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# Glossary cards — Chapter ᚏ(R): The gateway

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T1, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch01.md)

*Undefinable real numbers, the extended reals, and the max-plus tropical semiring. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Ruis — `ᚏ` {#g-og-ruis}

*Ogham letter*

Ogham letter name meaning ‘red’ or ‘redness’; its word-oghams are “most intense blushing,” “reddening of faces” and “glow of anger,” and the medieval glossators attached it to the elder.

Ruis is one of the letter names that is not a tree name. On the evidence of the kennings it means ‘red’ or ‘redness’. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give tindem rucci, “most intense blushing”; rúamnae drech, “reddening of faces”; and bruth fergae, “glow of anger.” The arboreal tradition reads that redness as a tree: the Auraicept na nÉces glosses ruis with trom, elder, and prints ruamma ruice ruis, “the redness of shame is ruis.” That gloss is medieval, and it is where the book’s elder comes from. What is said today about transition is later still: Graves’s calendar of 1948 makes Ruis its thirteenth and last month, 25 November to 22 December, and elder folklore makes the tree one to ask permission of, a shelter where the fairy court passes on Midsummer’s Eve, and the tree of the Judas legend. Scottish Gaelic ruis is a name for the elder that occurs only rarely in Scottish place names. The book’s own use of the letter is a separate matter: see Ruis numbers.

Cites: [2.15](ch02-refs.md#r-2.15), [2.16](ch02-refs.md#r-2.16), [2.17](ch02-refs.md#r-2.17), [2.19](ch02-refs.md#r-2.19), [2.20](ch02-refs.md#r-2.20)

See also: [Ruis numbers](ch01-cards.md#g-ruis)

### Rose’s number — `ᚏ_{4.16}` {#g-rose}

*Undefinable*

ᚏ_{4.16} := x/4.16: one chosen Ruis number, labeled with Rose’s April 16 birthday and fixed throughout this book.

Rose’s number is the name attached to the label 4.16 in the Ruis notation. Its value here is x/4.16 for this book’s chosen x. A formula outside this book may choose another x and use the same birthday label for a different value. Here, the original choice stays fixed wherever Rose’s number returns, including Chapter ᚋ(M). The date inspired the label; it does not tell us the number’s value. The label cannot smuggle in a definition: if x/4.16 had one in the book’s original language, then “that number times 4.16” would define x, and x has none. Many of the first things we can say about her number are things it is not, and the list is longer than it first looks. Zero has a definition, so it was never in the pool; but so does every fraction, every root of a polynomial with integer coefficients, every number a program could print digit by digit, and even Chaitin’s Ω, which no algorithm computes but one sentence specifies. So her number is irrational, transcendental, and uncomputable, without a single proof about it in particular: the fence around D does all the work, and hers is the only symbol in the equation with the Undefinable badge. The one ordinary question the book leaves open is its sign: x was chosen from ℝ ∖ D, not from the positive reals, and the arithmetic does not care, since a negative finite number over −∞ is 0 just the same. In fact the left side needs none of this; any finite number over −∞ comes home as 0. The undefinability is not there for the logic. It is there so that the one number in the piece that carries her name is also a number whose value our original language cannot specify without parameters. A side note, from a parent. Read the left side’s −∞ as the surreal −ω instead, in the field of Chapter ᚙ(X), and her number does not vanish: ᚏ_{4.16}/(−ω) is a nonzero infinitesimal, smaller than every positive real and never zero, and the equation would not balance until a standard part brought it home to 0. The book’s left side lives in the extended reals, where −∞ is not a field element and a/(−∞) = 0 does that job on its own. So through ordinary eyes, dividing her by infinity reduces her to 0. Through a parent’s eyes she lives in a richer field, and there even that division does not reduce her to nothing. Type: undefinable relative to the book’s fixed setting.

Cites: [1.1](ch01-refs.md#r-1.1), [1.10](ch01-refs.md#r-1.10), [1.2](ch01-refs.md#r-1.2), [1.4](ch01-refs.md#r-1.4), [1.9](ch01-refs.md#r-1.9), [26.9](ch26-refs.md#r-26.9)

See also: [the definable real numbers](ch01-cards.md#g-defreals), [the extended real numbers](ch01-cards.md#g-extreals), [an infinitesimal](ch26-cards.md#g-infinitesimal), [negative infinity](ch01-cards.md#g-neginf), [Chaitin’s constant](ch05-cards.md#g-omega), [surreal omega](ch26-cards.md#g-omegaS), [the real numbers](ch01-cards.md#g-reals), [Ruis numbers](ch01-cards.md#g-ruis), [the standard part](ch26-cards.md#g-stpart), [uncomputable](ch05-cards.md#g-uncomputable)

### negative infinity — `−∞` {#g-neginf}

*Infinity or limit device*

The value the tropical bracket hands to the extended reals; any finite number divided by −∞ is 0 by convention.

The value the tropical bracket hands to the extended reals, ℝ ∪ {−∞, +∞}, the number system the left side is read in. The extended reals are the standard setting of measure theory and analysis, and their arithmetic conventions, including finite/±∞ = 0, are what make the left side equal 0. It is chosen over +∞ so that the piece contains a negative number, which is why the max-plus semiring is used rather than min-plus, whose zero is +∞. It is not a surreal number: in Conway’s surreals, dividing by ω gives an infinitesimal rather than 0, and ∞ itself is a gap rather than a number. What that alternative would do to Rose’s number is a side note on her card. It is the one value the equation is sure is negative: nothing on the right side comes home negative. Some selected negative real values are read through absolute-value bars, including the double torus’s χ, Feigenbaum’s α, ζ(0), ζ(−1), and ψ(1). Every denominator is nonzero in the framework where its fraction is evaluated. Completed chapter contributions on the right are ordinary finite real numbers before joining the book’s product; their internal calculations can contain nonreal or infinite values. And the zero it produces is adopted rather than proved, unlike the settled zeros on the right; the book’s own model for such a convention is 0! = 1 in the garden, which everything else leans on. Type: infinity or limit device, a convention rather than a value.

Cites: [1.10](ch01-refs.md#r-1.10), [1.4](ch01-refs.md#r-1.4), [1.9](ch01-refs.md#r-1.9), [26.14](ch26-refs.md#r-26.14)

See also: [Euler characteristic of the double torus](ch12-cards.md#g-chiS2g), [the digamma function at one](ch17-cards.md#g-digamma1), [the extended real numbers](ch01-cards.md#g-extreals), [Feigenbaum’s α](ch05-cards.md#g-feigA), [positive infinity of the extended reals](ch05-cards.md#g-inf), [an infinitesimal](ch26-cards.md#g-infinitesimal), [the min-plus semiring](ch01-cards.md#g-minplus), [surreal omega](ch26-cards.md#g-omegaS), [the real numbers](ch01-cards.md#g-reals), [Rose’s number](ch01-cards.md#g-rose), [a gap in the surreals](ch26-cards.md#g-surgap), [the tropical zero](ch01-cards.md#g-tropical), [union](ch01-cards.md#g-union), [zero factorial](ch04-cards.md#g-zerofact), [zeta of zero](ch18-cards.md#g-zeta0), [zeta of minus one](ch18-cards.md#g-zetam1)

### the tropical zero — `𝟘` {#g-tropical}

*Non-real object*

In the max-plus tropical semiring, where addition is max and multiplication is +, the additive zero is −∞: an arithmetic whose “nothing” is infinite.

The additive identity of the max-plus tropical semiring, where a ⊕ b = max(a, b) and a ⊗ b = a + b. The identity for max is −∞, because max(x, −∞) = x for every x, so the tropical zero is literally negative infinity, and the tropical one is the ordinary 0. The min-plus version was introduced by the Brazilian computer scientist Imre Simon in 1978 for problems about finite automata; the adjective tropical was coined afterwards by French colleagues in his honor, Jean-Éric Pin crediting Dominique Perrin with it and Simon himself crediting Christian Choffrut, and means nothing deeper than the French view of Brazil. Max-plus algebra grew up separately from the early 1960s, with applications that include job scheduling. Tropical geometry proper, in which polynomials become piecewise-linear shapes, took off around 2000 with Mikhalkin, Sturmfels, and others. Type: non-real object, because 𝟘 lives in a semiring, and only its resolved value, −∞, comes out of the bracket into the extended reals.

Cites: [1.5](ch01-refs.md#r-1.5), [1.6](ch01-refs.md#r-1.6)

See also: [the extended real numbers](ch01-cards.md#g-extreals), [the min-plus semiring](ch01-cards.md#g-minplus), [negative infinity](ch01-cards.md#g-neginf), [the tropical one](ch01-cards.md#g-tropone), [tropical addition](ch01-cards.md#g-tropplus), [tropical multiplication](ch01-cards.md#g-troptimes)

### the real numbers — `ℝ` {#g-reals}

*Function or operator*

The complete ordered field: every decimal expansion, rational or not, and the line the whole ladder of numbers is drawn on.

ℝ, the real numbers, is the set of all points on the number line: the rationals together with every limit of rationals. Dedekind built it from cuts and Cantor from Cauchy sequences, both in 1872, and it is the unique complete ordered field. It is uncountable (Cantor, 1874), which is the fact the governing sentence rests on. The blackboard-bold letter is the usual notation, with ℕ, ℤ, ℚ, and ℂ for the naturals, integers, rationals, and complex numbers, whose examples fill Chapters ᚂ(L) and ᚔ(I). Type: a tool of the framework rather than a value.

Cites: [1.3](ch01-refs.md#r-1.3), [4.21](ch04-refs-1.md#r-4.21)

### the definable real numbers — `D` {#g-defreals}

*Function or operator*

The reals that a finite formula picks out uniquely, without parameters, in the fixed setting; counted from outside, they form a countable set within the uncountable real line.

D collects the reals that a finite formula picks out uniquely, without parameters, in the fixed formal language and mathematical setting used here. A parameter is an extra object supplied as input; allowing arbitrary reals as parameters would make every real definable using itself. Familiar integer and rational labels already have definitions, so they require no such extra input. With the language and setting fixed, there are only countably many finite formulas; viewed externally against the ordinary uncountable real line, D is countable and has measure zero. Familiar algebraic and computable numbers, and definable uncomputable numbers such as Chaitin’s Ω, are on this side. Familiar examples include π and e. The author proposes Ruis numbers as a name for the reals outside D. Hamkins, Linetsky, and Reitz explain why the external viewpoint matters: if ZFC is consistent, there are models in which every real is definable without parameters. A new constant naming x expands the language; it does not give a definition in the original one. The adopted setting. For this book, fix the set-theoretic structure ⟨V_{ω+ω}, ∈⟩ and use its first-order language with no parameters, counting its definable reals from outside against the ordinary uncountable real line. V_{ω+ω} is a fixed level of the cumulative hierarchy of sets, rich enough for the familiar constructions used here, including algebraic and computable reals and the halting probability of our fixed optimal universal prefix-free machine. Hamkins, Linetsky, and Reitz discuss this structure as a background for classical definitions; our use of it is the book’s declared choice. Type: a tool of the framework.

Cites: [1.1](ch01-refs.md#r-1.1), [1.17](ch01-refs.md#r-1.17), [1.2](ch01-refs.md#r-1.2)

See also: [Euler’s number](ch04-cards.md#g-e), [Chaitin’s constant](ch05-cards.md#g-omega), [pi](ch04-cards.md#g-pi), [the real numbers](ch01-cards.md#g-reals), [Ruis numbers](ch01-cards.md#g-ruis), [the standard axioms of set theory](ch27-cards.md#g-zfc)

### set difference — `∖` {#g-setminus}

*Function or operator*

A ∖ B is everything in A that is not in B; here ℝ ∖ D is the reals with the definable ones removed.

The set-difference symbol, A ∖ B, denotes the members of A that are not members of B. In the governing sentence ℝ ∖ D is the collection this book calls Ruis numbers; it is from that collection that this book’s fixed x is chosen. The related symbols of set theory, ∈ for membership, ⊆ for subset, ∪ and ∩ for union and intersection, and ∅ for the empty set, belong to the same language; ∅ is in Chapter ᚚ(P). Type: a tool of the framework.

Cites: [1.8](ch01-refs.md#r-1.8)

See also: [the definable real numbers](ch01-cards.md#g-defreals), [is an element of](ch01-cards.md#g-elem), [the real numbers](ch01-cards.md#g-reals), [Ruis numbers](ch01-cards.md#g-ruis), [union](ch01-cards.md#g-union)

### is an element of — `∈` {#g-elem}

*Function or operator*

x ∈ ℝ ∖ D reads: x is an element of the reals minus the definable reals.

The membership symbol, introduced by Peano in 1889 as an abbreviation for the Latin est, in a lunate epsilon rather than the stylized one now used, is the basic relation of set theory: x ∈ A means x is a member of A. The governing sentence uses it once, to say that the number is chosen from the undefinable reals. Type: a tool of the framework.

Cites: [1.12](ch01-refs.md#r-1.12)

See also: [the definable real numbers](ch01-cards.md#g-defreals), [the real numbers](ch01-cards.md#g-reals), [set difference](ch01-cards.md#g-setminus)

### the extended real numbers — `ℝ ∪ {−∞, +∞}` {#g-extreals}

*Function or operator*

The reals with two infinities added, the setting chosen for the left side of the equation.

The extended real line adds −∞ and +∞ to ℝ, ordered below and above every real, with conventions such as x + ∞ = ∞ and x/±∞ = 0 for finite x, while ∞ − ∞ and 0·∞ are left undefined here. It is the standard setting of limits in analysis and of measure theory—with the caveat that measure theory does give 0·∞ a value, setting it to 0, the natural choice for integration. This book does not, because nothing in the equation needs it. It is the framework the left side of the equation is evaluated in: a finite number over −∞ is 0. Type: a tool of the framework.

Cites: [1.4](ch01-refs.md#r-1.4)

See also: [negative infinity](ch01-cards.md#g-neginf), [the real numbers](ch01-cards.md#g-reals)

### is defined as — `:=` {#g-defsym}

*Function or operator*

The colon-equals defines the relationship ᚏ_{4.16} := x/4.16 using the chosen x; it does not specify x’s value.

The symbol := marks a definition rather than an equation: ᚏ_{4.16} := x / 4.16 introduces the name on the left for the thing on the right. In programming it was the assignment operator of Algol 58 and then of Algol 60; other mathematical forms are ≝ and the phrase “let.” In this piece it defines an individual Ruis symbol in terms of the chosen x. Labels used together in one formula share that x, and this book keeps its choice fixed throughout. A formula outside this book may choose another x. The relationship does not supply a parameter-free definition of the individual value in the original setting. Type: a tool of the framework.

Cites: [1.12](ch01-refs.md#r-1.12), [1.8](ch01-refs.md#r-1.8)

See also: [Ruis numbers](ch01-cards.md#g-ruis)

### tropical addition — `⊕` {#g-tropplus}

*Function or operator*

In the max-plus semiring a ⊕ b = max(a, b); its identity is the tropical zero, −∞.

Tropical addition, written ⊕, is the maximum: a ⊕ b = max(a, b). It is associative, commutative, and idempotent (a ⊕ a = a), and its identity element is −∞, which is why the tropical zero is infinite. In the min-plus version, the kind Simon put to work in 1978, ⊕ is the minimum and the zero is +∞. Type: an operator of the framework.

Cites: [1.5](ch01-refs.md#r-1.5), [1.6](ch01-refs.md#r-1.6)

See also: [the min-plus semiring](ch01-cards.md#g-minplus), [negative infinity](ch01-cards.md#g-neginf), [the tropical zero](ch01-cards.md#g-tropical)

### tropical multiplication — `⊗` {#g-troptimes}

*Function or operator*

In the tropical semiring a ⊗ b = a + b, ordinary addition; its identity, the tropical one, is 0.

Tropical multiplication, written ⊗, is ordinary addition: a ⊗ b = a + b. It distributes over ⊕ because a + max(b, c) = max(a + b, a + c). Its identity is 0, the tropical one, and every element except the zero has an inverse, −a, which makes the tropical semiring a semifield. Tropical polynomials such as x ⊗ x ⊕ 3 ⊗ x ⊕ 5 are piecewise-linear functions, which is the door into tropical geometry. Type: an operator of the framework.

Cites: [1.5](ch01-refs.md#r-1.5)

See also: [the tropical one](ch01-cards.md#g-tropone), [tropical addition](ch01-cards.md#g-tropplus)

### the tropical one — `𝟙` {#g-tropone}

*Integer or rational*

The multiplicative identity of the tropical semiring, which is the ordinary number 0.

In the max-plus semiring the multiplicative identity, written 𝟙, is the real number 0, since a ⊗ 0 = a + 0 = a. So the framework that puts its zero at −∞ puts its one at 0: both identities of ordinary arithmetic have moved down the number line. It is not used in the formula, but it is the reason the tropical zero had to be written 𝟘 rather than 0. Type: integer, exactly 0.

Cites: [1.5](ch01-refs.md#r-1.5)

See also: [negative infinity](ch01-cards.md#g-neginf), [the tropical zero](ch01-cards.md#g-tropical), [tropical multiplication](ch01-cards.md#g-troptimes)

### union — `∪` {#g-union}

*Function or operator*

A ∪ B is everything in A or in B; the extended reals are ℝ ∪ {−∞, +∞}.

The union A ∪ B collects the members of either set; the extended real line of the left side is the union of ℝ with the two-element set {−∞, +∞}. The symbols ∪ and ∩ were introduced by Peano in 1888. Type: a tool of the framework.

Cites: [1.12](ch01-refs.md#r-1.12), [1.4](ch01-refs.md#r-1.4), [1.8](ch01-refs.md#r-1.8)

See also: [the extended real numbers](ch01-cards.md#g-extreals), [negative infinity](ch01-cards.md#g-neginf), [the real numbers](ch01-cards.md#g-reals)

### Richard’s paradox — `Richard’s paradox` {#g-richard}

*Function or operator*

List every English phrase that defines a real number, then diagonalize: the number you build seems to be defined by that very sentence, yet it is not on the list. Jules Richard, 1905.

Jules Richard observed in 1905 that the phrases of a language that define real numbers can be listed, and that Cantor’s diagonal recipe then produces a real number differing from every listed one, although the recipe itself is a phrase that defines a real number. The resolution is that “defines a real number” is not itself a precise predicate of the language; it is the same fact the Ruis numbers card records, that definability has to be judged from outside the language. Type: a tool of the framework.

Cites: [1.7](ch01-refs.md#r-1.7)

See also: [the diagonal argument](ch27-cards.md#g-diagonal), [the real numbers](ch01-cards.md#g-reals)

### Berry’s paradox — `Berry’s paradox` {#g-berry}

*Function or operator*

”The smallest positive integer not definable in under sixty letters” has only fifty-seven letters. G. G. Berry’s paradox, first published by Russell in 1906.

Bertrand Russell published the paradox in 1906 and credited it to G. G. Berry, a junior librarian at the Bodleian. The sixty-letters form in the headline is the usual modern wording; Russell’s of 1908 is “the least integer not nameable in fewer than nineteen syllables,” itself a name of eighteen syllables. Like Richard’s, it shows that “definable” cannot be used inside the language it talks about; Chaitin turned it into a proof of Gödel’s incompleteness theorem in 1971. Type: a tool of the framework.

Cites: [1.11](ch01-refs.md#r-1.11)

See also: [Gödel’s incompleteness theorems](ch08-cards.md#g-godel), [Richard’s paradox](ch01-cards.md#g-richard)

### the min-plus semiring — `(ℝ ∪ {+∞}, min, +)` {#g-minplus}

*Function or operator*

The min-plus form, the kind Imre Simon put to work in 1978, whose zero is +∞; the alternate reading of the left side, with a positive infinity instead of a negative one.

The min-plus semiring takes minimum as addition and ordinary addition as multiplication, so its zero is +∞ and its one is 0; it is the kind Imre Simon put to work in 1978 and the one shortest-path algorithms compute in. Use min-plus to interpret the tropical symbol and the denominator becomes +∞; evaluate the quotient in the extended reals and its value is still 0. The piece loses its one negative number that way, which is why max-plus was chosen. Type: a tool of the framework.

Cites: [1.5](ch01-refs.md#r-1.5), [1.6](ch01-refs.md#r-1.6)

See also: [the extended real numbers](ch01-cards.md#g-extreals), [positive infinity of the extended reals](ch05-cards.md#g-inf), [the tropical zero](ch01-cards.md#g-tropical)

### Ruis numbers — `ᚏ_{b}` {#g-ruis}

*Undefinable*

The author’s proposed name for real numbers our fixed original language cannot individually define in the book’s chosen mathematical setting.

Ruis numbers are the reals outside D: those with no definition without parameters in the adopted original language and setting. A parameter is an extra object supplied to a formula; allowing an arbitrary real to be supplied would let “the number equal to r” identify any r. Our definitions must work without that extra input. There are only countably many finite formulas in the fixed language, so the definable collection has measure zero in the ordinary uncountable real line. Why the different words? Asaf Karagila uses undefinable and unnameable, while explaining that intuitive nameability and mathematical definability are not automatically the same. Anne Newstead and James Franklin use indefinables and compare them to the dark matter of mathematics. These descriptions emphasize definitions, names, and an analogy; they are collected here because they inform this chapter’s vocabulary. “Ruis numbers” has the specific fixed-language definition above. “The dark pool” is descriptive language in this book; “normal” here means typical in everyday speech, rather than the technical digit-frequency property of a normal number. The notation. The name covers the whole collection ℝ ∖ D; the symbol ᚏ_{b} := x/b presents an individual chosen member, using a nonzero rational label b. Labels used together in one formula share the same x. This book holds its choice fixed throughout the master equation and its later appearances, including Chapter ᚋ(M). A formula outside this book may choose another x, so equal labels in independently chosen formulas need not represent equal values. Dividing x by a nonzero rational preserves undefinability: a definition of x/b in the original setting would define x there by multiplication. Since x is nonzero, distinct labels sharing it give distinct values. The birthday label 4.16 gives Rose’s number. The setting matters. Hamkins, Linetsky, and Reitz explain that, if ZFC is consistent, some models have every real definable without parameters. This book adopts an external view of a fixed setting containing the ordinary uncountable real line. Adding a new constant interpreted as our chosen x changes the language; it does not supply a definition in the original one. Ruis is the Pictish Ogham letter associated with the elder tree. Type: undefinable relative to the book’s fixed setting.

Cites: [1.1](ch01-refs.md#r-1.1), [1.17](ch01-refs.md#r-1.17), [1.18](ch01-refs.md#r-1.18), [1.19](ch01-refs.md#r-1.19), [1.2](ch01-refs.md#r-1.2)

See also: [the definable real numbers](ch01-cards.md#g-defreals), [is defined as](ch01-cards.md#g-defsym), [the letter Ruis](ch01-cards.md#g-og-ruis), [Rose’s number](ch01-cards.md#g-rose), [the standard axioms of set theory](ch27-cards.md#g-zfc)

### the integers — `ℤ` {#g-ints}

*Function or operator*

ℤ is the whole numbers together with their negatives and zero, the smallest ring built from the naturals by adding subtraction.

ℤ = {…, −2, −1, 0, 1, 2, …} extends the natural numbers ℕ by giving every element an additive inverse, forming a ring rather than a mere semiring: closed under addition, subtraction, and multiplication, though not division. It sits between ℕ and the rational numbers ℚ, which sit inside the real numbers ℝ, the field the book’s governing sentence is written in. The blackboard-bold letter abbreviates the German Zahlen, numbers.

Cites: [1.14](ch01-refs.md#r-1.14)

See also: [the natural numbers](ch01-cards.md#g-nats), [the rational numbers](ch01-cards.md#g-rats), [the real numbers](ch01-cards.md#g-reals)

### the natural numbers — `ℕ` {#g-nats}

*Function or operator*

ℕ is the counting numbers, 0, 1, 2, 3, and so on, the starting point that ℤ, ℚ, and ℝ are built from.

Whether ℕ begins at 0 or at 1 is a convention that differs by author and by field; either way it is closed under addition and multiplication but not subtraction, which is exactly why the integers ℤ have to exist. Giuseppe Peano gave the natural numbers their modern axiomatic footing in 1889, defining them by a starting element and a successor operation rather than by counting objects. Every other number system in the chapter’s tower, ℤ, ℚ, ℝ, is built from ℕ by adjoining something it lacks: negatives, then ratios, then limits.

Cites: [1.15](ch01-refs.md#r-1.15)

See also: [the integers](ch01-cards.md#g-ints)

### the rational numbers — `ℚ` {#g-rats}

*Function or operator*

ℚ is every ratio of two integers, the numbers that can be written as a terminating or eventually repeating decimal.

Built as fractions a/b with integers a and b ≠ 0, ℚ is a field: every nonzero element has a multiplicative inverse, unlike the integers it is built from. It is countably infinite, unlike the real numbers ℝ it sits inside, even though between any two rationals there is always another; the definable real numbers of Chapter ᚏ(R) are also a countable set, because there are only countably many formulas. The letter Q abbreviates quotient, and the gaps the rationals leave on the number line, at numbers like √2, are exactly what forces the construction of ℝ.

Cites: [1.16](ch01-refs.md#r-1.16)

See also: [the definable real numbers](ch01-cards.md#g-defreals), [the integers](ch01-cards.md#g-ints), [the real numbers](ch01-cards.md#g-reals)

### 100%—probability 1 — `P(X ∉ D) = 1` {#g-probability-one}

*Integer or rational*

Under an ideal uniform draw from [0, 1], the probability of hitting a real outside the book’s definable collection is exactly 1, or 100%.

Why exactly 100%? In the book’s adopted setting, finite formulas in the fixed original language can be listed, and each formula picks out at most one real uniquely. So D is countable. Under the uniform continuous distribution on [0, 1], a single point has probability zero; countable additivity gives the entire countable collection probability zero too. Therefore P(X ∈ D) = 0 and P(X ∉ D) = 1 − 0 = 1. This is exact, rather than a percentage rounded up from something smaller. Probability 1 means “almost surely.” A definable outcome is still logically possible: 1/2 remains a point in the interval even though its individual probability is zero. The claim concerns the ideal continuous model. A physical dart or a computer choosing among finite-precision outputs uses a different model. What about a million languages? Their rules and interpretations must be fixed independently of the target, with each using a finite or countable stock of symbols and finite descriptions. A million countable collections together are still countable, so their combined describable targets still have probability zero. If we choose the target first and then invent a special name interpreted as that target, we have changed the question. More time and computers do not change the countability argument. The everyday word “normal.” Here it means typical. Mathematical normality is a separate property of digit frequencies; a normal number can be computable and definable. Sources and deduction. Hamkins, Linetsky, and Reitz, Pointwise Definable Models of Set Theory, introduction, supplies the language/setting distinction. Curtis McMullen’s Harvard Math 114 Real Analysis notes, §§1.1–1.2, gives an analogous countable-set probability argument for algebraic numbers and develops measure and countable additivity. The application to this book’s D is our deduction from those facts; neither source introduces Ruis notation.

Cites: [1.17](ch01-refs.md#r-1.17)

See also: [the definable real numbers](ch01-cards.md#g-defreals)