Fact-checked by Grok 2 weeks ago

Equinumerosity

Equinumerosity is a fundamental concept in set theory, denoting the relation between two sets that have the same cardinality, meaning there exists a bijection—a one-to-one correspondence—between their elements.^[1] This bijection ensures that every element in one set pairs uniquely with exactly one element in the other, without leftovers, capturing the intuitive notion of "same size" in a precise mathematical framework applicable to both finite and infinite collections.^[2] For example, the set of natural numbers and the set of perfect squares are equinumerous, as the function f(n) = n^2 establishes a bijection between them.^[3] The concept originated in the late 19th century through the work of Georg Cantor, who developed set theory to rigorously compare the sizes of infinite sets, resolving paradoxes noted earlier by thinkers like Galileo Galilei.^[3] Galileo observed in the early 17th century that the natural numbers and their squares seemed equally numerous despite the latter being a proper subset, a counterintuitive property of infinity that Cantor formalized using bijections to define equinumerosity.^[3] Cantor's approach, introduced in the 1870s, laid the groundwork for transfinite arithmetic and the hierarchy of infinite cardinalities, such as the countable infinity \aleph_0 for the naturals and the uncountable continuum c for the reals.^[3] Equinumerosity forms an equivalence relation on the class of all sets, satisfying reflexivity (every set is equinumerous to itself via the identity function), symmetry (if set A is equinumerous to B, then B to A via the inverse bijection), and transitivity (if A to B and B to C, then A to C via the composition of bijections).^[2]^[4] This relational structure partitions sets into equivalence classes by cardinality, enabling the assignment of cardinal numbers to measure "size" without relying on traditional counting, which fails for infinities.^[1] In practice, equinumerosity reveals profound distinctions among infinite sets, such as the natural numbers being equinumerous to the rationals but not to the reals, as proven by Cantor's diagonal argument.^[3] It underpins modern mathematics, influencing fields from topology to computer science, by providing a tool to analyze structures like countable unions or Hilbert's infinite hotel paradox, where infinite sets can absorb additional elements while remaining equinumerous to their originals.^[4]

Fundamentals

Definition

In set theory, a set is a well-defined collection of distinct objects, called elements, which may be finite or infinite in extent. A function from a set A to a set B, denoted f: A \to B, is a rule that assigns to each element of A exactly one element of B, establishing a mapping between the two collections.^[5] Two sets A and B are equinumerous, or have the same size in a precise sense, if there exists a bijection f: A \to B between them. A bijection is a function that is both injective (one-to-one) and surjective (onto): injectivity means that distinct elements in A map to distinct elements in B, ensuring no two elements of A share the same image; surjectivity means that every element in B is the image of at least one element in A, ensuring the mapping covers all of B.^[6] This notion captures the intuitive idea of sets having the "same number" of elements without relying on a pre-existing concept of number, allowing comparison even for infinite sets. The term "equinumerosity," introduced by George Boolos in the 1980s, describes the relation of sets having the same cardinality via a bijection, building on foundational ideas from Richard Dedekind's 1888 monograph Was sind und was sollen die Zahlen?, which emphasized one-to-one correspondences independent of numerical presuppositions. Equinumerosity partitions the class of all sets into equivalence classes, each defining a unique cardinal number.^[7]

Notation and Terminology

In mathematical literature, equinumerosity between two sets A and B is commonly denoted by |A| = |B|, indicating that they have the same cardinality.^[1] An alternative symbol, A \sim B, is also widely used to signify the existence of a bijection between the sets.^[8] The term "equinumerosity" itself derives from roots meaning equal in number, while synonymous variants include "equipotent," introduced by Georg Cantor to describe sets that can be placed in one-to-one correspondence, emphasizing equal potency or power.^[9] "Equipollent," a historical term predating Cantor, similarly conveys equal multitude or strength and was used in early set theory texts to denote sets of equivalent size via bijection.^[10] The etymology of "equipotent" traces to the English compounding of "equi-" (equal) and "potent" (powerful) in the 1870s, while "equipollent" originates from Latin aequipollēns (equal in power), borrowed into Middle English via French around 1420.^[11]^[12] In pure mathematics, notations like |A| = |B| directly invoke cardinality, presupposing numerical comparison, whereas in philosophy of mathematics, terms such as "equinumerosity" or "equipollent" are preferred to avoid implying pre-existing numbers, focusing instead on relational equivalence through correspondence.^[7] This distinction arises from foundational debates, where equinumerosity is analyzed as a primitive relation rather than a derived numerical property.^[13] Gottlob Frege significantly influenced the terminology by formalizing equinumerosity as the existence of a one-to-one correspondence between concepts, denoted in modern reconstructions as F \approx G, to ground arithmetic in logic without circular reference to numbers.^[7]

Historical Development

Pre-Cantorian Ideas

The ancient Greeks developed intuitive notions of equinumerosity through pairing elements in finite geometric and arithmetic contexts, eschewing the actual infinite in favor of potential infinity. In Euclid's Elements (c. 300 BCE), equality of magnitudes is established by superimposition or division into corresponding parts, implicitly relying on one-to-one pairings to demonstrate congruence without invoking infinite processes.^[14] Aristotle, in Physics (c. 350 BCE), further elaborated on numerical equality by pairing units to distinguish even from odd numbers, viewing numbers as discrete multitudes where such correspondences reveal parity, though he rejected actual infinities as unrealized potentials. These approaches grounded early ideas of sameness in quantity within finite bounds, influencing subsequent mathematical philosophy. Philosophical inquiries into infinite divisibility introduced tensions with equinumerosity for collections. David Hume, in A Treatise of Human Nature (1739), critiqued the notion that space or time could be infinitely divided into equal parts, arguing that such divisions imply an infinite number of indivisible components forming the whole, yet the mind perceives extensions as finite collections of simple ideas rather than endless equal subdivisions.^[15] Hume emphasized that equality in these collections is discerned through general appearance or superposition, not enumeration of infinite parts, highlighting the conceptual challenges of infinite equal divisions in perceptual sets. In the early modern period, paradoxes of infinite equinumerosity emerged, underscoring counterintuitive equalities. Galileo Galilei, in Dialogues Concerning Two New Sciences (1638), observed that the natural numbers appear more numerous than the perfect squares—a proper subset—yet each square corresponds uniquely to its root, suggesting an equality in quantity that defies finite intuition. This pairing revealed that infinite collections could match in multitude despite apparent disparities, prompting Galileo to conclude that terms like "equal" or "greater" fail for infinities.^[16] By the 19th century, precursors formalized these intuitions for infinite sets. Bernard Bolzano, in Paradoxien des Unendlichen (1851), defined equality in multitude for infinite collections through mutual transformability into one another via one-to-one correspondences, allowing proper subsets like even numbers to equal the whole set of naturals in quantity.^[17] Bolzano's treatment of multitudes as pluralities or sums extended pre-Cantorian ideas, resolving paradoxes by accepting such equalities without contradiction. These developments paved the way for rigorous set-theoretic analysis.

Cantor's Formalization and Beyond

Georg Cantor laid the foundations for modern set theory by developing the notion of one-to-one correspondence as the criterion for determining when two sets have the same size, even if infinite, in a series of publications spanning 1873 to 1895. In his 1873 letter to Richard Dedekind and subsequent 1874 paper "On a Property of the Collection of All Real Algebraic Numbers," Cantor demonstrated that the set of real numbers is uncountable by showing it cannot be placed in bijection with the natural numbers, thus establishing equinumerosity as equivalence under bijective mappings. Over the following decades, including works in 1883 and 1895's "Contributions to the Founding of the Theory of Transfinite Numbers," he formalized this into a theory of transfinite cardinal numbers, where equinumerosity classes define distinct infinite sizes, such as the countable infinity of the naturals and larger transfinites. Richard Dedekind played a pivotal role in refining these ideas, particularly through his 1872 supplement to Dirichlet's lectures on number theory and his 1888 monograph "The Nature and Meaning of Numbers," where he introduced the concept of equinumerosity—sets are equinumerous if there exists a one-to-one correspondence between them—and used it to rigorously define finite and infinite sets via the absence or presence of proper subsets in bijection with the whole. Dedekind's approach emphasized the logical construction of numbers from sets, bridging Cantor's combinatorial insights with arithmetic foundations. Complementing this, Giuseppe Peano provided an axiomatic framework in his 1889 "Arithmetices principia, nova methodo exposita," where he formalized the natural numbers with postulates that implicitly rely on equinumerosity for induction and successor functions, ensuring the structure supports Cantorian comparisons without paradoxes. The evolution of equinumerosity culminated in the transition to axiomatic set theory with Ernst Zermelo's 1908 "Investigations in the Foundations of Set Theory," which incorporated the concept implicitly through axioms like extensionality (sets are equal if they have the same elements) and the axiom of infinity, enabling the construction of infinite sets while avoiding contradictions like Russell's paradox; this framework treated equinumerosity as a derived relation for comparing cardinalities within a consistent system. A landmark application appeared in Cantor's 1891 paper "Über eine elementare Frage der Mannigfaltigkeitslehre," where he employed the diagonalization method to prove that the continuum of real numbers is uncountably infinite, demonstrating no bijection exists with the naturals and thus establishing a strictly larger cardinality, $2^{\aleph_0}.

Properties

As an Equivalence Relation

Equinumerosity, the relation where two sets A and B satisfy A \sim B if there exists a bijection between them, is an equivalence relation on the class of all sets.^[18] To verify this, the relation must satisfy the properties of reflexivity, symmetry, and transitivity.^[18] Reflexivity. For any set A, the identity function \mathrm{id}_A: A \to A given by \mathrm{id}_A(a) = a for all a \in A is a bijection, since it is both injective (distinct elements map to distinct elements) and surjective (every element in A is hit). Thus, A \sim A.^[18] Symmetry. Suppose A \sim B via a bijection f: A \to B. The inverse function f^{-1}: B \to A exists and is also a bijection, as the inverse of a bijection preserves injectivity and surjectivity. Therefore, B \sim A.^[18] Transitivity. If A \sim B via a bijection f: A \to B and B \sim C via a bijection g: B \to C, then the composition g \circ f: A \to C is a bijection. This follows because the composition of injective functions is injective, the composition of surjective functions is surjective, and bijections compose accordingly. Hence, A \sim C.^[18] These proofs establish the equivalence relation properties in ZF set theory without requiring the axiom of choice, as they rely solely on the definitions of bijections and function composition available in ZF. As a result, equinumerosity partitions the class of all sets into equivalence classes, with each class comprising all sets that share the same cardinality.^[18]

Behavior in Finite and Infinite Sets

In the case of finite sets, equinumerosity is straightforward and aligns with intuitive notions of size: two finite sets are equinumerous if and only if they have the same number of elements, with any bijection between them amounting to a permutation that rearranges the elements without altering the count.^[14] This property ensures that finite equinumerosity preserves the standard arithmetic of counting, where no set can be matched one-to-one with a proper subset of itself. For infinite sets, equinumerosity reveals counterintuitive behaviors that defy finite intuitions, allowing a set to be placed in one-to-one correspondence with a proper subset. A classic example is the equinumerosity between the set of natural numbers \mathbb{N} = \{1, 2, 3, \dots\} and the set of even natural numbers \{2, 4, 6, \dots\}, established by the bijection f(n) = 2n, which pairs each natural number n with a unique even number while covering all evens.^[14] This pairing demonstrates how infinite sets can "absorb" removals or rearrangements without reducing their effective size, a phenomenon first highlighted by Galileo but formalized by Cantor. The Hilbert's Grand Hotel paradox provides a memorable illustration of this infinite equinumerosity. Imagine a hotel with countably infinitely many rooms, all occupied by guests in rooms numbered $1, 2, 3, \dots. To accommodate one additional guest, the manager instructs each guest in room n to move to room n+1, freeing room $1without displacing anyone overall. For countably infinitely many new guests, guests shift from roomn

to room &#36;2n

, vacating all odd-numbered rooms for the newcomers. These maneuvers show that the original set of occupied rooms remains equinumerous to the expanded occupancy, underscoring the paradoxical capacity of infinite sets to match larger configurations through simple bijections.^[14] Richard Dedekind formalized this distinguishing feature by defining a set as Dedekind-infinite if it is equinumerous to one of its proper subsets, providing an intrinsic characterization of infinity independent of natural number embeddings (though equivalent under the axiom of choice).^[14] Without the axiom of choice, however, the situation is more nuanced: while equinumerosity can still hold between certain infinite sets—such as those provably bijective via ZF alone—explicit constructions of the bijections may not be possible, as choice is often required to select elements or define such mappings systematically.^[14]

Relation to Cardinality

Defining Cardinal Numbers via Equinumerosity

In set theory, the cardinal number of a set A, denoted |A|, is formally defined as the equivalence class of all sets equinumerous to A under the relation of bijection, capturing the intuitive notion of "size" independent of structure.^[9] This approach, originating from Cantor's work but refined in axiomatic frameworks, partitions the universe of sets into equivalence classes where two sets belong to the same class if and only if there exists a bijective function between them. However, in pure Zermelo-Fraenkel set theory (ZF), such equivalence classes are proper classes rather than sets, posing foundational challenges; to represent cardinals as actual sets while avoiding paradoxes like Russell's, additional axioms are invoked.^[19] A standard construction in Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) identifies cardinal numbers with initial ordinals, following John von Neumann's definition. For finite cardinals, the von Neumann ordinal n is the set \{0, 1, \dots, n-1\}, where each finite ordinal serves as its own cardinal, aligning with the natural numbers.^[9] For infinite cardinals, these are the smallest ordinals not equinumerous to any smaller ordinal, known as initial ordinals; for instance, the least infinite cardinal is the ordinal \omega, the set of all finite ordinals. This representation ensures that every set is equinumerous to a unique cardinal, which is an ordinal, via the well-ordering theorem.^[20] An alternative definition, applicable even without the Axiom of Choice in ZF, employs Scott's trick to construct cardinals as sets of sets with minimal rank in their equivalence class, thereby circumventing the proper class issue. Specifically, for an equivalence class under equinumerosity, the cardinal is the set of all sets in that class with the least possible rank (the smallest ordinal \alpha such that some set of rank \alpha belongs to the class).^[21] This method, due to Dana Scott, produces a set representative for each cardinal while preserving the equivalence relation's structure and avoiding impredicative definitions that could lead to paradoxes. Infinite cardinals are commonly denoted by Greek letters such as \kappa or \lambda, where |A| = \kappa means A is equinumerous to the cardinal \kappa. In ZFC, the class of all cardinals is well-ordered by the relation induced by ordinal inclusion, and equinumerosity preserves this order: if A \sim B and |A| < |C|, then |B| < |C|, reflecting the strict totality of the ordering on cardinals.^[9]^[20]

Comparing Set Sizes

To determine whether one set is smaller than, equal to, or larger than another in terms of cardinality, mathematicians rely on the existence of specific functions between the sets, particularly injections. An injection (one-to-one function) from set A to set B implies that |A| \leq |B|, meaning A can be embedded into B without overlapping elements, so A is at most as large as B.^[9] If such an injection exists but there is no bijection between A and B, then the inequality is strict: |A| < |B|, indicating B is genuinely larger.^[9] A key tool for establishing equality of cardinalities is the Schröder-Bernstein theorem, which states that if there exist injections f: A \to B and g: B \to A, then there is a bijection between A and B, so |A| = |B|.^[22] The theorem was proved independently by Ernst Schröder in 1898 and by Felix Bernstein between 1898 and 1903.^[22] A standard proof constructs the bijection by analyzing the functional graph formed by f and g^{-1} (where defined), decomposing it into disjoint chains—such as finite cycles, infinite rays, or two-way infinite paths—and defining the bijection to alternate mappings along each chain while fixing points outside these structures.^[22] The Schröder-Bernstein theorem is provable in Zermelo-Fraenkel set theory (ZF) without invoking the axiom of choice (AC), as its construction yields an explicit bijection from the given injections.^[23] However, when comparing cardinalities in practice, establishing the initial injections—especially for certain infinite sets—may require AC to guarantee their existence or to derive explicit bijections in applications.^[23] An important application demonstrating strict inequality is Cantor's theorem, which asserts that no set is equinumerous to its power set: for any set A, there is an injection from A to \mathcal{P}(A) (mapping each element to its singleton), but no injection from \mathcal{P}(A) to A, so |\mathcal{P}(A)| > |A|. This result, first proved by Georg Cantor in 1891 using a diagonal argument, establishes that the power set operation always produces a strictly larger cardinality and implies the existence of infinitely many distinct infinite cardinalities.

Examples and Applications

Finite Set Illustrations

Equinumerosity provides a foundational way to compare the sizes of finite sets through the existence of a bijection, which pairs each element of one set uniquely with an element of the other. Consider the sets A = \{1, 2, 3\} and B = \{a, b, c\}. These sets are equinumerous because there exists a bijection f: A \to B defined by f(1) = a, f(2) = b, and f(3) = c, establishing a one-to-one correspondence between their elements.^[18] In contrast, the sets C = \{1, 2\} and D = \{1, 2, 3\} are not equinumerous, as no bijection exists between them; any function from C to D cannot cover all elements of D without repetition or omission, reflecting their differing sizes.^[24] For finite sets A and B, equinumerosity holds if and only if both have the same cardinality n for some natural number n, meaning there is a bijection with the initial segment \{1, 2, \dots, n\}. This criterion aligns with the Peano axioms, where finite cardinals satisfy the arithmetic structure through such one-to-one correspondences.^[24]^[25] The concept extends to counting the possible pairings: for two finite sets each of size n, the number of distinct bijections between them equals n!, the factorial of n, representing all possible permutations of the elements.^[26] In practical terms, equinumerosity applies to tangible scenarios, such as determining if the number of knives equals the number of plates at a table setting by pairing each knife uniquely with a plate, verifiable through direct listing for small finite collections.^[18] Similarly, the populations of two cities are equinumerous if their inhabitant counts match, confirmed via a census-based bijection that lists and pairs individuals without leftovers.^[18]

Infinite Set Cases

Equinumerosity in infinite sets reveals counterintuitive properties, where sets of seemingly different "sizes" can be placed in one-to-one correspondence, and others cannot, leading to a hierarchy of infinities. Unlike finite sets, infinite sets can be equinumerous to proper subsets of themselves, challenging intuitive notions of size. A set is Dedekind-infinite if it is equinumerous to one of its proper subsets, a concept formalized by Richard Dedekind that aligns with the existence of bijections in such cases.^[27] The set of natural numbers \mathbb{N} is equinumerous to the set of integers \mathbb{Z}, despite \mathbb{Z} including negative numbers and zero. This is demonstrated by the explicit bijection f: \mathbb{N} \to \mathbb{Z} defined as f(n) = \frac{n}{2} if n is even, and f(n) = -\frac{n+1}{2} if n is odd. This function pairs each natural number with a unique integer: even inputs map to non-negative integers, while odd inputs map to negatives, covering all of \mathbb{Z} without repetition.^[28] Similarly, the set of rational numbers \mathbb{Q} is equinumerous to \mathbb{N}, even though the rationals are dense in the reals and appear more numerous. Georg Cantor established this by enumerating the positive rationals through a grid of numerators and denominators, traversing diagonals to list fractions in order while skipping duplicates (e.g., \frac{1}{1}, \frac{1}{2}, \frac{2}{1}, \frac{1}{3}, \frac{2}{2} skipped, \frac{3}{1}, etc.), and extending to negatives and zero via additional pairings. This countable enumeration confirms a bijection exists, placing \mathbb{Q} in the same infinite cardinality as \mathbb{N}.^[29] Hilbert's paradox of the Grand Hotel illustrates the flexibility of countably infinite sets under equinumerosity. Consider a hotel with infinitely many rooms, all occupied by guests numbered by natural numbers. To accommodate one new guest, shift each existing guest from room n to room n+1, freeing room 1; this bijection f(n) = n+1 on the room assignments shows the set of occupied rooms remains equinumerous to \mathbb{N} after expansion. The paradox, introduced by David Hilbert in 1924, highlights how infinite sets can absorb additional elements without increasing cardinality.^[30] In contrast, the set of real numbers \mathbb{R} is not equinumerous to \mathbb{N}, as shown by Cantor's diagonal argument. Assume for contradiction a bijection lists all reals in (0,1) as infinite decimals r_1 = 0.d_{11}d_{12}\dots, r_2 = 0.d_{21}d_{22}\dots, etc. Construct a new real r = 0.d_1 d_2 \dots where d_i \neq d_{ii} (e.g., differing by 1 modulo 10, avoiding 0-9 ambiguities). This r differs from each r_n in the n-th position, so it is unlisted, contradicting the bijection. Thus, no such bijection exists, proving \mathbb{R} is uncountable.^[31] The cardinality of the continuum, denoted |\mathbb{R}| = \mathfrak{c} = 2^{\aleph_0}, exceeds \aleph_0 = |\mathbb{N}| and equals the cardinality of the power set \mathcal{P}(\mathbb{N}), the set of all subsets of \mathbb{N}. Cantor proved |\mathbb{R}| \leq |\mathcal{P}(\mathbb{N})| by injecting reals into binary sequences (via expansions), and |\mathcal{P}(\mathbb{N})| \leq |\mathbb{R}| similarly, yielding equinumerosity. His theorem further shows $2^{\aleph_0} > \aleph_0, establishing a strict hierarchy.

References

[1]
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/06%3A_New_Page/6.01%3A_New_Page
[2]
Equinumerosity - Mathmatique
Two sets are equinumerous to one another if there exists a bijection between them. For example, the set A={a,b,c} ...
[3]
[PDF] Infinity and its cardinalities
1870's The Russian mathematician Georg Cantor proposes his ground- breaking theory of sets and an arithmetic for “transfinite” cardinal numbers. His work ...
[4]
[PDF] Exercises: Equinumerosity - - Logic Matters
We say that a set is countable iff it is either empty or equinumerous with some set of natural numbers (maybe all of them!). It is countably infinite iff it is ...
[5]
Frege's Theorem and Foundations for Arithmetic
Jun 10, 1998 · The notion of equinumerosity plays an important and fundamental role in the development of Frege's Theorem. After developing the definition of ...<|control11|><|separator|>
[6]
[PDF] MATH 3110: HOMEWORK 3 You will be graded on both the ...
Let A, B, and C be sets. For this problem only, we'll write A ∼ B to mean that A and B are equinumerous, meaning that there is a.<|separator|>
[7]
Set Theory (Stanford Encyclopedia of Philosophy)
### Summary of Cantor's Terminology for Equinumerosity
[8]
Equipollent -- from Wolfram MathWorld
Two sets and are said to be equipollent iff there is a one-to-one correspondence (i.e., a bijection) from onto.
[9]
equipotent, adj. meanings, etymology and more
equipotent is formed within English, by compounding. Etymons: equi- comb. form, potent adj.1 · See etymology ...<|separator|>
[10]
equipollent, adj. & n. meanings, etymology and more
equipollent is a borrowing from French. Etymons: French equipolent. See etymology. Nearby entries. equipendent, adj.a1640–81; equipensate, v.1717–; equiperiodic ...Missing: equipotent | Show results with:equipotent
[11]
Cardinality, Counting, and Equinumerosity
### Summary of Equinumerosity in Philosophy vs Mathematics, Notation Differences, and Synonyms
[12]
Infinity - Stanford Encyclopedia of Philosophy
Apr 29, 2021 · Greek mathematics generally avoids any recourse to the actual infinite, and scholars have spoken of a “horror of infinity” typical of Greek ...
[13]
[PDF] Treatise of Human Nature, Book 1 - Early Modern Texts
It has been objected to me that infinite divisibility requires only an infinite number of proportional parts, .... and that an infinite number of pro-.
[14]
[PDF] Philosophical Method and Galileo's Paradox of Infinity - PhilArchive
II. GALILEO Galileo points out in his last dialogue ([1638] 1954, 32) that the square numbers (1, 4, 9, 16,...) are clearly fewer than the “numbers” (the ...
[15]
[PDF] Bolzano's Mathematical Infinite - PhilArchive
Nov 16, 2020 · Since multitudes, pluralities and sums are the infinite collections. Bolzano concerns himself with, the identification of his infinite ...
[16]
[PDF] siz.1 Equinumerosity - Open Logic Project Builds
Equinumerosity is an equivalence relation. Proof. We must show that equinumerosity is reflexive, symmetric, and transi- tive. Let A, B, and C be sets ...
[17]
[PDF] Set Theory
An infinite product of cardinals is defined using infinite products of sets. ... Definition 8.18 and Theorem 8.20 are due to Jech [1984]. The ...
[18]
[PDF] 3. Cardinal Numbers - MIMUW
Cardinal numbers can be defined either using the Axiom of Regularity (via equivalence classes of (3.1)), or using the Axiom of Choice. In this chapter we define ...
[19]
[PDF] Set theory following Jech Contents
Set theory following Jech. J. D. Monk. September 13, 2024. These are largely self-contained notes developing set theory, following Jech. For Jech.
[20]
Cantor-Schroeder-Bernstein theorem in nLab
### Summary of Cantor-Schroeder-Bernstein Theorem (nLab)
[21]
The Cantor–Bernstein theorem: how many proofs? - Journals
Jan 21, 2019 · Dedekind [1] was the first to prove the theorem without appealing to Cantor's well-ordering principle in a manuscript from 1887. The proof was ...
[22]
[PDF] 5 Cardinality Definition 5.1. Two sets S and T are called ...
Definition 5.1. Two sets S and T are called equinumerous, denoted by S ∼ T, if there exists a bijection from S onto T. Equinumerosity defines an equivalence ...<|control11|><|separator|>
[23]
CARDINALITY, COUNTING, AND EQUINUMEROSITY - Project Euclid
Husserl famously argued that one should not explain the concept of number in terms of that of equinumerosity (or one-one correspondence), but should explain ...Missing: origin | Show results with:origin
[24]
[PDF] Homework 11
Prove that if A is a set such that |A| = n ∈ N, then the number of bijections from A to itself is n!. Recall that 0! = 1, and for n ∈ N, n! = n(n − 1)(n − 2)··· ...
[25]
Dedekind-infinite - PlanetMath
Mar 22, 2013 · A set A A is said to be Dedekind-infinite if there is an injective function f:ω→A f : ω → A , where ω ω denotes the set of natural numbers.
[26]
[PDF] Note 20
f is clearly one-to-one, since distinct natural numbers get mapped to distinct even natural numbers (because f(n) = 2n). f is also onto, since every n in ...
[27]
Countability of Rational Numbers
The set of rational numbers is countable. The most common proof is based on Cantor's enumeration of a countable collection of countable sets.
[28]
[PDF] The True (?) Story of Hilbert's Infinite Hotel - arXiv
The paper outlines the origin and early history of Hilbert's hotel paradox. At the same time it retracts the author's earlier conclusion that the paradox was ...
[29]
[PDF] Cantor's Other Proofs that R Is Uncountable
One of the best known proofs is Georg Cantor's diagonalization argument showing the uncountability of the real numbers R. Few people know, however, that this ...