<!-- Generated from https://www.teclamairme.com/#group-T26 by readviews.py; never edited by hand. -->

# Glossary cards — Chapter ᚙ(X): The bridge

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T26, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch26.md)

*Surreal numbers and the hyperreals of nonstandard analysis: two fields where you can divide by an infinity. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Eamhancholl — `ᚙ` {#g-og-emancholl}

*Ogham letter*

A supplementary letter named for its shape: ‘twin of Coll’, two Colls side by side.

Emancholl means ‘twin of coll’. The name is taken from the shape of the sign, which is two of the letter Coll set together, and perhaps also from its sound, the fricative of the sound Coll carries. Two lists give it a kenning, and both say the same thing: lúad sáethaig, “groan of a sick person” and mol galraig, “groan of a sick person,” which fits the consonant value it held before the decision that all five forfeda should be vowels. It is the one forfid the Auraicept does not gloss with a plant, and it does not appear in the orthodox inscriptions at all, though some later scholastic inscriptions use all five forfeda; the witch hazel of the letters table belongs to a modern list.

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), [2.18](ch02-refs.md#r-2.18), [2.19](ch02-refs.md#r-2.19)

See also: [the letter Coll](ch11-cards.md#g-og-coll)

### the infinite and the infinitesimal, canceled — `ε·ω − 1` {#g-epsomega}

*Integer or rational*

Exactly 0, because the surreals are a field.

The numerator: ε times ω, minus 1. Because the surreal numbers are a field and ε = 1/ω, the product is exactly 1 and the difference exactly 0. The value is the integer 0, which is why it is tinted as one even though it was computed among the surreals.

Cites: [26.3](ch26-refs.md#r-26.3), [26.5](ch26-refs.md#r-26.5)

See also: [the class of all surreal numbers](ch26-cards.md#g-No)

### one less than infinity — `ω − 1` {#g-omm1}

*Non-real object*

ω minus one, still infinite: a surreal that ordinal arithmetic cannot even write down.

One less than infinity: {0, 1, 2, … | ω}, born the day after ω. Among the ordinals there is no such thing, because ω has no immediate predecessor; among the surreals it is an ordinary infinite number, and ω − 1 + 1 = ω. A non-real object, and positive.

Cites: [26.4](ch26-refs.md#r-26.4), [26.6](ch26-refs.md#r-26.6)

See also: [the class of all surreal numbers](ch26-cards.md#g-No), [the ordinals after omega](ch27-cards.md#g-ordarith)

### the square root of omega — `√ω` {#g-sqrtomega}

*Non-real object*

A surreal whose square is ω.

A surreal whose square is ω, which exists because Conway’s field is real-closed: every positive surreal has a square root. It is infinite but smaller than ω/n for every n, and it has no counterpart among the ordinals. A non-real object.

Cites: [26.3](ch26-refs.md#r-26.3), [26.6](ch26-refs.md#r-26.6)

See also: [the ordinals after omega](ch27-cards.md#g-ordarith)

### omega to the omega — `ω^{ω}` {#g-omom}

*Non-real object*

ω raised to the power ω in the surreals: still a number, still nonzero.

ω raised to the power ω in the surreals, an infinite number vastly larger than ω, still a member of the same ordered field with a reciprocal of its own. The ordinal ω^ω exists too, in Chapter ᚚ(P)’s sense, but only the surreal one can be divided by. A non-real object, positive.

Cites: [26.3](ch26-refs.md#r-26.3), [26.6](ch26-refs.md#r-26.6)

See also: [the class of all surreal numbers](ch26-cards.md#g-No)

### one half, born on day two — `{0 | 1}` {#g-halfS}

*Integer or rational*

The simplest number between 0 and 1.

One half, written as Conway writes it: {0 | 1}, the simplest number strictly between 0 and 1, born on day two. It is the same ½ as everywhere else, which is why it is tinted as a rational; it sits in the denominator to show that the surreal field contains the ordinary numbers as well as the strange ones.

Cites: [26.2](ch26-refs.md#r-26.2)

See also: [the class of all surreal numbers](ch26-cards.md#g-No), [the birthday of a surreal](ch26-cards.md#g-sbirthday)

### half of infinity — `ω/2` {#g-omhalf}

*Non-real object*

Half of ω: a surreal that is still infinite.

Half of ω, {0, 1, 2, … | ω, ω − 1, ω − 2, …}, an infinite surreal that is exactly half of the first infinite number. Ordinal arithmetic has no ordinal that added to itself gives ω; surreal arithmetic can, because it is a field. A non-real object, positive.

Cites: [26.3](ch26-refs.md#r-26.3), [26.6](ch26-refs.md#r-26.6)

See also: [one less than infinity](ch26-cards.md#g-omm1), [the ordinals after omega](ch27-cards.md#g-ordarith)

### the surreal zero, born on day zero — `{ | }` {#g-surzero}

*Integer or rational*

The first surreal number, with nothing on either side; every other number is built from it, and it is 0.

The surreal number { | }, born on day 0: the empty set on the left and the empty set on the right. Conway’s construction, found around 1970 and set out in Conway’s book of 1976, makes every number from two sets of earlier numbers, so on the first day there are no earlier numbers and the only possibility is this one, which behaves as 0 (x + { | } = x for every x). It is the framework’s own zero, exactly 0, and it rides in the numerator as such.

Cites: [26.18](ch26-refs.md#r-26.18), [26.2](ch26-refs.md#r-26.2)

See also: [the birthday of a surreal](ch26-cards.md#g-sbirthday)

### a gap in the surreals {#g-surgap}

*Non-real object*

A cut {L | R} whose two sides use up the whole of No: at least one of them is then a proper class, so a gap is not a surreal number at all, and Conway’s ∞ is one of them.

Conway defines a gap in No as a pair {L | R} in which every member of L is less than every member of R and the two sides between them exhaust the surreal numbers; because at least one side is then a proper class, a gap is not a number. Gaps are where the surreals’ missing least upper bounds live. There is no least positive infinite surreal, but the gap between the surreals that lie below some whole number and those above every whole number is greater than every real and less than every positive infinite surreal, and it is the least upper bound of the reals in the completion of No; On = {No | } is the gap above every surreal, and ∞ is named among the gaps as well. That is the distinction this chapter turns on: ω is a surreal number and divides like any other, while Conway’s ∞ is a gap, not a number, and the extended-real +∞ of the book’s left side is an adjoined symbol, not a field element. A non-real object, and not a number.

Cites: [26.14](ch26-refs.md#r-26.14), [26.19](ch26-refs.md#r-26.19)

See also: [the class of all surreal numbers](ch26-cards.md#g-No), [Conway’s bracket](ch26-cards.md#g-lrnotation), [the real numbers](ch01-cards.md#g-reals)

### the hyperreal standard part — `st(1/H)` {#g-hyper}

*Integer or rational*

For an infinite hyperreal H, 1/H is infinitesimal and its standard part is 0; this H is a hyperreal, not the letter of a chapter.

The standard part of 1/H for an infinite hyperreal H. Abraham Robinson announced nonstandard analysis in a 1961 paper and gave the full treatment in his 1966 book Non-standard Analysis, making Leibniz’s infinitesimals rigorous nearly three centuries later; the hyperreals *ℝ are built as an ultrapower of ℝ, and Jerzy Łoś’s theorem of 1955 is what transfers first-order truth between ℝ and *ℝ, once the quantifiers are read as ranging over internal sets. 1/H is infinitesimal and the standard-part map st rounds every finite hyperreal to its nearest real, so st(1/H) = 0; this is how nonstandard analysis defines a derivative, as st((f(x + dx) − f(x))/dx). The value is the integer 0.

Cites: [26.7](ch26-refs.md#r-26.7), [26.8](ch26-refs.md#r-26.8), [26.9](ch26-refs.md#r-26.9)

See also: [the hyperreal numbers](ch26-cards.md#g-hyperreals), [an infinitesimal](ch26-cards.md#g-infinitesimal), [the symbols of logic](ch08-cards.md#g-logicsyms), [the real numbers](ch01-cards.md#g-reals), [the standard part](ch26-cards.md#g-stpart)

### a positive infinite hyperreal — `H` {#g-hyperH}

*Non-real object*

A fixed infinite element of the hyperreal field *ℝ; nonzero and invertible there, and not the extended-real symbol ∞.

H denotes a fixed positive infinite hyperreal, an element of *ℝ larger than every ordinary real, such as the class of (1, 2, 3, …) in the ultrapower construction. It is an ordinary nonzero element of an ordered field and so has a reciprocal 1/H, itself a positive infinitesimal; taking the standard part of that reciprocal is where this chapter’s second zero comes from. H is not the extended-real ∞, which is an adjoined symbol, not an element of a field, and it is not the letter of any chapter. A non-real object.

Cites: [26.14](ch26-refs.md#r-26.14), [26.7](ch26-refs.md#r-26.7)

See also: [the hyperreal numbers](ch26-cards.md#g-hyperreals), [an infinitesimal](ch26-cards.md#g-infinitesimal), [the standard part](ch26-cards.md#g-stpart)

### surreal omega — `ω` {#g-omegaS}

*Non-real object*

{0, 1, 2, … | }, the first infinite surreal number.

The first infinite surreal, ω = {0, 1, 2, … | }, born on day ω of Conway’s construction, which he found around 1970 while analyzing Go endgames and published in On Numbers and Games in 1976; Knuth’s 1974 novelette named the numbers surreal. Every natural number is on its left and nothing on its right, so it exceeds them all. Unlike the ordinal ω of Chapter ᚚ(P) it has a reciprocal, a square root, a half, and a predecessor, because the surreals form a field. A non-real object.

Cites: [26.1](ch26-refs.md#r-26.1), [26.18](ch26-refs.md#r-26.18), [26.2](ch26-refs.md#r-26.2), [7.16](ch07-refs.md#r-7.16)

See also: [the class of all surreal numbers](ch26-cards.md#g-No), [the birthday of a surreal](ch26-cards.md#g-sbirthday)

### surreal infinitesimal — `ε` {#g-epsS}

*Non-real object*

{0 | 1, ½, ¼, …} = 1/ω, smaller than every positive real but not zero.

The surreal infinitesimal ε = {0 | 1, ½, ¼, …}, born on day ω like its partner, greater than 0 and less than every positive real. Conway showed it equals 1/ω exactly, because the surreals form an ordered field, so ε·ω = 1. The letter ε is reused across the book: here a surreal, in Chapter ᚔ(I) a dual number with ε² = 0. A non-real object.

Cites: [26.10](ch26-refs.md#r-26.10), [26.5](ch26-refs.md#r-26.5)

See also: [the class of all surreal numbers](ch26-cards.md#g-No), [the dual-number zero](ch21-cards.md#g-dual), [an infinitesimal](ch26-cards.md#g-infinitesimal), [the birthday of a surreal](ch26-cards.md#g-sbirthday)

### Conway’s bracket — `{ L | R }` {#g-lrnotation}

*Function or operator*

Every surreal number is a pair of sets of earlier surreals, a left set and a right set, with every left member less than every right member.

Conway’s notation {L | R}: a surreal number is defined by a left set L and a right set R of previously created numbers with no member of L greater than or equal to any member of R; the number is the simplest one lying between them. So {0 | } = 1, { | 0} = −1, {0 | 1} = ½, and {0, 1, 2, … | } = ω. Set out by Conway in On Numbers and Games (1976); the construction first appeared in print in Knuth’s 1974 novelette, which presents the same two rules carved on a stone found on a beach. A notation of the framework.

Cites: [26.2](ch26-refs.md#r-26.2), [26.21](ch26-refs.md#r-26.21), [26.24](ch26-refs.md#r-26.24)

See also: [the class of all surreal numbers](ch26-cards.md#g-No), [one half, born on day two](ch26-cards.md#g-halfS), [surreal omega](ch26-cards.md#g-omegaS), [the sign expansion of a surreal](ch26-cards.md#g-signexp)

### the birthday of a surreal — `b(x)` {#g-sbirthday}

*Function or operator*

The first day on which a number can be written: b(0) = 0, b(±1) = 1, b(½) = 2, b(ω) = ω.

The birthday b(x) of a surreal number, the ordinal on which it is first created; numbers born earlier are simpler, and the number defined by {L | R} is always the earliest-born number between L and R. The naturals are born on their own day, dyadic rationals on finite days, every other real on day ω, and ω and ε on day ω as well; ω − 1 and ω/2 wait for later days. The word as Conway’s On Numbers and Games of 1976 uses it. A function of the framework.

Cites: [26.2](ch26-refs.md#r-26.2), [26.24](ch26-refs.md#r-26.24), [26.6](ch26-refs.md#r-26.6)

See also: [Conway’s bracket](ch26-cards.md#g-lrnotation), [half of infinity](ch26-cards.md#g-omhalf), [one less than infinity](ch26-cards.md#g-omm1), [the ordinals after omega](ch27-cards.md#g-ordarith)

### the class of all surreal numbers — `𝐍𝐨` {#g-No}

*Non-real object*

Too big to be a set: it contains every real, every ordinal, and the field they generate.

Conway’s name for the surreal numbers as a whole, No, which is a proper class rather than a set because it contains every ordinal. It is a real-closed ordered field—strictly a Field, capital F, since it is a proper class—and in von Neumann–Bernays–Gödel set theory with global choice it is universal: every ordered field whose universe is a set or a proper class embeds in it, and up to isomorphism No is the only homogeneous universal ordered field. The ordinals sit inside it as the numbers with empty right set. Conway 1976. A non-real object.

Cites: [26.11](ch26-refs.md#r-26.11), [26.16](ch26-refs.md#r-26.16), [26.24](ch26-refs.md#r-26.24)

### the omnific integers — `𝐎𝐳` {#g-omnific}

*Non-real object*

Conway’s integers among the surreals: ω and ω − 1 are integers here, while ½ and ε are not.

The omnific integers, Oz, first defined by Simon Norton and set out by Conway, are the generalization of the integers to the surreals: x is an omnific integer when x = {x − 1 | x + 1}. Every ordinal is one, so ω and ω^ω are integers in this sense—and so is ω − 1, which is no ordinal at all. Every surreal is the ratio of two of them, just as every rational is a ratio of two integers, so No is the field of fractions of Oz. Conway 1976. A non-real object.

Cites: [26.17](ch26-refs.md#r-26.17), [26.24](ch26-refs.md#r-26.24)

See also: [the class of all surreal numbers](ch26-cards.md#g-No), [the integers](ch01-cards.md#g-ints), [one less than infinity](ch26-cards.md#g-omm1), [omega to the omega](ch26-cards.md#g-omom), [the ordinals after omega](ch27-cards.md#g-ordarith)

### Conway normal form — `Σ ω^{y_{i}} r_{i}` {#g-cnf}

*Function or operator*

Every surreal is a decreasing sum of powers of ω with real coefficients, Cantor’s normal form for ordinals extended to a whole field.

The Conway normal form: every surreal number can be written uniquely as a sum of terms ω^y·r over a decreasing sequence of surreal exponents y with nonzero real coefficients r, possibly transfinitely many. It generalizes Cantor’s normal form for ordinals of 1897, and it is what makes ω − 1, ω/2, and √ω (which is ω^½) routine to write down. Conway 1976. A notation of the framework.

Cites: [26.11](ch26-refs.md#r-26.11), [26.22](ch26-refs.md#r-26.22), [26.24](ch26-refs.md#r-26.24), [27.20](ch27-refs.md#r-27.20)

See also: [half of infinity](ch26-cards.md#g-omhalf), [one less than infinity](ch26-cards.md#g-omm1), [the ordinals after omega](ch27-cards.md#g-ordarith), [the square root of omega](ch26-cards.md#g-sqrtomega)

### the sign expansion of a surreal — `sign expansion` {#g-signexp}

*Function or operator*

A surreal number as a transfinite string of + and −: ½ is +−, ω is + repeated ω times, ε is + followed by ω minuses.

The sign expansion: every surreal number corresponds to a sequence of plus and minus signs indexed by an ordinal, its birthday being the length of the sequence; ½ is (+, −), ω is ω pluses, and ε is a plus followed by ω minuses. It is the surreal analogue of a binary expansion, and Conway’s rule of simplicity says {L | R} is the number whose sign expansion is the shortest one between L and R. Conway 1976; developed by Gonshor in 1986. A notation of the framework.

Cites: [26.12](ch26-refs.md#r-26.12), [26.24](ch26-refs.md#r-26.24)

See also: [Conway’s bracket](ch26-cards.md#g-lrnotation), [the ordinals after omega](ch27-cards.md#g-ordarith), [the birthday of a surreal](ch26-cards.md#g-sbirthday)

### the hyperreal numbers — `*ℝ` {#g-hyperreals}

*Non-real object*

Robinson’s extension of the reals with infinitesimals and infinite numbers, built as an ultrapower and obeying the same first-order laws as the reals once the quantifiers are read as ranging over internal sets.

The hyperreal field *ℝ of Abraham Robinson’s nonstandard analysis (paper 1961, book 1966): an ordered field extending ℝ that contains infinitesimals and their infinite reciprocals. It is constructed as an ultrapower of ℝ, sequences of reals modulo a nonprincipal ultrafilter—whose existence needs the ultrafilter lemma, a weak form of the axiom of choice, and which cannot be constructed explicitly—an idea going back to Hewitt (1948) and Łoś (1955). Unlike the surreals it is a set, and, once the quantifiers are read as ranging over internal sets, it has the same first-order theory as ℝ by the transfer principle. A non-real object.

Cites: [26.11](ch26-refs.md#r-26.11), [26.24](ch26-refs.md#r-26.24), [26.7](ch26-refs.md#r-26.7), [26.8](ch26-refs.md#r-26.8)

See also: [the class of all surreal numbers](ch26-cards.md#g-No), [the axiom of choice](ch27-cards.md#g-ac), [an infinitesimal](ch26-cards.md#g-infinitesimal), [the real numbers](ch01-cards.md#g-reals), [the transfer principle](ch26-cards.md#g-transfer)

### the transfer principle — `transfer principle` {#g-transfer}

*Function or operator*

A first-order statement holds of the reals exactly when its starred version holds of the hyperreals, quantifiers ranging over internal sets: that is what lets a proof with infinitesimals count as a proof about ℝ.

The transfer principle of nonstandard analysis: a formula holds in ℝ if and only if its starred transform holds in *ℝ, with quantification bounded and ranging over internal sets. The qualification is not a formality. Read without it, the principle would make *ℝ Archimedean, since “every positive number exceeds 1/n for some positive integer n” is true in ℝ; what carries over is the version in which n may be a hyperinteger. It is a consequence of Łoś’s theorem on ultraproducts (1955) and was formulated by Robinson in 1961 as the rigorous version of Leibniz’s law of continuity. It is the license under which st(1/H) = 0 turns into ordinary calculus. A theorem of the framework.

Cites: [26.7](ch26-refs.md#r-26.7), [26.8](ch26-refs.md#r-26.8)

See also: [the hyperreal standard part](ch26-cards.md#g-hyper), [the hyperreal numbers](ch26-cards.md#g-hyperreals), [an infinitesimal](ch26-cards.md#g-infinitesimal), [the real numbers](ch01-cards.md#g-reals)

### the standard part — `st(x)` {#g-stpart}

*Function or operator*

Rounds every finite hyperreal to the unique real infinitely close to it; st(1/H) = 0 for infinite H.

The standard-part function st: every finite hyperreal x is infinitely close to exactly one real number, and st(x) is that real. It is the map that finishes a nonstandard calculation, discarding the infinitesimal error, so that the derivative is st((f(x + dx) − f(x))/dx). Robinson 1961–1966. A function of the framework.

Cites: [26.13](ch26-refs.md#r-26.13), [26.7](ch26-refs.md#r-26.7), [26.9](ch26-refs.md#r-26.9)

See also: [the hyperreal standard part](ch26-cards.md#g-hyper), [the hyperreal numbers](ch26-cards.md#g-hyperreals), [an infinitesimal](ch26-cards.md#g-infinitesimal)

### an infinitesimal — `dx` {#g-infinitesimal}

*Non-real object*

A number greater than 0 but smaller than every positive real: Leibniz’s dx, banished by Weierstrass and readmitted by Robinson.

An infinitesimal: a positive number smaller than 1/n for every natural n. Leibniz was using them by 1675, when he first wrote ∫ f(x)dx, and Euler freely; Weierstrass’s epsilon-delta limits, written down in his Berlin lectures of 1861 and anticipated by Bolzano in 1817 and Cauchy in the 1820s, removed them from analysis. Robinson’s hyperreals of 1961 and Conway’s surreals of 1976 both contain them legitimately, the surreal ε = 1/ω being one. The reals contain none, by the Archimedean property. A non-real object.

Cites: [26.10](ch26-refs.md#r-26.10), [26.15](ch26-refs.md#r-26.15), [26.20](ch26-refs.md#r-26.20), [26.24](ch26-refs.md#r-26.24), [26.7](ch26-refs.md#r-26.7)

See also: [the class of all surreal numbers](ch26-cards.md#g-No), [the hyperreal numbers](ch26-cards.md#g-hyperreals), [the real numbers](ch01-cards.md#g-reals)