# Chapter ᚏ(R): The gateway
*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T1, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*
[Guide for AI readers](index.md) · [Its glossary cards](ch01-cards.md) · [Its references](ch01-refs.md) · [Next chapter](ch02.md)
*A footnote such as [r-1.3] is reference entry 1.3 and links to it. Section 1 of 28; chapters are named by their Ogham letter, never by number.*
## At the gateway
ᚏ – [Ruis](ch01-cards.md#g-og-ruis): associated with the elder tree, Section 1
## Chapter ᚏ(R): The gateway
> “The essence of mathematics lies precisely in its freedom.”
> — Georg Cantor · *Grundlagen einer allgemeinen Mannigfaltigkeitslehre*, 1883 [r-1.13](ch01-refs.md#r-1.13)
Formula: ([ᚏ4.16](ch01-cards.md#g-rose))/(([𝟘](ch01-cards.md#g-tropical)))
#### Type of number
Undefinable · Non-real object ·
Key concepts: Undefinable real numbers, the extended reals, and the max-plus tropical semiring. It includes symbols you may not have seen before, such as: [ᚏ4.16](ch01-cards.md#g-rose), [:=](ch01-cards.md#g-defsym), and [𝟘](ch01-cards.md#g-tropical). Here “nothing” is infinite: the tropical zero is [−∞](ch01-cards.md#g-neginf), and a number nobody can pin down, divided by it, still comes home as 0.
What better way to begin a piece of mathematical art than with a number our chosen language cannot define?
As I built this formula out of more and more complicated numbers, it bothered me that all of them together are a drop in the bucket beside the numbers our chosen language can never describe. I really wanted some of those in the piece too. But how do I write down something that can’t even be expressed, let alone calculated, when nobody can know its value? What to do… what… to… do? Read on!
Field notes
### the left side
One chosen number over one [tropical zero](ch01-cards.md#g-tropical). The numerator comes from the gap between what can be defined and what exists; the denominator comes from an arithmetic in which adding means taking the larger.
#### Numerator
[ᚏ4.16](ch01-cards.md#g-rose) [:=](ch01-cards.md#g-defsym) x / 4.16, x [∈](ch01-cards.md#g-elem) [ℝ](ch01-cards.md#g-reals) [∖](ch01-cards.md#g-setminus) [D](ch01-cards.md#g-defreals), x ≠ 0: a finite, nonzero, undefinable real; a [Ruis number](ch01-cards.md#g-ruis). This particular one is called Rose’s number: named for my daughter, stamped with her birthday, and symbolized by the elder tree.
First, the numbers we do not choose from. Every integer... every fraction... every number a formula can pin down... every number a program could print digit by digit... and even a number no program can print but a single sentence can specify, like [Chaitin’s Ω](ch05-cards.md#g-omega)... has a definition we can point to. All belong to [D](ch01-cards.md#g-defreals), the collection of real numbers that can be pinned down by a finite, exact definition. [r-1.1](ch01-refs.md#r-1.1)
But Georg Cantor showed that the [real numbers](ch01-cards.md#g-reals) are [uncountable](ch04-cards.md#g-cantorcount): simply too many to be listed, even in a list that goes on forever. Our language is built from a finite alphabet, so its formulas *can* be listed, and we run out of formulas long before we run out of numbers. The numbers left over, [ℝ](ch01-cards.md#g-reals) [∖](ch01-cards.md#g-setminus) [D](ch01-cards.md#g-defreals), sit in the dark: no formula or equation in that original language (the formal language of set theory, fixed before we start) can single one out on its own, and no computer program can calculate one to any requested precision. Yet they make up almost the entirety of the [real line](ch01-cards.md#g-reals). [r-1.1](ch01-refs.md#r-1.1)
Let that sink in. Throw an ideal mathematical dart at the real number line between 0 and 1, with every part of that interval equally likely, and your chance of hitting a number outside [D](ch01-cards.md#g-defreals) is greater than 99.99999999999999%. In fact, [it is 100%—probability 1, lol.](ch01-cards.md#g-probability-one) The definable numbers are still there, but their whole countable collection has probability zero under this ideal draw.
Give us a million years, a million languages, and a million computers, and we would still have only a listable collection of finite descriptions. Almost every target would remain beyond all of them. This is the “normal” crowd in the everyday sense: the overwhelming majority. The numbers we can actually specify are the exceptions. [r-1.17](ch01-refs.md#r-1.17)
I wanted these “normal” numbers in the equation too. I’ve seen them described as [undefinable reals](ch01-cards.md#g-ruis), [unnameable](ch01-cards.md#g-ruis), and [indefinables](ch01-cards.md#g-ruis). Anne Newstead and James Franklin even compare them to the [dark matter of mathematics](ch01-cards.md#g-ruis). The words have different flavors: what we cannot define, what we cannot name, an enormous hidden population. For this book, I will call them **[Ruis numbers](ch01-cards.md#g-ruis)**, borrowing a letter from Pictish Ogham, a script with no numerals of its own. These numbers already exist among the reals. What I needed was a way to put one on the page and show how I could calculate with it. Here is the notation I chose.
We choose one x from that dark pool and hold it fixed throughout this book. For any nonzero rational label b, we write [ᚏb](ch01-cards.md#g-ruis) [:=](ch01-cards.md#g-defsym) x/b. Every ᚏ in this book is built from that same x; a formula outside this book may choose another x and give the same labels different values.
The symbol [:=](ch01-cards.md#g-defsym) means “is defined as”: it defines the relationship between our new symbol and the chosen x, without specifying x’s value. The symbol ᚏ is [Ruis](ch01-cards.md#g-og-ruis), the Pictish Ogham letter associated with the elder tree and, in modern tree lore, with transitions.
The label 4.16 is a personal choice. Once I needed a number to go with an R, who else would come to mind but Rose, my daughter? And what better number than her birthday, April 16? So this particular number, [ᚏ4.16](ch01-cards.md#g-rose), is her number: **[Rose’s number](ch01-cards.md#g-rose)**.
So now I have a symbol for a number whose value I cannot specify in our original language. Can I put it into the equation and calculate with it? You bet! I do not need its value when the calculation has the same answer whatever our chosen x may be.
The division itself works for any finite real number: over our upcoming denominator, they all come home as 0. I chose an undefinable one to make room in the artwork for **ALL** the real numbers: the “special” ones with celebrated histories, and the insanely large infinity of “normal” ones whose individual values our original language cannot express. One chosen number gives that dark crowd a place in the picture. We cannot even say exactly which one it is, ha ha ha.
The mechanics of the lock
For the rigorous reader, it is worth asking: does dividing by 4.16, or giving the number a name, accidentally rescue it from the dark pool? It does not. (Everyone else can skip to the denominator; nothing later depends on this.)
**The two-way street.** Division by a specific nonzero rational like 4.16 is perfectly reversible. If x/4.16 had a definition in the original setting, multiplying that result by 4.16 would give us a definition of x there. Our chosen x has no such definition, so x/4.16 cannot have one either. The same argument works for any nonzero rational label b: every number we write as ᚏb [:=](ch01-cards.md#g-defsym) x/b stays outside [D](ch01-cards.md#g-defreals). We can even know an exact relationship without knowing either value: 4.16 · [ᚏ4.16](ch01-cards.md#g-rose) = 4.03 · ᚏ4.03 = x.
**What counts as a definition?** The technical phrase is “without parameters”: a formula must pick out a number in the original language and setting without being handed extra objects as clues. If we allowed an arbitrary real r as a supplied value, “the number equal to r” would pick it out immediately, and every real would qualify. That is why [D](ch01-cards.md#g-defreals) counts definitions that work on their own; familiar rational labels such as 4.16 already have such definitions. Our chosen x is an ingredient in the new notation, not a definition the original language could produce on its own. [r-1.1](ch01-refs.md#r-1.1)
**The zero loophole.** Zero is about the easiest number in the universe to define, so it is already in [D](ch01-cards.md#g-defreals). Because x is drawn strictly from [ℝ](ch01-cards.md#g-reals) [∖](ch01-cards.md#g-setminus) [D](ch01-cards.md#g-defreals), x cannot be zero; writing x ≠ 0 simply says so out loud.
**The invisible properties.** Every fraction, every root of a polynomial with integer coefficients, and every number a program could print lives inside [D](ch01-cards.md#g-defreals), so the fence around [D](ch01-cards.md#g-defreals) does all the work: her number is irrational (not a fraction), transcendental (not such a root), and [uncomputable](ch05-cards.md#g-uncomputable) (no program prints it), without a single proof about it in particular. [r-1.1](ch01-refs.md#r-1.1) We do not even know its sign; it could be negative or positive. The math does not mind, since any finite number over [−∞](ch01-cards.md#g-neginf) is 0.
**The setting dependency.** Definability depends on the mathematical setting. As Hamkins, Linetsky, and Reitz explain, if [ZFC](ch27-cards.md#g-zfc), the standard axioms of set theory, is consistent, there are models of it (whole mathematical worlds that obey those axioms) in which every real can be defined by a formula alone. [r-1.2](ch01-refs.md#r-1.2) Seen from outside, though, such a world has only countably many reals: it is not the full real line. This book fixes one setting and counts [D](ch01-cards.md#g-defreals) from outside it, against the ordinary [uncountable](ch04-cards.md#g-cantorcount) [real line](ch01-cards.md#g-reals), and that is what guarantees the dark pool is not empty.
**The dart and the million languages.** Our dart is an ideal uniform draw from [0, 1]. A countable set of points has total length zero, so [D](ch01-cards.md#g-defreals) has probability zero and its complement has probability 1. The million languages must each have a finite or countable stock of symbols, with their rules and interpretations fixed independently of the target: even together, their finite descriptions remain countable. Everyday languages qualify too, once we fix what counts as an exact description. A name invented after the dart lands does not: it changes the language instead of describing the number in it. [Probability 1](ch01-cards.md#g-probability-one) does not make a definable outcome logically impossible. [r-1.17](ch01-refs.md#r-1.17)
**The naming loophole ([Richard’s paradox](ch01-cards.md#g-richard)).** By declaring [ᚏ4.16](ch01-cards.md#g-rose) [:=](ch01-cards.md#g-defsym) x/4.16, didn’t we just give x a definition? No. We expanded our alphabet with a new symbol; we did not write a formula in the original language that singles x out. [r-1.7](ch01-refs.md#r-1.7) Richard’s paradox of 1905 shows why that line matters: list every phrase that defines a real number, build a number that differs from each one on the list, and that recipe seems to be a phrase defining a number the list missed. Its cousin, [Berry’s paradox](ch01-cards.md#g-berry), does it in one line: “the smallest positive integer not definable in under sixty letters.” [r-1.11](ch01-refs.md#r-1.11) The way out is that “defines a real number” can only be judged from outside the language, and a new symbol changes which language we are using. In the expanded language the number can be named; it still has no parameter-free definition in the original language. [r-1.7](ch01-refs.md#r-1.7) Giving the void a nickname on the page does not illuminate its digits.
#### Denominator
([𝟘](ch01-cards.md#g-tropical)) = [−∞](ch01-cards.md#g-neginf); the tropical zero, an exotic number chosen to let the undefinable above communicate with the known world to the right.
I could have chosen several ways to do primitive calculations with a [Ruis number](ch01-cards.md#g-ruis). For art’s sake I chose an exotic zero, one that looks on the page as though we are illegally dividing by 0, but that instead sidesteps our not knowing the value of [ᚏ4.16](ch01-cards.md#g-rose) by “vanishing” it.
The symbol 𝟘 represents [the tropical zero](ch01-cards.md#g-tropical). The term “tropical” sounds like it belongs to geography, but it is actually an affectionate mathematical inside joke. When the Brazilian computer scientist Imre Simon put this kind of arithmetic, in its mirror-image [min-plus form](ch01-cards.md#g-minplus), to work on problems about finite automata in 1978, his French colleagues later dubbed it “tropical” in his honor, a word that meant nothing deeper than the French view of Brazil. [r-1.6](ch01-refs.md#r-1.6)
In this tropical world, the rules of arithmetic are bent. To [“add” two numbers](ch01-cards.md#g-tropplus) means simply to take the larger of the two: if you have 3 and 5, their tropical sum is 5. [r-1.5](ch01-refs.md#r-1.5) So what is the “zero” of this world, the number you can tropically add to anything without changing it? The answer is [negative infinity](ch01-cards.md#g-neginf). Because the larger of any number and [−∞](ch01-cards.md#g-neginf) is always the number itself, an arithmetic is born whose “nothing” is infinite. (To [“multiply”](ch01-cards.md#g-troptimes), you add in the ordinary way, so this world’s “one” is [the ordinary 0](ch01-cards.md#g-tropone).) Simon’s mirror image takes the smaller instead, and its zero is +∞; this book takes the larger, so that its zero is negative. [r-1.5](ch01-refs.md#r-1.5)
This gives us the final act of our equation. The parentheses around our [tropical zero](ch01-cards.md#g-tropical) tell it to evaluate to its true value: [−∞](ch01-cards.md#g-neginf). The fraction bar then carries that value back toward ordinary arithmetic, in [the extended reals](ch01-cards.md#g-extreals): the ordinary number line with −∞ and +∞ added at its two ends. We divide Rose’s unique, finite, undefinable number by an infinitely deep bottom, and any finite number divided by [negative infinity](ch01-cards.md#g-neginf) comes home as exactly 0.
The mechanics of the denominator
For the rigorous reader, the step from a [tropical zero](ch01-cards.md#g-tropical) to an ordinary zero happens in two strict, legal moves. (Everyone else can skip ahead; nothing later depends on this.)
**The tropical semiring.** In the max-plus tropical semiring, addition is a [⊕](ch01-cards.md#g-tropplus) b = max(a, b). The additive identity must satisfy max(x, 𝟘) = x for every x, and the only value that does is [−∞](ch01-cards.md#g-neginf). [r-1.5](ch01-refs.md#r-1.5)
**The extended-reals convention.** The fraction bar does not belong to tropical arithmetic; it belongs to the [extended real numbers](ch01-cards.md#g-extreals), [ℝ](ch01-cards.md#g-reals) [∪](ch01-cards.md#g-union) {[−∞](ch01-cards.md#g-neginf), +∞}. There we explicitly adopt the convention that a/([−∞](ch01-cards.md#g-neginf)) = 0 for every finite real a, which agrees with what happens to a/t as t tends to [−∞](ch01-cards.md#g-neginf). [r-1.4](ch01-refs.md#r-1.4)
**Avoiding the classic trap.** At no point do we divide by an ordinary zero. The denominator resolves to [−∞](ch01-cards.md#g-neginf) before the division takes place, so we step around the ancient trap of dividing by zero, and the math stays sound.
The only fraction is (a finite, nonzero, undefinable real) / ([−∞](ch01-cards.md#g-neginf)) = 0: a finite number over −∞ is 0 by the extended-real convention, whatever its sign.
**Fine print.** A bonus to all this swirly math: it looks as if we divide by zero, but the zero in plain sight under the fraction bar is [the tropical zero](ch01-cards.md#g-tropical), and what it hands the division is [−∞](ch01-cards.md#g-neginf). Two settings do the work, in order: tropical arithmetic gives the bracket its meaning, and the extended-real convention gives the quotient its meaning, bringing [Rose’s number](ch01-cards.md#g-rose) home to 0. Neither is ordinary arithmetic all the way: [tropical zero](ch01-cards.md#g-tropical) itself lives only in the semiring and has no multiplicative inverse, and the convention does not give [the extended reals](ch01-cards.md#g-extreals) all the division rules of an ordinary field. [r-1.5](ch01-refs.md#r-1.5) Exactly which French colleague coined the name is disputed; the [card](ch01-cards.md#g-tropical) has both accounts. Max-plus algebra, the version used here, grew up separately, from the early 1960s on, with applications in scheduling, and tropical geometry proper took off around 2000 with Mikhalkin, Sturmfels, and others. [r-1.6](ch01-refs.md#r-1.6)
**Another curious bonus.** The labels let us compare numbers whose values we cannot specify. Since 4.16 is greater than 4.03, 1/4.16 is less than 1/4.03, so with the same x, [ᚏ4.16](ch01-cards.md#g-rose) is closer to zero than ᚏ4.03, whatever x’s sign. If x is positive, Rose’s number is the smaller one; if x is negative, it is the larger one—but either way these are different numbers with a relationship we know exactly.
What this chapter puts on the left side: (a finite, nonzero, undefinable real) / ([−∞](ch01-cards.md#g-neginf)) = 0
We started our journey with [Ruis](ch01-cards.md#g-og-ruis), the elder tree. I like to imagine it as a gateway to a fairy world, and boy did we have some fantastical symbols. Let’s step through the equals sign and explore the other side. It’s a bit more tame, with all but the physical world returning the same answer. Logic is what makes it true, and the least of what makes it beautiful. On her side stands a number our original language cannot define, over a “tropical infinity”; on the other, everything humanity has measured, proven, computed, and named, multiplied together—and the two sides agree.
Watch and read
- [All the Numbers - Numberphile](https://www.youtube.com/watch?v=5TkIe60y2GI) Matt Parker climbs the whole ladder, from counting numbers to the reals that no description can reach; the left side of this book in fourteen minutes.
- [What is tropical geometry?](https://www.youtube.com/watch?v=aYqGanjjUdQ) A short introduction to the arithmetic where adding means taking the larger and multiplying means adding, which is where the denominator lives.
- [Definable real number](https://en.wikipedia.org/wiki/Definable_real_number) Wikipedia: what D is, and the subtlety about models
- [Tropical semiring](https://en.wikipedia.org/wiki/Tropical_semiring) Wikipedia: max-plus and min-plus, and where the name came from