The Book of Numbers
The Book of Numbers: A nonstandard ordering of the natural numbers
A fun example of nonstandard ordering of the natural numbers. Idea taken from Chapter 1 of The Book of Infinity, by Joel David Hamkins.
Take the natural numbers ℕ, to be 0, 1, 2, 3, … Note, many times the convention is to start with 1, but we choose to include 0 here.
Consider the following unusual ordering of the numbers up to 10 shown here:
8 5 4 9 1 7 6 10 6 3 2 0
Is this ordering random? Can you spot the underlying algorithm?
Maybe consider viewing it another way:
eight, five, four, nine, one, seven, six, ten, six, three, two, zero
That’s right, it’s ordering alphabetically using the English language.
The question is: What would an ordering of all the natural numbers according to this algorithm look like?
Imagine an infinite book of all the numbers, ordered according to this scheme. The ever larger numbers will appear eternally between and amongst the smaller nunbers already considered.
If you think about it a bit, you’ll soon realize that there are some rough edges that first need to be taken care of to unambiguously apply it. For example, is 100 written as “one hundred” or simply “hundred”? And what about numbers like a google, and the more rare ones that have been defined? We adopt a digit-pronounciation naming convention to remove the ambiguity. So, for example, the number 7216 is entered alphabetically according to the spelling it would have if written out as seven-two-one-six, and so on. Notice also, that 7216 would precede 72165, using the usual convention that initial segments of words come earlier in alphabetical order.
The First Number
Notice that there is a first number in the book, 8. This is because it begins with “e” and the only other numbers beginning with “e” also begin with 8, followed possibly by additional numbers.
The First Chapter
For a similar reason, the next number after 8 is 88, and then 888, and so on. There is no number alphabetically between any of these numbers. Thus, what might be seen as the first chapter of The Book of Numbers would be 8 88 888 8888, etc.
Successors
Pressing the idea a bit harder, you can see that every number (except 0) has an alphabetical successor, which is obtained simply by appending an 8 at the end of the decimal representation. Every number except zero in the alphabetical order has a successor, and a next successor, and so on, making a little copy of the natural numbers proceeding forward alphabetically just from that number.
Predecessors
What comes just before 532? Is there an immediate predecessor? The natural candidate would be 533, which certainly comes before 532 alphabetically. But so does 5338, 53388, and 5334568, and on and on. The same argument works for any number not ending in 8. If a number does not end in 8, then for any number preceding it alphabetically, we can tack an extra 8 at the end, and this number would come in between them. By contrast, if a number does end with 8, then it will have a predecessor simply by removing the 8.
Thus we see that every number (except zero) is sitting in its own little chain of successors, a chapter initiated by a number not ending in 8, followed by the iterated-8 successor sequence. For example, the entire chapter 532 appears before the chapter starting with 5325, like this:
Density of Chapters
Is there anything between these two chapters? Yes, there is. There is an entire iterated-8 successor sequence strictly between them. Simply consider the sequence starting with 52387:
And indeed, there are entire further chapters between any two of these, and so on infinitely.
Thus, the chapters of The Book of Numbers are themselves “densely” ordered, which means that strictly between any two of them, we can find another.
No Last Chapter Before 0
There is no last chapter before 0 because to any number before zero we can append a 7, 2, 5, or what have you, which will be later alphabetically, but since it doesn’t end in 8, it starts a new chapter, the beginning of an iterated-8 successor sequence. And the picture further unfolds to look like this:
The Overall Order Type
The analysis given so far is enough to conclude the overall order type of The Book of Numbers. Namely, we have divided the book into the iterated-8 successor sequence chapters, each starting with a number whose decimal representation does not end with 8, except for the very first chapter, which consists of 8, 88, 888, … and so on. After that first chapter, the chapters continue in a densely ordered manner, with no final chapter before the single final number 0 appears on the last page of the book.
From this information it follows that the order type of The Book of Numbers is given by:
ℕ · (1 + ℚ) + 1
Where ℚ refers to the rational numbers (all the fractions), which is a countable dense linear order, and 1 + ℚ is such a dense linear order with an initial point.
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