# References — Chapter ᚚ(P): The possible journeys *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-27, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch27.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.* ### 27.1 {#r-27.1} Cardinality is the size of a set, characterized by bijection; the empty set has cardinality 0. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, Basic Set Theory](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html) — defines cardinality by the least ordinal bijectable with a set and discusses finite cardinalities; the empty set is the zero case. ### 27.2 {#r-27.2} Aleph-null is the cardinality of the natural numbers and aleph-one is the first uncountable cardinal. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, Basic Set Theory](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html) — identifies ℵ₀ with the smallest infinite cardinal and ℵ₁ with the first uncountable cardinal. - [Wikipedia, Aleph number](https://en.wikipedia.org/wiki/Aleph_number) — identifies ℵ₀ with the natural numbers and describes the aleph sequence. ### 27.3 {#r-27.3} Cantor’s diagonal argument and power-set theorem show that the reals are uncountable and that every set has a strictly larger power set. ℶ_(α+1) = 2^ℶ_α, and by Cantor’s theorem the beth numbers strictly increase; the diagonal sequence takes its n-th digit complementary to the n-th digit of the n-th listed sequence, so it is not in the list. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, The Early Development of Set Theory](https://plato.stanford.edu/entries/settheory-early/) — dates Cantor’s diagonal proof of the uncountability of the reals to 1892 (“In the same short paper (1892), Cantor presented his famous proof” that the reals are “non-denumerable by the method of diagonalisation”) and describes the general power-set theorem. - [Stanford Encyclopedia of Philosophy, Basic Set Theory](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html) — gives the diagonal proof that a set cannot be bijected with its power set. - [Beth number, Wikipedia](https://en.wikipedia.org/wiki/Beth_number) — read as raw wikitext: “ℶ_(α+1) = 2^ℶ_α, and it follows by Cantor’s theorem and transfinite induction that the sequence of beth numbers is strictly increasing.” - [Cantor’s diagonal argument, Wikipedia](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument) — read as raw wikitext: s is built “choosing … for every n, the n-th digit as complementary to the n-th digit of sₙ”; from any enumeration “an element s of T can be constructed that doesn’t correspond to any sₙ in the enumeration.” ### 27.4 {#r-27.4} The continuum has cardinality 2^ℵ₀, and the continuum hypothesis 2^ℵ₀ = ℵ₁ is independent of ZFC through the consistency results of Gödel (announced 1938 in the Proceedings of the National Academy of Sciences, monograph 1940) and Cohen (1963). *Status: confirmed* - [Stanford Encyclopedia of Philosophy, The Continuum Hypothesis](https://plato.stanford.edu/archives/fall2016/entries/continuum-hypothesis/) — states CH as 2^ℵ₀ = ℵ₁ and records Gödel’s 1938 relative consistency result and Cohen’s 1963 result for ¬CH. - [Stanford Encyclopedia of Philosophy, Set Theory](https://plato.stanford.edu/entries/set-theory/) — “In 1938 Gödel proved the consistency of the CH with ZFC.”; its bibliography lists Gödel’s 1938 “The consistency of the axiom of choice and of the generalized continuum hypothesis,” Proceedings of the National Academy of Sciences, U.S.A. 24: 556–557; it states the CH formulation and the ZFC-independence conclusion; its bibliography lists Gödel, 1938, “The consistency of the axiom of choice and of the generalized continuum hypothesis,” Proceedings of the National Academy of Sciences, U.S.A. 24: 556–557. ### 27.5 {#r-27.5} The beth hierarchy starts at beth₀ = ℵ₀ and iterates powersets; bethω is its limit and is a strong-limit singular cardinal of cofinality ω. *Status: confirmed* - [Wikipedia, Beth number](https://en.wikipedia.org/wiki/Beth_number) — defines the beth recursion and its limit stages; “Beth omega ℶω (pronounced beth omega) is the smallest uncountable strong limit cardinal..” - [Wikipedia, Limit cardinal](https://en.wikipedia.org/wiki/Limit_cardinal) — states that bethω is a strong-limit cardinal of cofinality ω and is singular. ### 27.6 {#r-27.6} Alephω is the union/limit of the finite alephs and is singular because it is reached as a countable union of smaller cardinals; König’s theorem gives 2^ℵ₀ ≠ ℵω. Kőnig’s theorem shows that 2^ℵ₀ cannot equal ℵ_α for α of cofinality ω; for example 2^ℵ₀ ≠ ℵ_ω. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, The Continuum Hypothesis](https://plato.stanford.edu/archives/fall2016/entries/continuum-hypothesis/) — treats ℵω as a singular cardinal of countable cofinality and records König’s constraints on exponentiation. - [Wikipedia, Limit cardinal](https://en.wikipedia.org/wiki/Limit_cardinal) — describes ℵω as singular of cofinality ω and distinguishes it from bethω. - [Continuum hypothesis, Wikipedia (Kőnig’s theorem)](https://en.wikipedia.org/wiki/Continuum_hypothesis) — read as raw wikitext: Kőnig’s theorem “shows that 2^ℵ₀ cannot equal ℵ_α for α with cofinality ω. For example, 2^ℵ₀ ≠ ℵ_ω.” ### 27.7 {#r-27.7} For infinite cardinals, addition and multiplication give the maximum of the two cardinals, while 0 multiplied by any cardinal is 0; cardinal subtraction and division are not generally available. The product |X|·|Y| is the size of the Cartesian product X × Y, the set of all pairs (a, b) with a in X and b in Y, and 2^|X| is the size of the power set of X. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, The Continuum Hypothesis](https://plato.stanford.edu/archives/fall2016/entries/continuum-hypothesis/) — states κ + λ = κ · λ = max{κ,λ} for infinite cardinals and distinguishes exponentiation as the interesting operation. - [Stanford Encyclopedia of Philosophy, Basic Set Theory](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html) — provides the set/cardinality framework underlying products and cardinal arithmetic; “The Cartesian product A×B of two sets, A and B, is defined as the set of all ordered pairs (a, b) such that a∈A and b∈B.” - [Cardinal number, Wikipedia (cardinal arithmetic)](https://en.wikipedia.org/wiki/Cardinal_number) — read as raw wikitext: “Zero is a multiplicative absorbing element: κ·0 = 0·κ = 0”; for infinite σ a solution of μ + κ = σ exists iff μ ≤ σ, and “It will be unique (and equal to σ) if and only if μ < σ”; likewise for μ·κ = π with μ non-zero. “The product of cardinals comes from the Cartesian product”: |X|·|Y| = |X × Y|; and “2^|X| is the cardinality of the power set of the set X” (exponent as typeset on the page). ### 27.8 {#r-27.8} Ordinal multiplication is defined recursively with α·0 = 0, represents copies of the left ordinal, and is not commutative; for example 2·ω = ω while ω·2 = ω+ω. The first transfinite ordinal ω is the set of all natural numbers, followed by ω + 1, ω + 2, …, and ordinal addition is not commutative either (3 + ω = ω). *Status: confirmed* - [Wikipedia, Ordinal arithmetic](https://en.wikipedia.org/wiki/Ordinal_arithmetic) — gives the recursive definition α·0 = 0, the copies interpretation, and the examples 2·ω = ω and ω·2 = ω+ω; “The first transfinite ordinal is ω, the set of all natural numbers, followed by ω + 1, ω + 2, etc.” and “Ordinal addition is, in general, not commutative. For example, 3 + ω = ω.” - [Ordinal number, Wikipedia](https://en.wikipedia.org/wiki/Ordinal_number) — read as raw wikitext: “the cardinality of ω₀ = ω is ℵ₀, which is also the cardinality of ω² or ε₀ (all are countable ordinals).” ### 27.9 {#r-27.9} In the von Neumann representation, 0 is the empty set, 1 is {∅}, and 2 is {∅,{∅}}. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, Infinity](https://plato.sydney.edu.au/entries/infinity/index.html) — describes von Neumann’s 1923 representation and lists 0, 1, 2 and ω as sets of earlier ordinals. ### 27.10 {#r-27.10} The ordinals embed among the surreals, but ordinary ordinal addition is not the same as surreal addition: ordinal 1+ω = ω, whereas surreal addition is commutative and gives ω+1. The surreal ω − 1 exists but is not an ordinal, and the ordinal ω is not the successor of any ordinal. *Status: confirmed* - [Wikipedia, Surreal number](https://en.wikipedia.org/wiki/Surreal_number) — states that ordinals occur as particular surreals and explicitly contrasts ordinal 1+ω = ω with the surreal sum 1+ω = ω+1 > ω: “The sum of ordinals 1 + ω equals ω”; and “The surreal number ω − 1 is not an ordinal; the ordinal ω is not the successor of any ordinal.” ### 27.11 {#r-27.11} The empty-set symbol ∅ was introduced by Bourbaki, specifically André Weil, in 1939, inspired by the Norwegian/Danish letter Ø. *Status: judgement* - [Wikipedia, Empty set](https://en.wikipedia.org/wiki/Empty_set) — reports the Bourbaki/André Weil attribution and the 1939 date, with Ø as the inspiration. - [Jeff Miller, Earliest Uses of Symbols of Set Theory and Logic (MacTutor)](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/set/) — quotes Weil’s own account: “I had been personally responsible for its adoption … The symbol came from the Norwegian alphabet, with which I alone among the Bourbaki group was familiar” (André Weil, The Apprenticeship of a Mathematician, Birkhäuser 1992, p. 114). Editorial note: We found the 1939 Bourbaki/Weil attribution in the reference summary and a first-person Weil account of the Norwegian origin, but not a primary publication scan establishing the exact first printed use. We retain the chapter’s compact historical wording as a judgement call. ### 27.12 {#r-27.12} The empty set has one subset, so its power set has cardinality 1 and |P(∅)| − 1 = 0; Cantor’s theorem says every power set is strictly larger than its base set. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, Basic Set Theory](https://plato.stanford.edu/entries/set-theory/basic-set-theory.html) — proves that P(A) is not bijectable with A; the empty-set case has the single subset ∅. - [Wikipedia, Von Neumann universe](https://en.wikipedia.org/wiki/Von_Neumann_universe) — shows the cumulative hierarchy beginning with the empty set and its power set. ### 27.13 {#r-27.13} Eilenberg and Mac Lane introduced the general notions of category, functor, and natural transformation in their 1945 paper General Theory of Natural Equivalences. *Status: confirmed* - [Wikipedia, Category theory](https://en.wikipedia.org/wiki/Category_theory) — dates the general introduction of categories, functors, and natural transformations to the authors’ 1945 paper. ### 27.14 {#r-27.14} In Set, the empty set is an initial object and there is exactly one function from ∅ to every set X. *Status: confirmed* - [Wikipedia, Initial and terminal objects](https://en.wikipedia.org/wiki/Initial_and_terminal_objects) — identifies ∅ as the unique initial object in Set. - [Wikipedia, Function (mathematics)](https://en.wikipedia.org/wiki/Function_(mathematics)) — the empty function is the unique function with empty domain to any codomain. ### 27.15 {#r-27.15} ZFC is the usual Zermelo–Fraenkel framework with choice; the axiom of choice is associated with Zermelo’s 1904 well-ordering proof and is independent of ZF. Skolem and Fraenkel formalized Separation and introduced Replacement, and von Neumann added Foundation. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, Set Theory](https://plato.stanford.edu/entries/set-theory/) — describes ZFC, the role of AC, and the independence results for AC and CH; “Further work by Skolem and Fraenkel led to the formalization of the Separation axiom in terms of formulas of first-order, instead of the informal notion of property, as well as to the introduction of the axiom of Replacement,” and “A further addition, by von Neumann, of the axiom of Foundation, led to the standard axiom system of set theory, known as the Zermelo-Fraenkel axioms plus the Axiom of Choice, or ZFC.” - [Stanford Encyclopedia of Philosophy, Zermelo’s Axiomatization of Set Theory](https://plato.stanford.edu/entries/zermelo-set-theory/) — gives the historical setting for Zermelo’s well-ordering work and the axiomatization of set theory. ### 27.16 {#r-27.16} Russell’s paradox arises from the collection of sets that are not members of themselves and helped force axiomatization of set theory. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, Set Theory](https://plato.stanford.edu/entries/set-theory/) — discusses the paradoxes and the axiomatic development of set theory. - [Wikipedia, Russell’s paradox](https://en.wikipedia.org/wiki/Russell%27s_paradox) — states the self-membership contradiction and its historical consequences. ### 27.17 {#r-27.17} A strongly inaccessible cardinal is uncountable, regular, and a strong limit; its existence is not provable in ZFC under the usual consistency assumptions. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, Set Theory](https://plato.stanford.edu/entries/set-theory/) — defines strongly inaccessible cardinals by regularity and strong-limit behavior and explains why their existence is an additional axiom beyond ZFC: “One cannot prove in ZFC that there exists a regular limit cardinal κ, for if κ is such a cardinal, then L_κ is a model of ZFC, and so ZFC would prove its own consistency, contradicting Gödel’s second incompleteness theorem.” It defines the term: “An infinite cardinal κ is called regular if it is not the union of less than κ smaller cardinals.” (Symbols as rendered on the page.) ### 27.18 {#r-27.18} The ordinal ε₀ is the least positive fixed point of α ↦ ω^α, and Gentzen’s 1936 consistency proof for Peano arithmetic uses transfinite induction up to ε₀. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, Proof Theory](https://plato.stanford.edu/entries/proof-theory/) — defines ε₀ as the least α with ω^α = α and explains the descending-ordinal argument in Gentzen’s consistency proof for PA. - [Wikipedia, Gentzen’s consistency proof](https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof) — dates the published proof to 1936 and identifies transfinite induction up to ε₀ as its ordinal-theoretic ingredient. ### 27.19 {#r-27.19} Gregory H. Moore’s scholarly history traces the generalized continuum hypothesis through the work of Charles Sanders Peirce, who introduced the beth notation. Peirce used the beth notation in a letter to Cantor of December 1900. *Status: judgement* - [Gregory H. Moore, Early History of the Generalized Continuum Hypothesis: 1878–1938](https://doi.org/10.2178/bsl/1318855631) — the 2011 Bulletin of Symbolic Logic history; its abstract (the full text is not free): “This paper explores how the Generalized Continuum Hypothesis (GCH) arose from Cantor’s Continuum Hypothesis in the work of Peirce, Jourdain, Hausdorff, Tarski, and how GCH was used up to Gödel’s relative consistency result.” The beth notation and the December 1900 letter are on the Open Logic line. - [Open Logic Project, Set Theory, Cardinal arithmetic (chapter PDF)](https://builds.openlogicproject.org/content/set-theory/card-arithmetic/ch.pdf) — read from the PDF: “Peirce introduced the “ℶ” notation; this is beth, which is the second letter in the Hebrew alphabet,” with the footnote “Peirce used this notation in a letter to Cantor of December 1900.” Editorial note: Judgement retained: Moore is a specialist scholarly source and directly supports the attribution, but the primary letter scan was not available for inspection. The card therefore attributes the report to Moore rather than presenting an independently verified transcription. ### 27.20 {#r-27.20} Cantor’s transfinite set theory developed from his 1874 work through the 1883 introduction of transfinite ordinals and the 1895/1897 papers; he introduced aleph notation in 1895, while the diagonal paper appeared in 1891/92. Cardinality was in use in Cantor’s work from 1874 and 1878; the 1895 and 1897 papers were a well-organized presentation of his results on cardinals and ordinals. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, The Early Development of Set Theory](https://plato.stanford.edu/entries/settheory-early/) — dates Cantor’s foundational 1874 paper, the 1883 introduction of transfinite ordinals, the 1895 introduction of aleph notation, the 1895/1897 transfinite-number papers and the diagonal paper’s date, 1892: “In the same short paper (1892), Cantor presented his famous proof” that the reals are “non-denumerable by the method of diagonalisation”; the 1891 is on the next line. The “second number class” “was formed by ω and all numbers following it (including ω^ω, and many more) that have only a denumerable set of predecessors,” and “his theorems justified calling the cardinality of the “second number class” ℵ₁” (symbols as rendered on the page). - [Cantor’s diagonal argument, Wikipedia](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument) — “Georg Cantor published this proof in 1891,” its first reference giving the paper as Cantor (1891), “Ueber eine elementare Frage der Mannigfaltigkeitslehre,” Jahresbericht der Deutschen Mathematiker-Vereinigung 1: 75–78. - [Cantorsche Normalform, Wikipedia (German)](https://de.wikipedia.org/wiki/Cantorsche_Normalform) — read as raw wikitext: “Er wurde erstmals 1897 von Cantor für gewisse Ordinalzahlen bewiesen,” citing Cantor’s Beiträge zur Begründung der transfiniten Mengenlehre. ### 27.21 {#r-27.21} Hilbert’s 1926 paper Über das Unendliche contains the quoted defense of Cantor’s paradise, on p. 170. *Status: confirmed* - [Wikipedia (German), Cantors Paradies](https://de.wikipedia.org/wiki/Cantors_Paradies) — quotes Hilbert’s sentence and cites it to Über das Unendliche, Mathematische Annalen 95 (1926), p. 170: “Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können.” ### 27.22 {#r-27.22} Gödel’s completely detailed proof that the axiom of choice and the generalized continuum hypothesis, and so the continuum hypothesis itself, are consistent with the axioms of set theory is the 1940 monograph, which followed his briefer earlier announcements. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, Kurt Gödel](https://plato.stanford.edu/entries/goedel/) — names “The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory” as the 1940 monograph in which Gödel “gave a completely detailed proof of the two theorems,” the earlier announcement having been “lacking in a great many details”; its bibliography lists [1940] “The Consistency of the Continuum Hypothesis,” “Annals of Mathematics Studies, Volume 3, Princeton: Princeton University Press.” Editorial note: Neither source at 27.4 carries the year 1940: both give Gödel 1938, and the Continuum Hypothesis entry prints no Gödel bibliography entry at all. The chapter’s prose and the ch and cont cards all name the 1940 monograph, so the monograph is dated here. ### 27.23 {#r-27.23} ℵω is the first singular cardinal, because it is the union of countably many smaller cardinals. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, Set Theory](https://plato.stanford.edu/entries/set-theory/) — states “The first singular cardinal is ℵω, as it is the union of countably-many smaller cardinals, namely ℵω = ⋃{ℵn : n<ω},” and separately that one cannot prove in ZFC that a regular limit cardinal exists. Editorial note: 27.6’s two sources carry ℵω’s singularity and its cofinality ω but neither calls it the first singular cardinal; that superlative is carried here. It is a statement made inside ZFC, where every successor aleph is regular — in ZF alone ℵ1 can be singular. ### 27.24 {#r-27.24} A category consists of objects, arrows between them, an associative composition and an identity arrow on each object; objects are atomic in the definition and are characterized by their relations to other objects. *Status: confirmed* - [Wikipedia, Category theory](https://en.wikipedia.org/wiki/Category_theory) — gives the definition as a class ob(C) of objects, a class hom(C) of morphisms or arrows, a composition ∘: hom(b, c) × hom(a, b) → hom(a, c) and an identity morphism 1x on every object x, and states that objects are considered atomic so that “one can characterize these objects in terms of their relations to other objects, as given by the morphisms”; “A prototypical example of a category is Set, the category of sets and functions between them.” ### 27.25 {#r-27.25} An initial object of a category C is an object I such that for every object X there is precisely one morphism I → X, and any two initial objects are related by a unique isomorphism. *Status: confirmed* - [Wikipedia, Initial and terminal objects](https://en.wikipedia.org/wiki/Initial_and_terminal_objects) — states “An initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X,” that a terminal object has exactly one morphism the other way, that two different initial objects have a unique isomorphism between them, and that the empty set is the unique initial object in Set. ### 27.26 {#r-27.26} Two sets have the same cardinality exactly when there is a bijection between them, and Dedekind defined an infinite set as one that can be put in one-to-one correspondence with a proper subset of itself; the even numbers are as many as the whole numbers. Doubling, f(n) = 2n, is a bijection from the natural numbers to the even numbers. *Status: confirmed* - [Stanford Encyclopedia of Philosophy, Infinity](https://plato.sydney.edu.au/entries/infinity/index.html) — states “When there is a bijection between two sets, we say that they have the same cardinality,” gives Galileo’s paradox that “the squares of natural numbers are a proper subset of the natural numbers, but they have the same cardinality since they can be put in one-to-one correspondence,” and records Dedekind’s 1884 definition of an infinite set as one admitting a bijection with a proper subset of itself. - **Own computation** (the author's own work) — n ↦ 2n pairs every whole number with exactly one even number and leaves nothing over on either side, so the even numbers are equinumerous with the whole numbers; the source above carries the same fact for the squares. - [Cardinality, Wikipedia](https://en.wikipedia.org/wiki/Cardinality) — read as raw wikitext: the even numbers have the cardinality of the natural numbers “since the function f(n) = 2n is a bijection from ℕ to E”; Hilbert’s hotel moves “room n to room 2n.” Editorial note: 27.1’s source characterizes cardinality by bijection but contains no equinumerous-with-a-proper-subset statement and no even-numbers example, so the chapter’s opening paragraph is carried here instead. ### 27.27 {#r-27.27} A well-order is a total order in which every non-empty subset has a least element; every element except a possible greatest one has a unique successor, and some elements other than the least may have no predecessor. Two ordered sets have the same order type when there is an order isomorphism between them, and the order type of a well-ordered set is by definition an ordinal. ω is the smallest limit ordinal; 1 + ω = ω, which is not equal to ω + 1. *Status: confirmed* - [Well-order, Wikipedia](https://en.wikipedia.org/wiki/Well-order) — read as raw HTML: “a well-order… on a set S is a total ordering on S with the property that every non-empty subset of S has a least element… Every element s of a well-ordered set, except a possible greatest element, has a unique successor… There may be elements, besides the least element, that have no predecessor.” - [Order type, Wikipedia](https://en.wikipedia.org/wiki/Order_type) — two ordered sets share an order type exactly when an order isomorphism exists between them, and the order types of well-ordered sets are the ordinals. - [Well Ordered Set, MathWorld](https://mathworld.wolfram.com/WellOrderedSet.html) — the same least-element condition. - [Limit ordinal, Wikipedia](https://en.wikipedia.org/wiki/Limit_ordinal) — read as raw wikitext: “Limit ordinals are those that are non-zero and have no predecessor, such as ω or ω²”; “the smallest limit ordinal is ω, the smallest ordinal greater than every natural number.” - [Infinity, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/infinity/) — “1 + ω is the order type of a single element, followed by a copy of the natural numbers, which is in fact the same as the order type of the natural numbers! Thus, 1 + ω = ω, which is not equal to ω + 1.” ### 27.28 {#r-27.28} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: The continuum hypothesis is Cantor’s conjecture of 1878; ZFC grew from Zermelo’s 1908 axiomatization. *Status: confirmed* - [Continuum hypothesis, Wikipedia](https://en.wikipedia.org/wiki/Continuum_hypothesis) — the page’s words: “Georg Cantor advanced the continuum hypothesis in 1878.” - [The Early Development of Set Theory, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/settheory-early/) — read raw: “As a result of his incisive analysis, in 1908 he published his axiom system, showing how it blocked the known paradoxes.” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 27.34 {#r-27.34} States that Cantor’s second number class is the set of the infinite countable ordinals, itself uncountable, with cardinality ℵ₁. Georg Cantor introduced it in 1883; together with the first number class it makes up ω₁, the set of all countable ordinals. With countable choice, the supremum of every countable set of countable ordinals is countable, so no countable sequence is cofinal in ω₁. *Status: confirmed* - [Ordinal number, Wikipedia](https://en.wikipedia.org/wiki/Ordinal_number) — describes Cantor’s number classes: the second number class is the countably infinite ordinals, and together with the finite ordinals it makes up the countable ordinals, of cardinality ℵ₁. - [Aleph number, Wikipedia](https://en.wikipedia.org/wiki/Aleph_number) — carries ω₁ as the set of all countable ordinals: “ℵ1 is the cardinality of the set of all countable ordinal numbers. This set is denoted by ω1 (or sometimes Ω).” - [First uncountable ordinal, Wikipedia](https://en.wikipedia.org/wiki/First_uncountable_ordinal) — “the elements of ω1 are the countable ordinals (including finite ordinals), of which there are uncountably many”; and, if the axiom of countable choice holds, “the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal,” so no countable sequence of countable ordinals reaches ω₁. ### 27.35 {#r-27.35} States that for every set X there is exactly one function from the empty set to X, the empty function, making ∅ the initial object of Set. *Status: confirmed* - [Function (mathematics), Wikipedia](https://en.wikipedia.org/wiki/Function_(mathematics)) — states that the empty function, with empty domain, is the unique function from the empty set to any set. - [Empty set, Wikipedia](https://en.wikipedia.org/wiki/Empty_set) — carries the empty set as the unique initial object of Set: “As a result, the empty set is the unique initial object of the category of sets and functions.” It calls the empty topological space, in the category of topological spaces, a strict initial object: “only the empty set has a function to the empty set.” - [Initial and terminal objects, Wikipedia](https://en.wikipedia.org/wiki/Initial_and_terminal_objects) — carries that Set has no zero object: “The empty set is the unique initial object in Set, the category of sets. Every one-element set (singleton) is a terminal object in this category; there are no zero objects.”