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Infinite set

An infinite set is a set in mathematics that is equinumerous with at least one of its proper subsets, distinguishing it from finite sets which have a fixed cardinality expressible as a non-negative integer.^[1] This definition, formalized by Richard Dedekind in 1888 and building on Georg Cantor's foundational work in set theory during the 1870s, resolves classical paradoxes about infinity by treating it rigorously rather than as a vague limit or metaphor.^[2] Cantor's development of transfinite numbers revolutionized mathematics by showing that not all infinities are equal; for instance, the set of natural numbers \mathbb{N} is countably infinite, meaning its elements can be listed in a sequence via a bijection with \mathbb{N} itself, while the set of real numbers \mathbb{R} is uncountably infinite and has a strictly larger cardinality, as proven by Cantor's diagonal argument in 1891.^[3] Examples of countably infinite sets include the integers \mathbb{Z}, rational numbers \mathbb{Q}, and even the set of all ordered pairs of natural numbers \mathbb{N} \times \mathbb{N}, each equinumerous with \mathbb{N} through explicit bijections.^[2] Uncountable sets, like \mathbb{R} or the power set \mathcal{P}(S) of any infinite set S, exceed countable infinity, with Cantor's theorem establishing that |\mathcal{P}(S)| > |S| for any set S, implying an unending hierarchy of infinities.^[3] The theory of infinite sets underpins modern mathematics, from analysis and topology to computer science and logic, enabling precise treatments of continuity, limits, and computability, though it initially faced resistance due to its counterintuitive implications, such as Galileo's paradox that the natural numbers and their squares have the same "size."^[2] Key open questions, like the Continuum Hypothesis—which posits no set cardinality exists between |\mathbb{N}| and |\mathbb{R}|—remain independent of standard axioms like ZFC, highlighting the depth and ongoing relevance of infinite set theory.^[3]

Definition and Distinction from Finite Sets

Definition of Infinite Sets

In set theory, a set is defined as finite if it is equinumerous to some natural number n, meaning there exists a bijection between the set and the set \{0, 1, \dots, n-1\} (or \{1, 2, \dots, n\}, depending on the convention for indexing natural numbers).^[4] The empty set is finite with cardinality 0, as it is equinumerous to itself via the empty function, which is a bijection.^[5] Singleton sets, containing exactly one element, are finite with cardinality 1, equinumerous to \{0\} via a bijection mapping the element to 0.^[5] Consequently, a set is infinite if it is not finite, i.e., there is no natural number n such that the set is equinumerous to \{0, 1, \dots, n-1\}.^[6] Equinumerosity, or having the same cardinality, is the relation between two sets A and B if there exists a bijection f: A \to B, a function that is both injective (one-to-one) and surjective (onto).^[5] This bijection establishes a one-to-one correspondence preserving the "size" of the sets, serving as the foundational measure of cardinality in set theory.^[4] An equivalent characterization of infinite sets, due to Richard Dedekind, defines a set S as infinite if it admits a proper subset equinumerous to itself, i.e., there exists a bijection between S and a proper subset T \subsetneq S.^[7] This is closely related to the notion of Dedekind-infinite sets, where S is infinite if there exists an injection from S to a proper subset of S, allowing S to be mapped one-to-one into itself without covering the entire set.^[7] A key result in set theory states that every infinite set contains a countably infinite subset, assuming the axiom of countable choice (a weakened form of the axiom of choice).^[8] Without the axiom of choice, there exist models of ZF set theory where infinite sets lack countably infinite subsets, though such sets are Dedekind-finite.^[8] Cardinality provides a generalized notion of size for infinite sets, extending beyond finite counting via these equinumerosity classes.^[4]

Properties Distinguishing Infinite from Finite

One of the most intuitive illustrations of the counterintuitive nature of infinite sets is Hilbert's paradox of the Grand Hotel, which demonstrates how an infinite set can accommodate additional elements without increasing in "size." Imagine a hotel with infinitely many rooms, all occupied by guests. To accommodate a new guest, the management shifts every current guest from room n to room n+1, freeing up room 1 for the newcomer. This process can even extend to infinitely many new guests by reassigning guests to even-numbered rooms and placing the new arrivals in odd-numbered rooms, showing that the set of natural numbers can be placed in one-to-one correspondence with a proper subset of itself.^[9] This property formalizes as the Dedekind definition of infinity: a set is Dedekind-infinite if it admits a bijection with one of its proper subsets, a characteristic impossible for finite sets, where any proper subset has strictly fewer elements. For finite sets, removing elements always reduces the size, but infinite sets defy this intuition by maintaining equinumerosity with subsets obtained via systematic shifting, such as mapping each natural number n to n+1.^[7] Another distinguishing feature is the failure of additivity in measuring size for disjoint unions. For finite disjoint sets A and B, the size of their union satisfies |A \cup B| = |A| + |B|, but this does not hold for infinite sets; for instance, the set of even natural numbers and the set of odd natural numbers are both equinumerous to the full set of natural numbers, yet their disjoint union is precisely the natural numbers.^[10] Infinite sets thus permit injections into themselves that are not surjective, injecting the entire set into a proper subset without exhausting it, whereas any injection from a finite set into itself must be bijective.^[11]

Cardinality of Infinite Sets

Aleph-Null and Countable Infinity

A set is countable if it is finite or countably infinite. A finite set has a bijection with \{1, \dots, n\} for some natural number n, while a countably infinite set has a bijection with the set of natural numbers \mathbb{N}.^[12]^[13] The cardinality of the natural numbers, denoted \aleph_0 (aleph-null), represents the smallest infinite cardinality, such that |\mathbb{N}| = \aleph_0.^[14] This notation, introduced by Georg Cantor, distinguishes countable infinity from larger transfinite cardinals. Sets equinumerous to \mathbb{N}—meaning they share the same cardinality via a bijection—also have cardinality \aleph_0. The Schröder–Bernstein theorem provides a key tool for establishing such equinumerosity: if there are injections from set A to B and from B to A, then a bijection exists between A and B.^[15] For instance, the integers \mathbb{Z} and rationals \mathbb{Q} both have cardinality \aleph_0. An injection from \mathbb{N} to \mathbb{Z} maps even numbers to non-negative integers and odd numbers to negatives, while an injection from \mathbb{Z} to \mathbb{N} uses a zigzag enumeration; the theorem then implies |\mathbb{Z}| = \aleph_0. Similarly, injections between \mathbb{N} and \mathbb{Q} (e.g., via reduced fractions) yield |\mathbb{Q}| = \aleph_0.^[16]^[17] A fundamental property of countable sets is their closure under countable unions: the union of countably many countable sets is countable, assuming the axiom of countable choice.^[18] This follows from enumerating each set via bijections to \mathbb{N} and then pairing elements using a bijection from \mathbb{N} \times \mathbb{N} to \mathbb{N}, such as the Cantor pairing function. In particular, for two disjoint countably infinite sets A and B, their union A \cup B satisfies |A \cup B| = \aleph_0, as one can interleave their enumerations to form a single countable sequence. This arithmetic property highlights how countable infinities behave additively like \aleph_0 + \aleph_0 = \aleph_0.^[13]

Continuum and Uncountable Infinity

Uncountable sets are those whose cardinality exceeds that of the natural numbers \mathbb{N}, meaning there exists no bijection between the set and \mathbb{N}, and thus their size is strictly greater than \aleph_0. This distinction arises from Georg Cantor's foundational work in set theory, where he demonstrated that certain infinite sets surpass countable infinity in magnitude. Cantor's diagonal argument provides a rigorous proof that the set of real numbers \mathbb{R} is uncountable.^[19] Assume, for contradiction, that \mathbb{R} is countable, so its elements can be listed as an infinite sequence r_1, r_2, r_3, \dots, where each r_n is represented by an infinite decimal expansion (or equivalently, an infinite binary sequence for subsets of \mathbb{N}). Construct a new real number r by differing from r_n in the n-th decimal place for each n; this r cannot appear in the original list, yielding a contradiction.^[19] The argument extends to show that the power set of \mathbb{N}, \mathcal{P}(\mathbb{N}), is also uncountable, as its elements correspond to infinite binary sequences.^[19] The cardinality of the continuum, denoted c = |\mathbb{R}|, equals $2^{\aleph_0}, the cardinality of \mathcal{P}(\mathbb{N}).^[20] Cantor established that \aleph_0 < c \leq 2^{\aleph_0}, with the strict inequality following from the uncountability of \mathbb{R}, the lower bound $2^{\aleph_0} \leq c from the injection of \mathcal{P}(\mathbb{N}) into \mathbb{R} via binary expansions, and the upper bound c \leq 2^{\aleph_0} from the injection of \mathbb{R} into $10^{\mathbb{N}} via decimal expansions.^[20] In fact, c = 2^{\aleph_0} holds exactly, as |\mathbb{R}| equals |\mathbb{Q}^{\mathbb{N}}| = \aleph_0^{\aleph_0} = 2^{\aleph_0}, the cardinality of the power set.^[20] The continuum hypothesis (CH) posits that c = \aleph_1, where \aleph_1 is the smallest cardinal greater than \aleph_0. Introduced by Cantor in 1878, CH asserts no cardinal lies between \aleph_0 and c.^[21] However, Kurt Gödel proved in 1940 that CH is consistent with the Zermelo-Fraenkel axioms plus the axiom of choice (ZFC), assuming ZFC itself is consistent.^[22] Paul Cohen later showed in 1963 that the negation of CH is also consistent with ZFC, establishing CH's independence from standard set theory axioms.^[23] The aleph hierarchy extends beyond countable infinity, with \aleph_1 as the least uncountable cardinal, \aleph_2 the next successor, and so on, forming an ordinal-indexed sequence of infinite cardinals.^[24] Each \aleph_{\alpha+1} is the smallest cardinal larger than \aleph_\alpha, while limit cardinals arise at limit ordinals.^[24] This structure, developed by Cantor, underpins the transfinite cardinal arithmetic and reveals the vastness of infinite sizes.^[24]

Mathematical Properties

Infinite Arithmetic

In infinite set theory, arithmetic operations on cardinal numbers differ significantly from those on finite numbers, as they are defined via the cardinalities of unions, products, and function sets rather than counting elements sequentially. For infinite cardinals, these operations often simplify due to the ability to absorb smaller infinities, leading to counterintuitive results like the sum or product of two infinite sets having the same size as the larger one.^[25] Cardinal addition, denoted κ + λ, is the cardinality of the disjoint union of sets of sizes κ and λ. When at least one of κ or λ is infinite, κ + λ equals the maximum of κ and λ; for example, the countable infinity ℵ₀ added to any larger infinite cardinal κ (with κ ≥ ℵ₀) yields κ, as ℵ₀ + κ = κ. This absorption law holds more generally: if κ is infinite and μ ≤ κ, then κ + μ = κ.^[25]^[26] Cardinal multiplication, denoted κ · λ, is the cardinality of the Cartesian product of sets of sizes κ and λ. For infinite cardinals κ and λ (with neither zero), κ · λ also equals max(κ, λ); similarly, if κ is infinite and μ ≤ κ, then κ · μ = κ. A key consequence is that for any infinite cardinal κ, κ · κ = κ, illustrating how the plane ℕ × ℕ has the same cardinality as the natural numbers ℕ.^[25]^[26] Cardinal exponentiation, denoted κ^λ, is the cardinality of the set of all functions from a set of size λ to a set of size κ. For infinite λ, this operation is more complex and depends on additional assumptions, but under the generalized continuum hypothesis or when 2 ≤ κ ≤ 2^λ, it simplifies to κ^λ = max(κ, 2^λ). A foundational case is 2^ℵ₀ = 𝔠, where 𝔠 is the cardinality of the continuum (the real numbers), establishing the uncountable infinity of the reals as the power set of the naturals.^[25]^[26]

Subsets and Bijection Properties

One defining characteristic of infinite sets is that they admit a bijection with one of their proper subsets, meaning the set can be placed in one-to-one correspondence with a subset that excludes at least one element. This property, introduced by Richard Dedekind, distinguishes infinite sets from finite ones, where no such bijection exists, as removing any element reduces the cardinality. For example, the natural numbers \mathbb{N} are infinite because they biject with the proper subset of even numbers via the mapping n \mapsto 2n. The power set \mathcal{P}(S) of a set S, consisting of all subsets of S, has cardinality $2^{|S|}, where the exponent denotes the cardinality of the power set. This relation holds for any set S, finite or infinite. Cantor's theorem establishes that for any set S, the inequality |S| < |\mathcal{P}(S)| is strict, implying no injection from \mathcal{P}(S) into S exists, and thus the power set is always larger. For infinite S, this theorem demonstrates the existence of strictly larger infinities, as iterating the power set operation generates an unending hierarchy of cardinalities. The proof relies on a diagonal argument: assume a surjection f: S \to \mathcal{P}(S); then the set D = \{ x \in S \mid x \notin f(x) \} is in \mathcal{P}(S) but not in the image of f, yielding a contradiction.^[27] The axiom of choice (AC), formulated by Ernst Zermelo, states that for any collection of nonempty disjoint sets, there exists a set containing exactly one element from each. AC enables the well-ordering theorem, asserting that every set can be well-ordered, which is essential for comparing infinite cardinalities via ordinal assignments. Without AC, some infinite cardinals may be incomparable, meaning neither injects into the other nor vice versa. Zorn's lemma, proved equivalent to AC by Max Zorn, states that if a partially ordered set has every chain bounded above, then it contains a maximal element; this applies to infinite chains in posets, facilitating proofs in algebra and topology that rely on AC. For instance, Zorn's lemma underpins the existence of bases for vector spaces over infinite-dimensional spaces.^[28] Hartogs' theorem provides an AC-independent result: for every set S, there exists a cardinal \kappa (the Hartogs number of S) such that no injection from \kappa into S exists, ensuring that the cardinals form a proper class with no largest member. This theorem, proved without AC, guarantees that beyond any given infinite set, there is always a larger cardinal not embeddable into it, reinforcing the unbounded hierarchy of infinities.

Historical Context

Pre-Modern Conceptions

The concept of infinity posed significant philosophical challenges in ancient Greek thought, particularly through Zeno of Elea's paradoxes in the 5th century BCE, which highlighted issues of infinite divisibility and motion. Zeno's paradox of Achilles and the tortoise, for instance, argues that the swift Achilles can never overtake the slower tortoise because he must first cover an infinite number of diminishing distances, thereby questioning the coherence of infinite processes in physical reality.^[29] These paradoxes influenced subsequent Greek philosophers by underscoring the tension between finite experience and infinite division, without resolving whether such infinities could exist.^[30] Aristotle, in the 4th century BCE, provided a foundational distinction between potential and actual infinity to address these issues, as detailed in his Physics. He accepted potential infinity as a process that can continue indefinitely, such as the endless divisibility of a line, but rejected actual infinity as a completed totality, arguing that it leads to absurdities like indivisible units composing a continuum.^[31] This framework allowed Aristotle to reconcile Zeno's challenges with empirical observation, positing that infinity exists only as a limit approached but never attained.^[32] In medieval Islamic philosophy, thinkers like al-Kindi (c. 801–873 CE) engaged with Aristotelian ideas, using mathematical arguments to demonstrate the impossibility of actual infinity, particularly in cosmological contexts such as the eternity of the world. Al-Kindi contended that an actual infinite series, whether of causes or events, would violate principles of unity and order, as it implies a completed multitude without a beginning or end.^[33] This perspective extended to early explorations of infinite series in mathematics, where Islamic scholars grappled with convergence but generally avoided endorsing actual infinities.^[34] Western medieval thinkers, influenced by Aristotle via Islamic transmissions, similarly rejected actual infinity; Thomas Aquinas (1225–1274) in his Summa Theologica argued that infinite causal series are impossible because they lack a necessary first cause, rendering the universe unintelligible without a finite origin.^[35] Aquinas distinguished between accidental infinities, like an endless lineage of humans, which he deemed possible, and essential infinities in hierarchical causes, which he denied as they undermine explanatory power.^[36] The 17th century marked a shift toward mathematical applications of infinity in the development of calculus, beginning with Galileo's paradox in his Two New Sciences (1638), which observed that the infinite set of natural numbers can be put into one-to-one correspondence with the proper subset of perfect squares, yet their sum remains finite, challenging intuitive notions of size.^[37] This paradox illustrated how infinite collections defy finite comparisons, prompting reflections on non-standard magnitudes.^[38] Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus in the late 17th century, employing infinitesimals—quantities smaller than any finite but greater than zero—to handle limits and rates of change, effectively navigating infinite processes without fully endorsing actual infinities. Newton's fluxions treated infinitesimals as evanescent moments in motion, while Leibniz's notation formalized differentials as ratios approaching zero.^[39] These methods resolved practical problems in physics and geometry but sparked debates over the ontological status of infinitesimals.^[40] In the 1670s, Jakob Bernoulli advanced the study of infinite series and products, demonstrating in works like his Ars Conjectandi (published posthumously in 1713) that certain infinite products diverge, hinting at the existence of unbounded infinities beyond convergent sums. Bernoulli's analysis of the harmonic series, for example, proved its divergence by grouping terms to exceed any finite bound, providing early rigorous handling of infinite aggregates.^[41] This contributed to distinguishing convergent from divergent infinities, laying groundwork for later probability and analysis.^[42]

Development in Set Theory

The formalization of infinite sets began in the late 19th century with Georg Cantor's pioneering work on set theory. In the 1870s, Cantor introduced the notion of transfinite numbers to quantify different sizes of infinity, distinguishing countable from uncountable infinities. His seminal 1874 paper demonstrated the uncountability of the real numbers through a proof relying on the Bolzano-Weierstrass theorem, showing that no bijection exists between the natural numbers and the reals.^[43] By the 1890s, Cantor further developed these ideas, including the diagonal argument in later publications, which rigorously established the existence of infinities larger than the countable, laying the groundwork for transfinite arithmetic and the hierarchy of cardinalities.^[44] Independently, Richard Dedekind offered a precise definition of infinite sets in 1888, characterizing a set as infinite if it admits an injection onto one of its proper subsets, thereby providing a set-theoretic criterion free from reliance on the natural numbers.^[45] This definition highlighted the structural property of infinite sets allowing self-similarity through bijections with subsets, influencing subsequent axiomatic approaches. However, Cantor's naive set theory encountered foundational issues, exemplified by Bertrand Russell's 1901 paradox of the set of all sets that do not contain themselves, which revealed contradictions in unrestricted set comprehension and prompted the need for axiomatic restrictions on infinite collections.^[46] In response to these paradoxes, Ernst Zermelo proposed the first axiomatic system for set theory in 1908, incorporating the axiom of infinity to guarantee the existence of an infinite set, alongside axioms for extensionality, pairing, union, power set, separation, and choice, which collectively enabled the construction of transfinite sets while avoiding inconsistencies.^[47] Abraham Fraenkel and Thoralf Skolem refined Zermelo's system in the early 1920s by replacing the axiom of separation with a schema to limit comprehension to bounded formulas and introducing the axiom schema of replacement; these changes culminated in the Zermelo-Fraenkel axioms (ZF). The axiom of foundation, to prevent infinite descending membership chains, was later introduced by Zermelo in 1930. ZF was extended to ZFC with the explicit inclusion of the axiom of choice as a standard component.^[48] A key milestone in understanding infinite sets came in the mid-20th century with proofs of the independence of the continuum hypothesis (CH), which posits no cardinality between that of the naturals and the continuum. Kurt Gödel demonstrated in 1940 that CH is consistent with ZF by constructing the inner model of constructible sets L, where CH holds.^[49] Complementing this, Paul Cohen proved in 1963 that the negation of CH is also consistent with ZFC using his innovative forcing method, which extends models of set theory to incorporate generic sets, thereby showing CH's undecidability within standard axioms and underscoring the flexibility in handling infinite cardinalities.^[23]

Examples

Countable Examples

The natural numbers \mathbb{N} = \{0, 1, 2, 3, \dots \} form the archetypal example of a countably infinite set, with cardinality \aleph_0, as they are in one-to-one correspondence with themselves via the identity function f(n) = n.^[50] The set of integers \mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots \} is also countably infinite, equinumerous to \mathbb{N}, via the explicit bijection that enumerates them in a zig-zag pattern: f(0) = 0, f(1) = 1, f(2) = -1, f(3) = 2, f(4) = -2, and so on, defined more formally by

f(n) = \begin{cases} \frac{n}{2} & \text{if } n \text{ even}, \\ -\frac{n+1}{2} & \text{if } n \text{ odd}. \end{cases}

This mapping pairs each natural number with a unique integer, covering all of \mathbb{Z} without repetition.^[50] The rational numbers \mathbb{Q} constitute another countably infinite set, with |\mathbb{Q}| = \aleph_0, as demonstrated by Cantor's enumeration method, which lists positive rationals by arranging fractions p/q (in lowest terms, p, q > 0) in a grid ordered by increasing sum p + q, then traversing diagonals while skipping non-reduced fractions to avoid duplicates. For instance, the sequence begins $1/1, 1/2, 2/1, 1/3, 2/2 (skipped), $3/1, \dots, establishing a bijection with \mathbb{N} after extending to negatives and zero.^[50]^[43] The algebraic numbers, comprising all roots of non-zero polynomials with rational coefficients, form a countably infinite set, as shown by Cantor through their representation as a countable union of finite sets grouped by "height" N, where for each fixed N, only finitely many such polynomials (and thus roots) exist. Specifically, an algebraic number satisfies an irreducible equation a_0 x^n + \cdots + a_n = 0 with integer coefficients and height N = n - 1 + |a_0| + \cdots + |a_n|; enumerating by increasing N and then by value within each finite group yields a bijection with \mathbb{N}.^[43] A fundamental result underscoring these examples is that any countable union of finite sets is countable, since each finite set can be injected into \mathbb{N}, and the countable union of such injections composes to a surjection from \mathbb{N} \times \mathbb{N} (itself countable) onto the union.

Uncountable Examples

The real numbers \mathbb{R} form a fundamental example of an uncountable infinite set, with cardinality denoted by \mathfrak{c}, the cardinality of the continuum. Georg Cantor established the uncountability of \mathbb{R} using his diagonal argument, which shows that no enumeration of the reals can be complete by constructing a real number differing from each listed one in at least one decimal place.^[19] Specifically, the closed interval [0,1] has the same cardinality \mathfrak{c} as \mathbb{R}, as there exists a bijection between them via the tangent function or similar mappings.^[19] Another key example is the power set \mathcal{P}(\mathbb{N}) of the natural numbers \mathbb{N}, which consists of all subsets of \mathbb{N} and has cardinality |\mathcal{P}(\mathbb{N})| = 2^{\aleph_0} = \mathfrak{c}. Cantor's theorem demonstrates that no set is in bijection with its power set, implying |\mathcal{P}(\mathbb{N})| > \aleph_0, and the equality to \mathfrak{c} follows from identifying subsets with their characteristic functions, which correspond to infinite binary sequences representable as real numbers in [0,1].^[19] The Euclidean plane \mathbb{R}^2 also has cardinality \mathfrak{c}, matching that of \mathbb{R}, despite its higher dimension. Cantor constructed an explicit bijection by interleaving the decimal expansions of pairs (x, y) \in \mathbb{R}^2 to form a single real number, handling non-unique representations to ensure injectivity and surjectivity.^[51] This result generalizes to \mathbb{R}^n for any finite n, showing that dimensionality does not increase cardinality beyond the continuum in finite dimensions. The set of all functions from \mathbb{N} to \mathbb{N}, denoted \mathbb{N}^\mathbb{N}, provides yet another uncountable example with cardinality |\mathbb{N}^\mathbb{N}| = \aleph_0^{\aleph_0} = 2^{\aleph_0} = \mathfrak{c}. This follows from the fact that \mathbb{N}^\mathbb{N} can be injected into \mathcal{P}(\mathbb{N} \times \mathbb{N}) via graphs of functions, and by Cantor's theorem, the latter has cardinality at most $2^{\aleph_0}, while an injection from \mathcal{P}(\mathbb{N}) into \mathbb{N}^\mathbb{N} establishes the lower bound.^[52]

References

[1]
Infinite Set -- from Wolfram MathWorld
A set of elements S is said to be infinite if the elements of a proper subset S^' can be put into one-to-one correspondence with the elements of S.
[2]
[PDF] Georg Cantor (1845-1918): - Department of Mathematics
A set is infinite if and only if it is equi- pollent with some proper subset of itself. Next, some of Cantor's proofs. 15. Page 16. Theorem. |N| = |N. 2.
[3]
[PDF] 6.042J Chapter 13: Infinite sets - MIT OpenCourseWare
Georg Cantor was the mathematician who first developed the theory of infinite sizes (because he thought he needed it in his study of Fourier series). Cantor ...
[4]
[PDF] The Size of Sets - Open Logic Project Builds
They are slightly more advanced and use a difference definition of enumerability more suitable in a set theory context (i.e., bijection with N or an initial ...<|control11|><|separator|>
[5]
[PDF] 3 Functions
Jan 23, 2006 · Definition 4.1. Two sets A and B are equinumerous, written A ∼ B, iff there exists a bijection f : A → B. Example 4.2 ...
[6]
infinite set in nLab
May 28, 2022 · A set is infinite if it is not finite. The existence of an infinite set is usually given by an axiom of infinity. The main example is the set of natural ...Missing: authoritative | Show results with:authoritative
[7]
Dedekind's Contributions to the Foundations of Mathematics
Apr 22, 2008 · The definition is as follows: A set of objects is infinite—“Dedekind-infinite”, as we now say—if it can be mapped one-to-one onto a proper ...<|control11|><|separator|>
[8]
[PDF] Set Theory (MATH 6730) The Axiom of Choice. Cardinals and ...
Countable Axiom of Choice (ACω): Every countable set of nonempty sets has a choice ... (ii) ACω implies that every infinite set has a countably infinite subset ( ...<|control11|><|separator|>
[9]
[1403.0059] The True (?) Story of Hilbert's Infinite Hotel - arXiv
Mar 1, 2014 · The paper outlines the origin and early history of Hilbert's hotel paradox. At the same time it retracts the author's earlier conclusion ...
[10]
[PDF] infinite sets We recall the definition of the cardinality - UCSD Math
The even integers and odd integers have the same cardinality. One way to show this is simply to show that the even integers have the same cardinality as the ...
[11]
[PDF] Infinite Sets - Open Logic Project Builds
We have just shown that, given any Dedekind infinite set, we can define a set which will behave just like we want N to behave. Obviously, then, we cannot ...
[12]
4.7 Cardinality and Countability
We say a set A is countably infinite if N≈A, that is, A has the same cardinality as the natural numbers. We say A is countable if it is finite or countably ...
[13]
[PDF] 4. Countability
A set A is said to be countably infinite if |A| = |N|, and simply countable if. |A|≤|N|. In words, a set is countable if it has the same cardinality as some ...
[14]
HowToCount
Sets with cardinality ℵ0 or less are called countable; sets with cardinality exactly ℵ0 are countably infinite. That there are larger cardinalities is a ...<|control11|><|separator|>
[15]
[PDF] CHAPTER FIVE: INFINITIES
The Schröder-Bernstein Theorem: For any sets A and B, if A B and B A, then A ≈ B. Proof: By definition of , there are injections f : A → B and g: B → A. Let C ...
[16]
[PDF] MATH1050 Schröder-Bernstein Theorem
According to the Schröder-Bernstein Theorem, N∼Q. We also have N∼Z and Z∼Q. Remark. Hence there are as many natural numbers as there are integers or rational ...Missing: equinumerosity | Show results with:equinumerosity
[17]
[PDF] Philosophy 422 R. Rynasiewicz Axiomatic Set Theory Spring 2023 ...
Prove that the set of all integers and the set of all rational numbers are both countable. Proof . For the case of the integers, define f : Z → N so that f ...
[18]
The Axiom of Choice
The union of countably many countable sets is countable. Every infinite set has a denumerable subset. The Loewenheim-Skolem Theorem: Any first-order theory ...
[19]
[PDF] Ein Beitrag zur Mannigfaltigkeitslehre.
Ein Beitrag zur Mannigfaltigkeitslehre. Cantor, G. pp. 242 - 258. Terms and ... The PDF-version contains the table of contents as bookmarks, which allows easy ...
[20]
(PDF) A Translation of G. Cantor's “Ueber eine elementare Frage ...
Aug 23, 2019 · PDF | A Translation of G. Cantor's “Ueber eine elementare ... Über eine elementare Frage der Mannigfaltigkeitslehre. Article. G. Cantor · View ...
[21]
The Consistency of the Axiom of Choice and of the Generalized ...
Gödel,. The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis, Proc. Natl. Acad. Sci. U.S.A. 24 (12) 556-557, https://doi.org ...
[22]
THE INDEPENDENCE OF THE CONTINUUM HYPOTHESIS - PNAS
This is the first of two notes in which we outline a proof of the fact that the Con- tinuum Hypothesis cannot bederived from the other axioms of set theory, ...
[23]
cardinal arithmetic - PlanetMath
Mar 22, 2013 · In the following, κ κ , λ λ , μ μ and ν ν are arbitrary cardinals, unless otherwise specified. Cardinal arithmetic obeys many of the same ...
[24]
[PDF] Handbook of Set Theory
Jan 2, 2006 · Cardinal Arithmetic. U. Abraham and M. Magidor. 1. Introduction. Cardinal arithmetic is the study of rules and properties of arithmetic op-.
[25]
Ueber eine elementare Frage der Mannigfaltigketislehre. - EuDML
Ueber eine elementare Frage der Mannigfaltigketislehre. Georg Cantor · Jahresbericht der Deutschen Mathematiker-Vereinigung (1890/91). Volume: 1, page 72-78 ...Missing: pdf | Show results with:pdf
[26]
Beweis, daß jede Menge wohlgeordnet werden kann
Beweis, daß jede Menge wohlgeordnet werden kann. Aus einem an Herrn Hilbert gerichteten Briefe. Download PDF.
[27]
[PDF] Aristotelian Potential Infinity - PhilArchive
Jun 15, 2023 · Zeno's paradoxes as the real reason Aristotle rejects actually infinite totalities. ... the infinite is potential, not actual? • Aristotle ...
[28]
[PDF] ZENO'S PARADOXES.ACARDINAL PROBLEM - PhilSci-Archive
This is confirmed by Aristotle's own interpretation: For in two ways it can be said that a distance or a period or any other continuum is infinite [apeiron], ...
[29]
7 Aristotle, Zeno, and the Potential Infinite - Oxford Academic
This essay argues that Aristotle misdescribes his own position when he sums it up as the claim that infinity can only be potential and never actual. He readily ...
[30]
Aristotle's Notion of Continuity: The Structure Underlying Motion ...
Sep 28, 2020 · The potential infinity of division is crucial for the refutation of Zeno's first argument, since the dichotomy paradox would indeed be ...
[31]
[PDF] AL KINDI'S AND W. L. CRAIG'S COSMOLOGICAL ARGUMENTS1
Except in his most famous work, On the First Philosophy, al-Kindi also gives mathematical proofs for the impossibility of actual infinity in some of his shorter ...
[32]
Mathematics and Her Sisters in Medieval Islam - ScienceDirect.com
A discussion of some medieval Islamic thought on infinity and questions ... The history of medieval Islamic mathematics and her sister sciences continues.<|control11|><|separator|>
[33]
[PDF] Why Thomas Aquinas Rejects Infinite, Essentially Ordered, Causal ...
Sep 5, 2013 · Abstract. Several of Thomas Aquinas's proofs for the existence of God rely on the claim that causal series cannot proceed in infinitum.Missing: Al- Kindi
[34]
[PDF] Aquinas on Infinite Multitudes - Cornell eCommons
To my knowledge, it is only in ST I, 7 , 4 that Aquinas considers quite on its own the question whether actually infinite multitudes are possible.Missing: Kindi scholarly sources<|control11|><|separator|>
[35]
[PDF] Philosophical Method and Galileo's Paradox of Infinity - PhilArchive
For any finite set, Cantor observed, Anzahl and power coincide and determine each other, but not so for the infinite; two infinite sets can have the same pow-.Missing: scholarly | Show results with:scholarly
[36]
(PDF) Galileo's Paradox and Numerosities - ResearchGate
Jul 20, 2020 · Galileo's paradox of infinity involves comparing the set of natural numbers, N, and the set of squares, {n 2 : n ∈ N}.Missing: source | Show results with:source
[37]
Infinitesimals in the foundations of Newton's mechanics
This paper discusses two concepts of “moment” (infinitesimal) used successively by Newton in his calculus and relates these two concepts to the two concepts ...
[38]
[1205.0174] Leibniz's infinitesimals: Their fictionality, their modern ...
We argue that Leibniz's defense of the infinitesimal calculus - both philosophical and mathematical - guided his successors toward an infinitesimal analysis ...
[39]
[PDF] Jakob Bernoulli's Method for Finding Exact Sums of Infinite Series ...
Oct 19, 2021 · Over a period of almost twenty years, from the late 1680s to the early 1700s, Jakob Bernoulli wrote five treatises on the theory of infinite ...
[40]
[PDF] The Bernoullis and the Harmonic Series - Mathematics
Jakob Bernoulli was certainly convinced of the importance of his brother's deduction and emphasized its salient point when he wrote: The sum of an infinite ...
[41]
[PDF] On a Property of the Class of all Real Algebraic Numbers.
On a Property of the Class of all Real Algebraic Numbers. by Georg Cantor. Crelle's Journal for Mathematics, Vol. 77, pp. 258–262 (1874).<|separator|>
[42]
Georg Cantor (1845 - 1918) - Biography - MacTutor
He had proved that the real numbers were not countable by December 1873 and published this in a paper in 1874. It is in this paper that the idea of a one-one ...
[43]
Was sind und was sollen die Zahlen? : Richard Dedekind
Feb 9, 2013 · Die 1888 in erster Auflage erschienene Schrift ''Was sind und was sollen die Zahlen ... PDF download · download 1 file · SINGLE PAGE PROCESSED ...
[44]
[PDF] Letter to Frege - BERTRAND RUSSELL - (1902) - Daniel W. Harris
Bertrand Russell discovered what be- came known as the Russell paradox in. June 1901 (see 1944, p. 13). In the letter below, written more than a year later and.
[45]
Untersuchungen über die Grundlagen der Mengenlehre. I
Download PDF · Mathematische Annalen Aims and scope ... Zermelo, E. Untersuchungen über die Grundlagen der Mengenlehre. I. Math. Ann. 65, 261–281 (1908).
[46]
The Consistency Of The Axiom Of Choice and Of The Generalized ...
Oct 21, 2020 · The Consistency Of The Axiom Of Choice and Of The Generalized Continuum Hypothesis ... PDF WITH TEXT download · download 1 file · SINGLE PAGE ...
[47]
[PDF] Chapter 20 Countability
This chapter covers infinite sets and countability. 20.1 The rationals and the reals. You're familiar with three basic sets of numbers: the integers, the ...
[48]
[PDF] SET THEORY
This book provides an introduction to relative consistency proofs in axiomatic set theory, and is intended to be used as a text in beginning.
[49]
[PDF] Cantor's Proof of the Nondenumerability of Perfect Sets
Abstract. This paper provides an explication of mathematician Georg Cantor's 1883 proof of the nondenumerability of perfect sets of real numbers.