# References — Chapter ᚏ(R): The gateway *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-1, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch01.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 1.1 {#r-1.1} In the rich fixed mathematical setting described on D’s card, reals uniquely definable without parameters in the countable original language form an externally countable set. Every rational, every algebraic, and every computable real is definable there; a real outside that set is irrational, transcendental, and uncomputable. The author proposes Ruis numbers as a name for the entire collection outside D. The notation ᚏ_{b} := x/b presents a chosen member using a nonzero rational label b and a shared x within one formula; this book holds x fixed throughout. Being countable, D has Lebesgue measure zero. Chaitin’s fixed-machine Ω, though uncomputable, is definable. The Ogham letter Ruis is glossed as the elder; Ogham was also used, less often, to write Pictish. *Status: confirmed* - [Definable real number](https://en.wikipedia.org/wiki/Definable_real_number) — defines definable reals by unique specification in a formal language, states there are only countably many formulas, and states that the reals are uncountable. Re-read raw on 17 September 2026 for the containment the chapter’s inference rests on: “All analytical numbers, and in particular all computable numbers, are definable in the language of set theory. Thus the real numbers definable in the language of set theory include all familiar real numbers such as 0, 1, π, e, et cetera, along with all algebraic numbers.” - [Cantor’s diagonal argument](https://en.wikipedia.org/wiki/Cantor's_diagonal_argument) — gives Cantor’s 1891 diagonal proof that the real numbers are uncountable: “Georg Cantor published this proof in 1891, but it was not his first proof of the uncountability of the real numbers, which appeared in 1874.” - **Author’s declaration and own computation** (the author's own work) — The notation dates to V17, 11 September 2026; labels were widened from positive to nonzero rationals on 17 September 2026. Clarified 30 September 2026: “Ruis numbers” names ℝ ∖ D, while ᚏ_{b} := x/b presents a chosen member. Labels used together in one formula share x, and this book keeps that choice fixed throughout; a formula outside this book may choose another x. If x/b were definable without parameters in the original setting, multiplying by the definable rational b would define x there. Also x ≠ 0, so x/b = x/c implies b = c. Thus distinct rational labels sharing x give distinct undefinable values. These calculations establish relationships without specifying those values. - [Null set, Wikipedia](https://en.wikipedia.org/wiki/Null_set) — “For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null.” - [Chaitin’s constant, Wikipedia](https://en.wikipedia.org/wiki/Chaitin%27s_constant) — states that Omega is definable though uncomputable: “Ω is among the few definable algorithmically random numbers and is the best-known algorithmically random number, but it is not at all typical of all algorithmically random numbers.” - [Ogham, Wikipedia](https://en.wikipedia.org/wiki/Ogham) — “Ruis, Old Irish Ruis means ‘red’ or ‘redness’, glossed as trom ‘elder’.”; “It was also used much less frequently to record the Pictish language in what is now Scotland.” ### 1.2 {#r-1.2} Definability without parameters depends on the formal language and model. Pointwise definability is not a first-order property; if ZFC is consistent, there are models of ZFC in which every real is definable without parameters, and such models are countable. *Status: confirmed* - [Hamkins, Linetsky and Reitz, Pointwise Definable Models of Set Theory](https://arxiv.org/abs/1105.4597) — abstract and bibliographic record confirm the pointwise-definable-model result; posted to arXiv in 2011 and published in The Journal of Symbolic Logic 78(1) (2013), 139–156. Introduction, pp. 1–3: definitions are relative to a language and structure and use no parameters. Page 2 discusses ⟨V_{ω+ω}, ∈⟩ as a rich background for classical mathematical definitions. This book adopts that fixed set structure and counts its definable reals externally; this choice is not the paper’s definition of “Ruis.” The pointwise-definable-model result explains why the argument cannot be transferred indiscriminately to every model. p. 2: “In any fixed structure M in a countable language, including the higher-order set-theoretic structures ⟨Vα, ∈⟩, the math-tea argument seems fine”; p. 3: “Clearly, pointwise definable models in a countable language must be countable, since there are only countably many definitions.” and “The first two claims have fairly soft proofs and might be considered to be a part of the mathematical folklore. Statement (1) could be credited to Myhill [Myh52].” - [Definable real number](https://en.wikipedia.org/wiki/Definable_real_number) — lines on model-relative definability, non-expressibility in ZFC, and countable models in which all reals are definable: “there are countable models of ZFC in which all real numbers, all sets of real numbers, functions on the reals, etc. are definable.” ### 1.3 {#r-1.3} The real numbers form the complete ordered field and are commonly denoted by ℝ; the inclusion of naturals, integers, rationals, and complex numbers is standard notation. Two independent definitions were published in 1872, Dedekind’s by cuts and Cantor’s by equivalence classes of Cauchy sequences. *Status: confirmed* - [Real number](https://en.wikipedia.org/wiki/Real_number) — confirms the real-number system, its order/completeness presentation, and standard notation. Re-read raw on 17 September 2026 for the dates: “Beginning with Richard Dedekind in 1858, several mathematicians worked on the definition of the real numbers … leading to the publication in 1872 of two independent definitions of real numbers, one by Dedekind, as Dedekind cuts, and the other one by Georg Cantor, as equivalence classes of Cauchy sequences.” ### 1.4 {#r-1.4} The extended real line is ℝ ∪ {−∞,+∞}, with finite x divided by either infinity conventionally equal to 0 and indeterminate expressions such as ∞−∞ left undefined. Measure theory sets 0·∞ = 0, a convention chosen for integration. In the min-plus semiring the additive identity, its zero, is +∞. *Status: confirmed* - [Extended real number line](https://en.wikipedia.org/wiki/Extended_real_number_line) — confirms adjoining the two infinities and the standard arithmetic conventions and undefined forms. - [A. Sengupta, Lecture 2. Measure and Integration, Math 7312 notes, Louisiana State University](https://www.math.lsu.edu/~sengupta/7312s02/msreinteg.pdf) — “0 · ∞ = 0” among the conventions; “The convention 0·∞ = 0 might seem like an arbitrary choice, but it is the really the right one : it will be useful in integration and there it will appear as the natural choice.” - [Tropical semiring, Wikipedia](https://en.wikipedia.org/wiki/Tropical_semiring) — min-plus semiring: “The identity element for ⊕ is +∞, and the identity element for ⊗ is 0.” ### 1.5 {#r-1.5} In the max-plus tropical semiring, addition is maximum, multiplication is ordinary addition, the additive identity is −∞, and the multiplicative identity is 0; min-plus uses minimum and +∞. In the extended reals a finite number divided by +∞ or −∞ is 0. Tropical addition, like the addition of every semiring, is associative and commutative. Tropical multiplication distributes over tropical addition, the tropical semiring is a semifield (every nonzero element invertible), and tropical polynomials are piecewise-linear functions. *Status: confirmed* - [Tropical semiring](https://en.wikipedia.org/wiki/Tropical_semiring) — gives both min-plus and max-plus operations and their identities, and identifies tropical geometry as an application. - [Extended real number line, Wikipedia](https://en.wikipedia.org/wiki/Extended_real_number_line) — “The arithmetic operations of ℝ can be partially extended to ℝ̄ as follows:” including a/±∞ = 0 for a ∈ ℝ. - [Semiring, Wikipedia](https://en.wikipedia.org/wiki/Semiring) — definition: “(R, +) is a commutative monoid with an identity element called 0,” listing (a + b) + c = a + (b + c) and a + b = b + a. - [Tropical mathematics, David Speyer and Bernd Sturmfels, arXiv](https://arxiv.org/abs/math/0408099) — states distributivity (min-plus convention): “The distributive law holds for tropical addition and tropical multiplication: x ⊙ (y ⊕ z) = x ⊙ y ⊕ x ⊙ z.” - [Semifield, Wikipedia](https://en.wikipedia.org/wiki/Semifield) — “a semifield is a semiring (S,+,·) in which all nonzero elements have a multiplicative inverse.” and “(R, max, +)” ... “These are called the tropical semiring.” - [Tropical geometry, Wikipedia](https://en.wikipedia.org/wiki/Tropical_geometry) — “Thus a tropical polynomial F is the minimum of a finite collection of affine-linear functions in which the variables have integer coefficients, so it is concave, continuous, and piecewise linear.” ### 1.6 {#r-1.6} Imre Simon introduced the min-plus semiring in 1978, in the context of automata theory, in Limited subsets of a free monoid, Proc. 19th Annual Symposium on Foundations of Computer Science, IEEE, 1978, pp. 143–150. The adjective tropical was coined afterwards by French colleagues in his honor — he lived and worked in Brazil — with Jean-Éric Pin crediting Dominique Perrin and Simon himself crediting Christian Choffrut. Tropical mathematics has applications including scheduling and shortest paths, and tropical geometry consolidated as a field from the late 1990s and took off around 2000, with Kontsevich, Mikhalkin, Speyer, and Sturmfels among those who built it. Tropical (idempotent) mathematics, the setting of max-plus algebra, dates back to the early 1960s. *Status: confirmed* - [Tropical semiring](https://en.wikipedia.org/wiki/Tropical_semiring) — attributes the name to Imre Simon and states applications and the connection with tropical geometry. - [Tropical geometry](https://en.wikipedia.org/wiki/Tropical_geometry) — dates the consolidation of the field to the late 1990s, names Mikhalkin and Kontsevich, and records that Jean-Éric Pin credits the adjective to Dominique Perrin while Simon credited it to Christian Choffrut, and describes Simon as “the Hungarian-born Brazilian computer scientist.” - [Jean-Éric Pin, Tropical Semirings, in J. Gunawardena (ed.), Idempotency, Publications of the Newton Institute 11, Cambridge University Press, 1998, pp. 50–69](https://www.irif.fr/~jep/PDF/Tropical.pdf) — read as extracted PDF text. §2.2: “The Min-Plus semiring is M = (N ∪ {∞}, min, +) … This semiring was introduced by Simon [41] in the context of automata theory (it is also a familiar semiring in Operations Research).” Its reference [41] is “I. Simon, Limited subsets of a free monoid, in Proc. 19th Annual Symposium on Foundations of Computer Science, Piscataway, N.J., 1978, pp. 143–150, IEEE,” which Crossref confirms at doi 10.1109/sfcs.1978.21, October 1978. §1: “These semirings were baptized tropical semirings by Dominique Perrin in honour of the pioneering work of our brazilian colleague and friend Imre Simon, but are also commonly known as (min, +)-semirings.” - [David Speyer and Bernd Sturmfels, Tropical mathematics, Mathematics Magazine 82 (2009), 163–173](https://arxiv.org/abs/math/0408099) — the expository paper behind the chapter’s naming of Sturmfels among the founders; posted to the arXiv in 2004 and published in 2009. Its introduction (arXiv version, read as extracted PDF text): “The adjective ‘tropical’ was coined by French mathematicians, including Jean-Eric Pin, in the honor of their Brazilian colleague Imre Simon … There is no deeper meaning in the adjective ‘tropical’. It simply stands for the French view of Brazil.” - [N. Krivulin, Algebraic solutions of tropical optimization problems, arXiv](https://arxiv.org/abs/1406.1777) — introduction: “Tropical (idempotent) mathematics, which is concerned with the theory and applications of semirings with idempotent addition, dates back to the early 1960’s, when a few innovative works by Pandit [47], Cuninghame-Green [11], Giffler [24], Hoffman [29], Vorob’ev [51] and Romanovskiĭ [48] made their appearance.” - [Bernd Sturmfels, Wikipedia](https://en.wikipedia.org/wiki/Bernd_Sturmfels) — “Bernd Sturmfels has made contributions to a variety of areas of mathematics, including algebraic geometry, commutative algebra, discrete geometry, Gröbner bases, toric varieties, tropical geometry, algebraic statistics, and computational biology.” - [Min-plus matrix multiplication, Wikipedia](https://en.wikipedia.org/wiki/Min-plus_matrix_multiplication) — read as raw wikitext: the distance product “is standard matrix multiplication for the semi-ring of tropical numbers in the min convention,” and “This operation is closely related to the shortest path problem.” Editorial note: The chapter also dates max-plus algebra, as distinct from Simon’s 1978 min-plus semiring, to the early 1960s. We keep that: Krivulin’s survey dates tropical (idempotent) mathematics to the early 1960s. Extended 17 September 2026. Until then the 1978 date was asserted in four places — the chapter prose, #tropical, #tropplus, and #minplus — and appeared in no claim line and in no cited source; Pin’s survey now carries the year, the min-plus attribution, and the automata context in one sentence. The chapter’s “took off around 2000 with Mikhalkin, Sturmfels, and others” and this entry’s “late 1990s” were never in conflict, but the entry named different people from the page; since the compendium is the source, the entry widened to it rather than the other way round. ### 1.7 {#r-1.7} Richard’s paradox uses a list of defining phrases and diagonalization; Richard’s observation dates to 1905. Its resolution is that the property of defining a unique real number is not expressible in the formal language itself and belongs to the metatheory, outside the language. *Status: confirmed* - [Richard’s paradox](https://en.wikipedia.org/wiki/Richard's_paradox) — confirms the diagonal construction and Jules Richard’s 1905 publication. Re-read on 29 September 2026 for the resolution the chapter now states: “The property D(φ) that a formula φ of ZFC defines a unique real number is not itself expressible by ZFC, but must be considered as part of the metatheory used to formalize ZFC.” ### 1.8 {#r-1.8} The notation of the governing sentence has the meanings used in the chapter: the set-theoretic symbols ∖, ∈, and ∪, and := for a definition rather than an asserted equality. The forms ≜, ≝, and := each abbreviate the phrase “let x = E.” The definable reals D form a countable subset of ℝ. *Status: confirmed* - [Set (mathematics)](https://en.wikipedia.org/wiki/Set_(mathematics)) — confirms the standard set-membership and set-operation meanings. - [Set difference](https://en.wikipedia.org/wiki/Set_difference) — confirms A∖B as the elements of A not in B. - [Defined, Wolfram MathWorld](https://mathworld.wolfram.com/Defined.html) — gives A := B as one of the standard notations for A is defined to be B. - [Glossary of mathematical symbols, Wikipedia](https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbols) — entry for ≜, ≝ (=def), and :=: “Any of these is sometimes used for naming a mathematical object.” The forms x ≜ E, x ≝ E, x := E and E =: x “are each an abbreviation of the phrase “let x = E.”” - [Definable real number, Wikipedia](https://en.wikipedia.org/wiki/Definable_real_number) — “there are only countably many formulas, and so only countably many elements of M can be definable over M.” ### 1.9 {#r-1.9} In ordinal arithmetic α·β consists of β copies of α, while the surreal numbers contain the ordinals and have different arithmetic; in particular the surreal reading uses 1+ω=ω+1. *Status: confirmed* - [Ordinal arithmetic](https://en.wikipedia.org/wiki/Ordinal_arithmetic) — confirms ordinal multiplication as repeated copies and the noncommutative behavior of ordinal arithmetic. - [Surreal number](https://en.wikipedia.org/wiki/Surreal_number) — confirms that ordinals embed in the surreals and that surreal arithmetic differs from ordinal arithmetic. ### 1.10 {#r-1.10} Read in the surreal numbers under the piece’s own convention ε = 1/ω, taking −∞ as −ω makes (x/4.16)/(−ω) = −(x/4.16)·ε: an infinitesimal whose size depends on x, and not ε itself. *Status: confirmed* - **Own computation** (the author's own work) — Using the glossary’s ε = 1/ω, (x/4.16)/(−ω) = −(x/4.16)·(1/ω) = −(x/4.16)·ε. Equality with ε would force x = −4.16, which is definable and therefore excluded by the governing sentence. ### 1.11 {#r-1.11} Berry’s paradox was first published by Bertrand Russell in 1906 and credited by him to G. G. Berry, a junior librarian at the Bodleian. The wording usually quoted, in which the least integer not nameable in fewer than nineteen syllables is itself a name of eighteen syllables, is Russell’s of 1908. The letters version quoted on the card, the smallest positive integer not definable in under sixty letters, has fifty-seven letters. Chaitin built a proof of Gödel’s incompleteness theorem on the paradox in 1971. *Status: confirmed* - [Berry paradox](https://en.wikipedia.org/wiki/Berry_paradox) — G. G. Berry (1867–1928), junior librarian at the Bodleian; Russell the first to discuss the paradox in print; cites Russell, Les paradoxes de la logique, Revue de métaphysique et de morale 14 (1906) 627–650. - [Russell, Mathematical Logic as Based on the Theory of Types (1908), p. 223](https://en.wikisource.org/wiki/Page:Mathematical_Logic_as_Based_on_the_Theory_of_Types_-_Bertrand_Russell_(1908).pdf/3) — the least integer not nameable in fewer than nineteen syllables denotes 111,777 and is itself a name consisting of eighteen syllables, credited to Berry in a footnote. - [Berry Paradox, Wolfram MathWorld](https://mathworld.wolfram.com/BerryParadox.html) — gives the nineteen/eighteen syllable form and attributes it to Russell, American Journal of Mathematics 30 (1908), 223. - [G. J. Chaitin, Computational Complexity and Gödel’s Incompleteness Theorem, ACM SIGACT News No. 9 (April 1971), pp. 11–12 (PDF, Charles University)](https://www.karlin.mff.cuni.cz/~krajicek/chaitin71.pdf) — header “ACM SIGACT News, No. 9 (April 1971), pp. 11–12”; abstract: “The proof resembles Berry’s paradox, not the Epimenides nor Richard paradoxes.” Editorial note: The card had given the nineteen-syllable wording as Russell’s of 1906. Russell published the paradox in 1906 in a different wording and gave the quoted one in 1908; the card now says both. The fifty-seven letters are countable from the phrase itself. ### 1.12 {#r-1.12} Peano introduced ∪ and ∩ in 1888 and ∈ in 1889; := is the assignment operator of Algol, from Algol 58 onward. *Status: confirmed* - [Earliest Uses of Symbols of Set Theory and Logic (Jeff Miller, MacTutor)](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/set/) — ∈ in Peano, Arithmetices principia nova methodo exposita (1889), as an abbreviation for est — re-read raw on 17 September 2026, the page says “He stated that the symbol was an abbreviation for est; the entire work is in Latin” and “Peano’s symbol for membership appears to be a lunate (or uncial) epsilon, and not the stylized epsilon that is now used”; ∪ and ∩ in Peano, Calcolo geometrico secondo l’Ausdehnungslehre di H. Grassmann (1888). - [Assignment (computer science)](https://en.wikipedia.org/wiki/Assignment_(computer_science)) — x := expr is given as originally Algol 1958, popularized by Pascal. Editorial note: The card for := had credited the notation to Algol 60 in 1960. The notation is Algol 58’s; Algol 60 carried it on, and the card now says so. ### 1.13 {#r-1.13} The chapter epigraph is Cantor’s, from the Grundlagen einer allgemeinen Mannigfaltigkeitslehre of 1883: the essence of mathematics lies in its freedom. *Status: confirmed* - [Georg Cantor, Wikiquote](https://en.wikiquote.org/wiki/Georg_Cantor) — read as raw HTML: under the heading “Grundlagen einer allgemeinen Mannigfaltigkeitslehre [Foundations of a General Theory of Aggregates] (1883)” it gives “The essence of mathematics lies entirely in its freedom,” with the variant translation “The essence of mathematics is in its freedom.” Editorial note: Added 17 September 2026 to source the chapter epigraph, which carried no marker. One wording note: the German is “Das Wesen der Mathematik liegt gerade in ihrer Freiheit,” and the page prints “lies precisely in its freedom” where this source gives “lies entirely in its freedom.” Both are current renderings of gerade; the page’s is the more usual one and is kept. ### 1.14 {#r-1.14} Defines the integers as the natural numbers with negatives adjoined, a ring closed under addition, subtraction, and multiplication. *Status: confirmed* - [Integer, Wikipedia](https://en.wikipedia.org/wiki/Integer) — defines the integers as the natural numbers together with their negatives and zero, forming a ring, and gives the blackboard-bold notation Z from German Zahlen; “The integers form the smallest group and the smallest ring containing the natural numbers.” - [Semiring, Wikipedia](https://en.wikipedia.org/wiki/Semiring) — carries the ring/semiring distinction the card draws: “Any field is also a ring, which in turn is a semiring in which also additive inverses exist.” ### 1.15 {#r-1.15} Defines the natural numbers as the counting numbers and notes Peano’s 1889 axiomatization. *Status: confirmed* - [Natural number, Wikipedia](https://en.wikipedia.org/wiki/Natural_number) — describes the natural numbers as the counting numbers, notes the convention split on whether 0 is included, and dates Peano’s axioms to 1889. ### 1.16 {#r-1.16} Defines the rational numbers as ratios of integers forming a field, gives them as exactly the reals whose decimal expansion terminates or eventually repeats, states that between any two rationals there is another, and states their countability. *Status: confirmed* - [Rational number, Wikipedia](https://en.wikipedia.org/wiki/Rational_number) — defines the rationals as quotients of integers forming a field and states that the set is countably infinite; the set is “closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations”; “The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over,” and “The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones.” - [Definable real number, Wikipedia](https://en.wikipedia.org/wiki/Definable_real_number) — carries the countability of the definable reals: “This follows from the fact that there are only countably many formulas, and so only countably many elements of M can be definable over M.” - [Earliest Uses of Symbols of Number Theory, MacTutor (Jeff Miller)](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/nth/) — carries the origin of the letter Q: “Q for the set of rational numbers and Z for the set of integers are apparently due to N. Bourbaki. ... The letters stand for the German Quotient and Zahlen.” ### 1.17 {#r-1.17} In the book’s adopted setting, D is countable. Under a uniform continuous draw from [0, 1], a countable set has probability zero, so its complement has probability 1. A million languages with finite descriptions and interpretations fixed independently of the target still supply only countably many describable targets in total. These conclusions are the author’s deductions from the counting and measure facts below; the sources do not introduce Ruis notation. “Normal” in the chapter means typical in everyday speech, not the technical digit-frequency property. *Status: confirmed* - [Hamkins, Linetsky and Reitz, Pointwise Definable Models of Set Theory](https://arxiv.org/pdf/1105.4597) — introduction, pp. 1–3: fixed languages/structures, parameter-free definitions, and the external viewpoint. - [Curtis McMullen, Harvard Math 114 Real Analysis notes](https://people.math.harvard.edu/~ctm/home/text/class/harvard/114/07/html/home/course/course.pdf) — §1.1 gives an analogous countable-set probability argument for algebraic numbers; §1.2 develops measure and countable additivity. - [Becher and Figueira, An example of a computable absolutely normal number](https://glyc.dc.uba.ar/santiago/papers/absnor.pdf) — abstract and introduction: normality concerns digit-block frequencies and can hold for a computable number. Editorial note: Probability 1 is exact in this ideal model and does not make a definable outcome logically impossible. A physical dart or a computer selecting finite-precision outputs uses a different model. ### 1.18 {#r-1.18} “Unnameable” as descriptive vocabulary. *Status: confirmed* - [Asaf Karagila, Name that number, 21 September 2015](https://karagila.org/2015/name-that-number/) — the paragraph beginning “And we can prove, now…” uses unnameable alongside undefinable. The later discussion distinguishes intuitive nameability from mathematical definability and explains dependence on the language and setting. This supports the vocabulary, not an absolute category beyond every possible language. ### 1.19 {#r-1.19} “Indefinables” and the dark-matter analogy. *Status: confirmed* - [Anne Newstead and James Franklin, On the Reality of the Continuum, Philosophy 83 (2008), pp. 117–128](https://web.maths.unsw.edu.au/~jim/newsteadcontinuum.pdf) — printed p. 121 / fifth PDF page uses indefinables and compares them to the dark matter of mathematics. The latter is an analogy, not a formal class called “dark numbers.” This citation supports those descriptions; the book’s mathematical definition remains the one declared on D’s card.