatoes is the same numerical size as a sack of three apples because we can index both of them with the set {1, 2, 3}. (It should be clear that it does not matter which potato we're pointing to when we call out a particular number; all that matters that we point to each potato only once and we call out the numbers in order only once.) Now so far this sounds like a ridiculously overdone way of stating the utterly obvious (and I've not even pulled out real mathematical terminology or notation!), but the utility of such abstraction will become more and more evident as we journey along.