Editing Cardinality

{{Short description|Size of a set}}
{{Other uses}}
[[File:Apples and Oranges v2.png|thumb|318x318px|A bijection, comparing a set of apples to a set of oranges, showing they have the same cardinality.]]
The '''cardinality''' of a [[finite set]] is the number of its elements, and is therefore a measure of size of the set. Since the discovery by [[Georg Cantor]], in the late 19th century, of different sizes of [[infinite set]]s, the term ''cardinality'' was coined for generalizing to [[infinite sets]] the concept of the number of elements.

More precisely, two sets have the same cardinality if there exists a [[one-to-one correspondence]] between them. In the case of finite sets, the common operation of ''counting'' consists of establishing a one-to-one correspondence between a given set and the set of the {{tmath|n}} first [[natural number]]s, for some natural number {{tmath|n}}. In this case, {{tmath|n}} is the cardinality of the set.

A set is ''infinite'' if the counting operation never finishes, that is, if its cardinality is not a natural number. The basic example of an infinite set is the set of all natural numbers.

The great discovery of Cantor is that infinite sets of apparently different size may have the same cardinality, but that there are infinitely many possible cardinalities. For example, the [[even number]]s, the [[prime number]]s and the [[polynomial]]s whose coefficients are [[rational number]] have the same cardinality as the natural numbers. The set of the [[real number]]s has a greater cardinality than the natural numbers, and has the same cardinality as the interval {{tmath|[0,1]}} and as every [[Euclidean space]] of any dimension. For every set, its [[power set]] (the set of all its subsets) has a greater cardinality.

Cardinalities are represented with [[cardinal number]]s, which are specific sets of a given cardinality, which have been chosen once for all. Some infinite cardinalities have been given a specific name, such as {{tmath|\aleph_0}} for the cardinality of the natural numbers and {{tmath|\mathfrak c}}, the [[cardinality of the continuum]], for the cardinality of the real numbers.

== Etymology ==
In English, the term ''cardinality'' originates from the [[Post-Classical Latin language|post-classical Latin]] ''cardinalis'', meaning "principal" or "chief", which derives from ''cardo'', a noun meaning "hinge". In Latin, ''cardo'' referred to something central or pivotal, both literally and metaphorically. This concept of centrality passed into [[medieval Latin]] and then into English, where ''cardinal'' came to describe things considered to be, in some sense, fundamental, such as ''[[cardinal virtues]]'', ''[[cardinal sins]]'', ''[[cardinal directions]]'', and (in the grammatical sense) ''[[Cardinal numbers (linguistics)|cardinal numbers]].<ref>[[Oxford English Dictionary]], "cardinal (adj.), Etymology," March 2025, https://doi.org/10.1093/OED/1490074521.</ref>''<ref>Harper Douglas, "Etymology of cardinal," [[Etymonline]], Online Etymology Dictionary, accessed April 20, 2025, https://www.etymonline.com/word/cardinal.</ref> The last of which referred to numbers used for counting (e.g., one, two, three),<ref>''[[Oxford English Dictionary]]'', "cardinal number (''n.''), sense 1," July 2023, https://doi.org/10.1093/OED/3193437451.</ref> as opposed to ''[[Ordinal numbers (linguistics)|ordinal numbers]]'', which express order (e.g., first, second, third),<ref>[[Oxford English Dictionary]], "ordinal (n.2)," June 2024, https://doi.org/10.1093/OED/6032173309.</ref> and [[Nominal number|''nominal numbers'']] used for labeling without meaning (e.g., [[jersey numbers]] and [[serial numbers]]).<ref>{{cite journal |last1=Woodin |first1=Greg |last2=Winter |first2=Bodo |year=2024 |title=Numbers in Context: Cardinals, Ordinals, and Nominals in American English |journal=Cognitive Science |volume=48 |doi=10.1111/cogs.13471 |pmc=11475258 |pmid=38895756 |doi-access=free |article-number=e13471 |number=6}}</ref>

In mathematics, the notion of cardinality was first introduced by [[Georg Cantor]] in the late 19th century, wherein he used the used the term ''Mächtigkeit'', which may be translated as "magnitude" or "power", though Cantor credited the term to a work by [[Jakob Steiner]] on [[projective geometry]].<ref name=":2">{{Cite book |last=Ferreirós |first=José |url=https://link.springer.com/book/10.1007/978-3-7643-8350-3 |title=Labyrinth of Thought |date=2007 |publisher=[[Birkhäuser]] |isbn=978-3-7643-8349-7 |edition=2nd |location=Basel |pages=24 |doi=10.1007/978-3-7643-8350-3}}</ref><ref name=":3">{{Cite journal |last=Cantor |first=Georg |author-link=Georg Cantor |date=1932 |editor-last=Zermelo |editor-first=Ernst |editor-link=Ernst Zermelo |title=Gesammelte Abhandlungen |url=https://link.springer.com/book/10.1007/978-3-662-00274-2 |journal=Springer |location=Berlin |pages=151 |doi=10.1007/978-3-662-00274-2 |isbn=978-3-662-00254-4}}</ref><ref name=":4">{{Cite book |last=Steiner |first=Jacob |url=https://archive.org/details/bub_gb_jCgPAAAAQAAJ/ |title=Vorlesungen über synthetische Geometrie / 1 Die Theorie der Kegelschnitte in elementarer Form |date=1867 |publisher=Leipzig : Teubner |others=Ghent University}}</ref> The terms ''cardinality'' and ''cardinal number'' were eventually adopted from the grammatical sense, and later translations would use these terms.<ref>Harper Douglas, "Etymology of cardinality," [[Etymonline]], Online Etymology Dictionary, accessed April 20, 2025, https://www.etymonline.com/word/cardinality.</ref><ref>[[Oxford English Dictionary]], "cardinality (n.2), Etymology," March 2025, https://doi.org/10.1093/OED/5444748676.</ref> Similarly, the terms for ''countable'' and ''uncountable sets'' come from [[Countable noun|''countable'']] and ''[[uncountable nouns]]''.{{citation needed|date=April 2025|reason=parallels are obvious, but is the use of "countable" +/- "un" for sets derived from words used for nouns, or is it original research?}}

==History==

=== Prehistory ===
A crude sense of cardinality, an awareness that groups of things or events compare with other groups by containing more, fewer, or the same number of instances, is observed in a variety of present-day animal species, suggesting an origin millions of years ago.<ref>Cepelewicz, Jordana   ''[https://www.quantamagazine.org/animals-can-count-and-use-zero-how-far-does-their-number-sense-go-20210809/ Animals Count and Use Zero. How Far Does Their Number Sense Go?]'', [[Quanta Magazine|Quanta]], August 9, 2021</ref> Human expression of cardinality is seen as early as {{val|40000}} years ago, with equating the size of a group with a group of recorded notches, or a representative collection of other things, such as sticks and shells.<ref>{{Cite web|url=https://mathtimeline.weebly.com/early-human-counting-tools.html|title=Early Human Counting Tools|website=Math Timeline|access-date=2018-04-26}}</ref> The abstraction of cardinality as a number is evident by 3000 BCE, in Sumerian [[History of mathematics|mathematics]] and the manipulation of numbers without reference to a specific group of things or events.<ref>Duncan J. Melville (2003). [http://it.stlawu.edu/~dmelvill/mesomath/3Mill/chronology.html Third Millennium Chronology] {{Webarchive|url=https://web.archive.org/web/20180707213616/http://it.stlawu.edu/~dmelvill/mesomath/3Mill/chronology.html |date=2018-07-07 }}, ''Third Millennium Mathematics''. [[St. Lawrence University]].</ref>

=== Ancient history ===
[[File:AristotlesWheelLabeledDiagram.svg|thumb|252x252px|Diagram of Aristotle's wheel as described in ''Mechanica''.]]
From the 6th century BCE, the writings of Greek philosophers show hints of infinite cardinality. While they considered generally infinity as an endless series of actions, such as adding 1 to a number repeatedly, they considered rarely infinite sets ([[actual infinity]]), and, if they did, they considered infinity as a unique cardinality.<ref name="Allen2">{{Cite web |last=Allen |first=Donald |date=2003 |title=The History of Infinity |url=https://www.math.tamu.edu/~dallen/masters/infinity/infinity.pdf |url-status=dead |archive-url=https://web.archive.org/web/20200801202539/https://www.math.tamu.edu/~dallen/masters/infinity/infinity.pdf |archive-date=August 1, 2020 |access-date=Nov 15, 2019 |website=Texas A&M Mathematics}}</ref> The ancient Greek notion of infinity also considered the division of things into parts repeated without limit. 

One of the earliest explicit uses of a one-to-one correspondence is recorded in [[Aristotle]]'s [[Mechanics (Aristotle)|''Mechanics'']] ({{Circa|350 BC}}), known as [[Aristotle's wheel paradox]]. The paradox can be briefly described as follows: A wheel is depicted as two [[concentric circles]]. The larger, outer circle is tangent to a horizontal line (e.g. a road that it rolls on), while the smaller, inner circle is rigidly affixed to the larger.  Assuming the larger circle rolls along the line without slipping (or skidding) for one full revolution, the distances moved by both circles are the same: the [[circumference]] of the larger circle. Further, the lines traced by the bottom-most point of each is the same length.<ref name=":0">{{Cite journal |last=Drabkin |first=Israel E. |date=1950 |title=Aristotle's Wheel: Notes on the History of a Paradox |journal=Osiris |volume=9 |pages=162–198 |doi=10.1086/368528 |jstor=301848 |s2cid=144387607}}</ref> Since the smaller wheel does not skip any points, and no point on the smaller wheel is used more than once, there is a one-to-one correspondence between the two circles.

=== Pre-Cantorian set theory ===
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[[Galileo Galilei]] presented what was later coined [[Galileo's paradox]] in his book ''[[Two New Sciences]]'' (1638), where he attempts to show that infinite quantities cannot be called greater or less than one another. He presents the paradox roughly as follows: a [[square number]] is one which is the product of another number with itself, such as 4 and 9, which are the squares of 2 and 3, respectively. Then the [[square root]] of a square number is that multiplicand. He then notes that there are as many square numbers as there are square roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square. But there are as many square roots as there are numbers, since every number is the square root of some square. He, however, concluded that this meant we could not compare the sizes of infinite sets, missing the opportunity to discover cardinality.<ref>{{Cite book |last=Galilei |first=Galileo |author-link=Galileo Galilei |url=https://dn790007.ca.archive.org/0/items/dialoguesconcern00galiuoft/dialoguesconcern00galiuoft.pdf |title=Dialogues Concerning Two New Sciences |publisher=[[The Macmillan Company]] |year=1914 |location=New York |pages=31–33 |language=en |translator-last=Crew |translator-first=Henry |orig-year=1638 |translator-last2=De Salvio |translator-first2=Alfonso}}</ref>

[[Bernard Bolzano]]'s ''[[Paradoxes of the Infinite]]'' (''Paradoxien des Unendlichen'', 1851) is often considered the first systematic attempt to introduce the concept of sets into [[mathematical analysis]]. In this work, Bolzano defended the notion of [[actual infinity]], examined various properties of infinite collections, including an early formulation of what would later be recognized as one-to-one correspondence between infinite sets, and proposed to base mathematics on a notion similar to sets. He discussed examples such as the pairing between the [[Interval (mathematics)|intervals]] <math>[0,5]</math> and <math>[0,12]</math> by the relation <math>5y =  12x.</math> Bolzano also revisited and extended Galileo's paradox. However, he too resisted saying that these sets were, in that sense, the same size. Thus, while ''Paradoxes of the Infinite'' anticipated several ideas central to later set theory, the work had little influence on contemporary mathematics, in part due to its [[posthumous publication]] and limited circulation.<ref>{{Citation |last=Ferreirós |first=José |title=The Early Development of Set Theory |date=2024 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/settheory-early/ |access-date=2025-01-04 |archive-url=https://web.archive.org/web/20210512135148/https://plato.stanford.edu/entries/settheory-early/ |archive-date=2021-05-12 |url-status=live |edition=Winter 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri |encyclopedia=The Stanford Encyclopedia of Philosophy}}</ref><ref>{{Citation |last=Bolzano |first=Bernard |title=Einleitung zur Größenlehre und erste Begriffe der allgemeinen Größenlehre |volume=II, A, 7 |page=152 |year=1975 |editor-last=Berg |editor-first=Jan |series=Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al. |location=Stuttgart, Bad Cannstatt |publisher=Friedrich Frommann Verlag |isbn=3-7728-0466-7 |author-link=Bernard Bolzano}}</ref><ref>{{Cite book |last=Bolzano |first=Bernard |url=https://archive.org/details/dli.ernet.503861/ |title=Paradoxes Of The Infinite |date=1950 |publisher=Routledge and Kegan Paul |location=London |translator-last=Prihonsky |translator-first=Fr.}}</ref>

Other, more minor contributions incude [[David Hume]] in ''[[A Treatise of Human Nature]]'' (1739), who said ''"When two numbers are so combined, as that the one has always a unit answering to every unit of the other, we pronounce them equal",<ref>{{cite book |last=Hume |first=David |date=1739–1740 |title=A Treatise of Human Nature |chapter=Part III. Of Knowledge and Probability: Sect. I. Of Knowledge |chapter-url=https://gutenberg.org/cache/epub/4705/pg4705-images.html#link2H_4_0021 |via=Project Gutenberg}}</ref>'' now called ''[[Hume's principle]]'', which was used extensively by [[Gottlob Frege]] later during the rise of set theory.<ref>{{cite book |last=Frege |first=Gottlob |date=1884 |title=Die Grundlagen der Arithmetik |chapter=IV. Der Begriff der Anzahl § 63. Die Möglichkeit der eindeutigen Zuordnung als solches. Logisches Bedenken, dass die Gleichheit für diesen Fall besonders erklärt wird |quote=§63. Ein solches Mittel nennt schon Hume: »Wenn zwei Zahlen so combinirt werden, dass die eine immer eine Einheit hat, die jeder Einheit der andern entspricht, so geben wir sie als gleich an.« |chapter-url=https://gutenberg.org/cache/epub/48312/pg48312-images.html#para_63 |via=Project Gutenberg}}</ref> [[Jakob Steiner]], whom [[Georg Cantor]] credits the original term, ''Mächtigkeit'', for cardinality (1867).<ref name=":2" /><ref name=":3" /><ref name=":4" /> [[Peter Gustav Lejeune Dirichlet]] is commonly credited for being the first to explicitly formulate the [[pigeonhole principle]] in 1834,<ref>Jeff Miller, Peter Flor, Gunnar Berg, and Julio González Cabillón. "[http://jeff560.tripod.com/p.html Pigeonhole principle]". In Jeff Miller (ed.) ''[http://jeff560.tripod.com/mathword.html Earliest Known Uses of Some of the Words of Mathematics]''. Electronic document, retrieved November 11, 2006</ref> though it was used at least two centuries earlier by [[Jean Leurechon]] in 1624.<ref name="leurechon">{{cite journal |last1=Rittaud |first1=Benoît |last2=Heeffer |first2=Albrecht |year=2014 |title=The pigeonhole principle, two centuries before Dirichlet |url=https://biblio.ugent.be/publication/4115264 |journal=The Mathematical Intelligencer |volume=36 |issue=2 |pages=27–29 |doi=10.1007/s00283-013-9389-1 |mr=3207654 |s2cid=44193229 |hdl-access=free |hdl=1854/LU-4115264}}</ref>

=== Early set theory ===

==== Georg Cantor ====
[[File:Georg_Cantor3.jpg|alt=refer to caption|thumb|339x339px|[[Georg Cantor]], {{spaces|4|hair}}{{circa}} 1870]]
The concept of cardinality, as a formal measure of the size of a set, emerged nearly fully formed in the work of Georg Cantor during the 1870s and 1880s, in the context of [[mathematical analysis]]. In a series of papers beginning with ''[[Cantor's first set theory article|On a Property of the Collection of All Real Algebraic Numbers]]'' (1874),<ref>{{Citation |last=Cantor |first=Herrn |title=Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen |date=1984 |work=Über unendliche, lineare Punktmannigfaltigkeiten: Arbeiten zur Mengenlehre aus den Jahren 1872–1884 |pages=19–24 |editor-last=Cantor |editor-first=Georg |orig-date=1874 |url=https://link.springer.com/chapter/10.1007/978-3-7091-9516-1_2 |access-date=2025-05-24 |place=Vienna |publisher=Springer |language=de |doi=10.1007/978-3-7091-9516-1_2 |isbn=978-3-7091-9516-1}}</ref> Cantor introduced the idea of comparing the sizes of infinite sets, through the notion of one-to-one correspondence. He showed that the set of [[real numbers]] was, in this sense, strictly larger than the set of natural numbers [[Cantor's first set theory article#Second theorem|using a nested intervals argument]]. This result was later refined into the more widely known [[Cantor's diagonal argument|diagonal argument]] of 1891, published in ''Über eine elementare Frage der Mannigfaltigkeitslehre,''<ref>{{Cite journal |last=Cantor |first=Georg |date=1890 |title=Ueber eine elementare Frage der Mannigfaltigketislehre. |url=https://eudml.org/doc/144383 |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |volume=1 |pages=72–78 |issn=0012-0456}}</ref> where he also proved the more general result (now called [[Cantor's Theorem]]) that the [[power set]] of any set is strictly larger than the set itself. 

Cantor introduced the notion [[cardinal numbers]] in terms of [[ordinal numbers]]. He viewed cardinal numbers as an abstraction of sets, introduced the notations, where, for a given set <math display="inline">M</math>, the [[order type]] of that set was written <math display="inline">\overline{M}</math>, and the cardinal number was <span style="border-top: 3px double;"><math display="inline">M</math></span>, a double abstraction. He also introduced the [[Cardinality#Aleph numbers|Aleph sequence]] for infinite cardinal numbers. These notations appeared in correspondence and were formalized in his later writings, particularly the series ''Beiträge zur Begründung der transfiniten Mengenlehre'' (1895{{En dash}}1897).<ref>{{Cite journal |last=Cantor |first=Georg |date=1895-11-01 |title=Beiträge zur Begründung der transfiniten Mengenlehre |url=https://link.springer.com/article/10.1007/BF02124929 |journal=Mathematische Annalen |language=de |volume=46 |issue=4 |pages=481–512 |doi=10.1007/BF02124929 |issn=1432-1807}}</ref> In these works, Cantor developed an [[Cardinal arithmetic|arithmetic of cardinal numbers]], defining addition, multiplication, and exponentiation of cardinal numbers based on set-theoretic constructions.  This led to the formulation of the [[Continuum Hypothesis]] (CH), the proposition that no set has cardinality strictly between <math>\aleph_0</math> and the [[cardinality of the continuum]], that is <math>|\R| = \aleph_1</math>. Cantor was unable to resolve CH and left it as an [[open problem]].

==== Other contributors ====
Parallel to Cantor’s development, [[Richard Dedekind]] independently formulated [[Dedekind-infinite set|a definition of infinite set]] as one that can be placed in bijection with a proper subset of itself, which was shown to be equivalent with Cantor’s definition of cardinality (given the [[axiom of choice]]). Dedekind’s ''[[Was sind und was sollen die Zahlen?]]'' (1888) emphasized structural properties over extensional definitions, and supported the bijective formulation of size and number. Dedekind was in correspondence with Cantor during the development of set theory; he supplied Cantor with a proof of the countability of the [[algebraic numbers]], and gave feedback and modifications on Cantor's proofs before publishing.

After Cantor's 1883 proof that all finite-dimensional [[manifolds]] have the same cardinality,<ref>{{Cite journal |last=Cantor |first=Georg |date=1883-12-01 |title=Ueber unendliche, lineare Punktmannichfaltigkeiten |url=https://doi.org/10.1007/BF01446819 |journal=Mathematische Annalen |language=de |volume=21 |issue=4 |pages=545–591 |doi=10.1007/BF01446819 |issn=1432-1807}}</ref>{{clarify|reason=Cantor dis not know the modern notion of a manifols. Using "manifold" here seem a mistranslation.|date=June 2025}} in 1890, [[Giuseppe Peano]] introducted the [[Peano curve]], which  was a more visual proof that the [[unit interval]] <math>[0,1]</math> has the same cardinality as the [[unit square]] on <math>\R^2.</math><ref>{{Cite journal |last=Peano |first=G. |date=1890-03-01 |title=Sur une courbe, qui remplit toute une aire plane |url=https://doi.org/10.1007/BF01199438 |journal=Mathematische Annalen |language=fr |volume=36 |issue=1 |pages=157–160 |doi=10.1007/BF01199438 |issn=1432-1807 |archive-url=https://archive.org/details/PeanoSurUneCurve |archive-date=2018-07-22}}</ref> This created a new area of mathematical analysis studying what is now called [[space-filling curves]].<ref>{{citation |last=Gugenheimer |first=Heinrich Walter |title=Differential Geometry |page=3 |year=1963 |url=https://books.google.com/books?id=CSYtkV4NTioC&pg=PA |publisher=Courier Dover Publications |isbn=9780486157207}}.</ref> 

German logician [[Gottlob Frege]] sought to ground the concept of number in logic, defining numbers using Cantor's theory of cardinality, connecting the notion to [[Hume's principle]]. In ''[[Die Grundlagen der Arithmetik]]'' (1884) and the subsequent ''Grundgesetze der Arithmetik'' (1893, 1903), Frege attempted to derive arithmetic from logical principles, treating cardinality and cardinal number as a [[primitive notion]]. However, Frege's approach to set theory was undermined by the discovery of [[Russell's paradox]] in 1901. The paradox played a crucial role in the [[foundational crisis in mathematics]] and especially the [[Logicism#History|logicist program]]. 

This was eventually resolved by [[Bertrand Russell]] himself in ''[[Principia Mathematica]]'' (1910{{En dash}}1913, vol. II),{{Sfn|Russell|Whitehead}} co-authored with [[Alfred North Whitehead]], which introduced a [[Type theory#History|theory of types]] to avoid such paradoxes, defining cardinal numbers at each level of the type hierarchy. Cardinal numbers were treated as [[equivalence classes]] of sets under equinumerosity, but only within a type-theoretic framework. Though Russell initially had difficulties understanding Cantor's and Frege’s intuitions of cardinality, shown in his 1905 manuscript ''On Some Difficulties in the Theory of Transfinite Numbers and Order Types.''<ref>{{Cite journal |last=Russell |first=B. |date=1907 |title=On Some Difficulties in the Theory of Transfinite Numbers and Order Types |url=https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s2-4.1.29?doi=10.1112%2Fplms%2Fs2-4.1.29 |journal=Proceedings of the London Mathematical Society |language=en |volume=s2-4 |issue=1 |pages=29–53 |doi=10.1112/plms/s2-4.1.29 |issn=1460-244X}}</ref>

==Comparing sets==

=== Introduction ===
[[File:Aplicación 2 inyectiva sobreyectiva04.svg|thumb|250px|A one-to-one correspondence from '''N''', the set of all non-negative integers, to the set '''E''' of non-negative [[even number]]s. Although '''E''' is a proper subset of '''N''', both sets have the same cardinality.]]
The basic notions of [[Set (mathematics)|sets]] and [[Function (mathematics)|functions]] are used to develop the concept of cardinality, and technical terms therein are used throughout this article. A [[Set (mathematics)|set]] can be understood as any collection of objects, usually represented with [[curly braces]]. For example, <math>S = \{1,2,3\}</math> specifies a set, called <math>S</math>, which contains the numbers 1, 2, and 3. The symbol <math>\in</math> represents set membership, for exmaple <math>1 \in S</math> says "1 is a member of the set <math>S</math>" which is true by the definition of <math>S</math> above.

A [[Function (mathematics)|function]] is an association that maps elements of one set to the elements of another, often represented with an arrow diagram. For example, the adjacent image depicts a function which maps the set of [[natural numbers]] to the set of [[even numbers]] by multiplying by 2. If a function does not map two elements to the same place, it is called [[injective]]. If a function covers every element in the output space, it is called [[surjective]]. If a function is both injective and surjective, it is called [[bijective]]. (For further clarification, see [[Bijection, injection and surjection|''Bijection, injection and surjection'']].)

===Equinumerosity===
{{Main|Equinumerosity}}
The intuitive property of two sets having the "same size" is that their objects can be paired one-to-one. For example, in [[Cardinality#top|the image above]], a set of apples is paired with a set of oranges, proving the sets are the same size without referring to these sets' "number of elements" at all. A one-to-one pairing between two sets defines a bijective function between them by mapping each object to its pair. Similarly, a bijection between two sets defines a pairing of their elements by pairing each object with the one it maps to. Therefore, these notions of "pairing" and "bijection" are intuitively equivalent. In fact, it is often the case that functions are defined literally as this set of pairings (see ''{{slink|Function (mathematics)|Formal definition}}''). Thus, the following definition is given:

Two sets <math>A</math> and <math>B</math> are said to have the ''same cardinality'' if their elements can be paired one-to-one. That is, if there exists a function <math>f:A \mapsto B</math> which is bijective. This is written as <math>A \sim B,</math> <math>A \approx B,</math> <math>A \equiv B,</math> and eventually <math> |A| = |B| ,  </math> once <math>|A|</math> has been defined. Alternatively, these sets, <math>A </math> and <math> B,</math> may be said to be ''similar'', [[Equinumerous|''equinumerous'']], or ''equipotent''. For example, the set <math>E = \{0, 2, 4, 6, \text{...}\}</math> of non-negative [[even number]]s has the same cardinality as the set <math>\N = \{0, 1, 2, 3, \text{...}\}</math> of [[natural numbers]], since the function <math>f(n) = 2n</math> is a bijection from {{tmath|\N}} to {{tmath|E}} (see picture above).

For finite sets {{tmath|A}} and {{tmath|B}}, if ''some'' bijection exists from {{tmath|A}} to {{tmath|B}}, then ''each'' injective or surjective function from {{tmath|A}} to {{tmath|B}} is a bijection. This property is no longer true for infinite {{tmath|A}} and {{tmath|B}}. For example, the function {{tmath|g}} from {{tmath|\N}} to {{tmath|E}}, defined by <math>g(n) = 4n</math> is injective, but not surjective since 2, for instance, is not mapped to, and {{tmath|h}} from {{tmath|\N}} to {{tmath|E}}, defined by <math>h(n) = 2 \operatorname{floor}(n/2)</math> (see: ''[[floor function]]'') is surjective, but not injective, since 0 and 1 for instance both map to 0. Neither {{tmath|g}} nor {{tmath|h}} can challenge <math>|E| = |\N|,</math> which was established by the existence of {{tmath|f}}.

==== Equivalence ====
[[File:Example for a composition of two functions.svg|thumb|Example for a composition of two functions.|282x282px]]
A fundamental result necessary in developing a theory of cardinality is showing it is an [[equivalence relation]]. A binary [[Relation (mathematics)|relation]] is an equivalence relation if it satisfies the three basic properties of equality: [[Reflexive relation|reflexivity]], [[Symmetric relation|symmetry]], and [[Transitive relation|transitivity]]. A relation <math>R</math> is reflexive if, for any <math>a,</math> <math>aRa</math> (read: <math>a</math> is <math>R</math>-related to <math>a</math>); symmetric if, for any <math>a</math> and <math>b,</math> if <math>aRb,</math> then <math>bRa</math> (read: if <math>a</math> is related to <math>b,</math> then <math>b</math> is related to <math>a</math>); and transitive if, for any <math>a,</math> <math>b,</math> and <math>c,</math> if <math>aRb</math> and <math>bRc,</math> then <math>aRc.</math>

Given any set <math>A,</math> there is a bijection from <math>A</math> to itself by the [[identity function]], therefore cardinality is reflexive. Given any sets <math>A</math> and <math>B,</math> such that there is a bijection <math>f</math> from <math>A</math> to <math>B,</math> then there is an [[inverse function]] <math>f^{-1}</math> from <math>B</math> to <math>A,</math> which is also bijective, therefore cardinality is symmetric. Finally, given any sets <math>A,</math> <math>B,</math> and <math>C</math> such that there is a bijection <math>f</math>  from <math>A</math> to <math>B,</math> and <math>g</math> from <math>B</math> to <math>C,</math> then their [[Function composition|composition]] <math>g \circ f</math> (read: <math>g</math> after <math>f</math>) is a bijection from <math>A</math> to <math>C,</math> and so cardinality is transitive. Thus, cardinality forms an equivalence relation. This means that cardinality [[Partition of a set|partitions sets]] into [[equivalence classes]], and one may assign a representative to denote this class. This motivates the notion of a [[Cardinality#Cardinal numbers|cardinal number]].

Somewhat more formally, a relation must be a certain set of [[ordered pairs]]. Since there is no [[set of all sets]] in standard set theory (see: ''{{section link||Cantor's paradox}}''), cardinality is not a relation in the usual sense, but a [[Predicate (logic)|predicate]] or a relation over [[Class (set theory)|classes]].

=== Inequality ===
[[File:Cantor-Bernstein.png|thumb|256x256px|[[Gyula Kőnig]]'s definition of a bijection {{color|#00c000|''h''}}:''A''&nbsp;→&nbsp;''B'' from the given injections {{color|#c00000|''f''}}:''A''&nbsp;→&nbsp;''B'' and {{color|#0000c0|''g''}}:''B''&nbsp;→&nbsp;''A''. ]]
A set <math>A</math> is not larger than a set <math>B</math> if it can be mapped into <math>B</math> without overlap. That is, the cardinality of <math>A</math> is less than or equal to the cardinality of <math>B</math> if there is an [[injective function]] from <math>A</math> to '''<math>B</math>'''. This is written <math>A \preceq B,</math> <math>A \lesssim B</math> and eventually <math>|A| \leq |B|.</math> If <math>A \preceq B,</math> but there is no injection from <math>B</math> to <math>A,</math> then <math>A</math> is said to be ''strictly'' smaller than <math>B,</math> written without the underline as <math>A \prec B</math> or <math>|A| < |B|.</math> For example, if <math>A</math> has four elements and <math>B</math> has five, then the following are true <math>A \preceq A,</math> <math>A \preceq B,</math> and <math>A \prec B.</math>

The basic properties of an inequality are reflexivity (for any <math>a,</math> <math>a \leq a</math>), transitivity (if <math>a \leq b</math> and <math>b \leq c,</math> then <math>a \leq c</math>) and [[Antisymmetric relation|antisymmetry]] (if <math>a \leq b</math> and <math>b \leq a,</math> then <math>a = b</math>) (See [[Inequality (mathematics)#Formal definitions and generalizations|''Inequality § Formal definitions'']]). Cardinal inequality <math>(\preceq)</math> as defined above is reflexive since the [[identity function]] is injective, and is transitive by function composition. Antisymmetry is established by the [[Schröder–Bernstein theorem]]. The proof roughly goes as follows. 

Given sets <math>A</math> and <math>B</math>, where <math>f:A \mapsto B</math> is the function that proves <math>A \preceq B</math> and <math>g: B \mapsto A</math> proves <math>B \preceq A</math>, then consider the sequence of points given by applying <math>f</math> then <math>g</math> over and over. Then we can define a bijection <math>h: A \mapsto B</math> as follows: If a sequence forms a cycle, begins with an element <math>a \in A</math> not mapped to by <math>g</math>, or extends infinitley in both directions, define <math>h(x) = f(x)</math> for each <math>x \in A</math> in those sequences. The last case, if a sequence begins with an element <math>b \in B</math>, not mapped to by <math>f</math>, define <math>h(x) = g^{-1}(x)</math> for each <math>x \in A</math> in that sequence. Then <math>h</math> is a bijection.

The above shows that cardinal inequality is a [[partial order]]. A [[total order]] has the additional property that, for any <math>a</math> and <math>b</math>, either <math>a \leq b</math> or <math>b \leq a.</math> This can be established by the [[well-ordering theorem]]. Every well-ordered set is uniquely [[order isomorphic]] to a unique [[ordinal number]], called the [[order type]] of the well-ordered set. Then, by comparing their order types, one can show that <math>A \preceq B</math> or <math>B \preceq A</math>. This result is equivalent to the [[axiom of choice]].<ref>{{citation | author=Friedrich M. Hartogs | author-link=Friedrich M. Hartogs | editor=Felix Klein | editor-link=Felix Klein |editor2=Walther von Dyck |editor2-link=Walther von Dyck |editor3=David Hilbert |editor3-link=David Hilbert |editor4=Otto Blumenthal |editor4-link=Otto Blumenthal | title=Über das Problem der Wohlordnung | journal=[[Mathematische Annalen]] | volume=76 | number=4 | publisher=B.&nbsp;G. Teubner | location=Leipzig | year=1915 | pages=438–443 | issn=0025-5831 |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0076&DMDID=DMDLOG_0037&L=1 | doi=10.1007/bf01458215| s2cid=121598654 }}</ref><ref>{{citation | author=Felix Hausdorff | author-link=Felix Hausdorff | editor=Egbert Brieskorn | editor-link=Egbert Brieskorn |editor2=Srishti D. Chatterji| title=Grundzüge der Mengenlehre | edition=1. | publisher=Springer | location=Berlin/Heidelberg | year=2002 | pages=587 | isbn=3-540-42224-2| url=https://books.google.com/books?id=3nth_p-6DpcC|display-editors=etal}} - [https://jscholarship.library.jhu.edu/handle/1774.2/34091 Original edition (1914)]</ref>

=== Countable sets ===
{{Main|Countable set}}
A set is called ''[[countable]]'' if it is [[Finite set|finite]] or has a bijection with the set of [[natural number]]s <math>(\N),</math> in which case it is called ''countably infinite''. The term ''[[denumerable]]'' is also sometimes used for countably infinite sets. For example, the set of all even natural numbers is countable, and therefore has the same cardinality as the whole set of natural numbers, even though it is a [[proper subset]]. Similarly, the set of [[square numbers]] is countable, which was considered paradoxical for hundreds of years before modern set theory (see: ''{{section link||Pre-Cantorian Set theory}}''). However, several other examples have historically been considered surprising or initially unintuitive since the rise of set theory.

{{Multiple image
| direction         = horizontal
| image1            = Diagonal argument.svg
| image2            = Straight counter-clockwise spiral path in square grid.png
| total_width       = 330
| footer            = Two images of a visual depiction of a function from <math>\N</math> to <math>\Q.</math> On the left, a version for the positive rational numbers. On the right, a spiral for all pairs of integers <math>p/q.</math>
}}

The [[rational numbers]] <math>(\Q)</math> are those which can be expressed as the [[quotient]] or [[Fraction (mathematics)|fraction]] {{tmath|\tfrac p q}} of two [[integer]]s. The rational numbers can be shown to be countable by considering the set of fractions as the set of all [[ordered pairs]] of integers, denoted <math>\Z\times\Z,</math> which can be visualized as the set of all [[Integer lattice|integer points]] on a grid. Then, an intuitive function can be described by drawing a line in a repeating pattern, or spiral, which eventually goes through each point in the grid. For example, going through each diagonal on the grid for positive fractions, or through a lattice spiral for all integer pairs. These technically over cover the rationals, since, for example, the rational number <math display="inline">\frac{1}{2}</math> gets mapped to by all the fractions <math display="inline">\frac{2}{4},\, \frac{3}{6}, \, \frac{4}{8}, \, \dots ,</math> as the grid method treats these all as distinct ordered pairs. So this function shows <math>|\Q| \leq |\N|</math> not <math>|\Q| = |\N|.</math> This can be corrected by "skipping over" these numbers in the grid, or by designing a function which does this naturally, but these methods are usually more complicated.
[[File:Algebraicszoom.png|thumb|273x273px|Algebraic numbers on the [[complex plane]], colored by degree]]
A number is called [[Algebraic number|algebraic]] if it is a solution of some [[polynomial]] equation (with integer [[coefficient]]s). For example, the [[square root of two]] <math>\sqrt2</math> is a solution to <math>x^2 - 2 = 0,</math> and the rational number <math>p/q</math> is the solution to <math>qx - p = 0.</math> Conversely, a number which cannot be the root of any polynomial is called [[Transcendental number|transcendental]]. Two examples include [[Euler's number]] (''{{mvar|e}}'') and [[Pi|pi ({{pi}})]]. In general, proving a number is transcendental is considered to be very difficult, and only a few classes of transcendental numbers are known. However, it can be shown that the set of algebraic numbers is countable (for example, see ''{{slink|Cantor's first set theory article|The proofs}}''). Since the set of algebraic numbers is countable while the real numbers are uncountable (shown in the following section), the transcendental numbers must form the vast majority of real numbers, even though they are individually much harder to identify. That is to say, [[almost all]] real numbers are transcendental.

=== Uncountable sets ===
{{hatnote group|{{Main|Uncountable set}}{{Further information|#Cardinality of the continuum}}}}
[[File:Diagonal argument powerset svg.svg|thumb|250px|<math>\N</math> does not have the same cardinality as its [[power set]] <math>\mathcal{P}(\N)</math>: For every function ''f'' from '''<math>\N</math>''' to <math>\mathcal{P}(\N)</math>, the set <math>T = \{n \in N : n \notin f(n) \}</math> disagrees with every set in the [[range of a function|range]] of <math>f</math>, hence <math>f</math> cannot be surjective. The picture shows an example <math>f</math> and the corresponding <math>T</math>; {{color|#800000|'''red'''}}: <math>n \notin T</math>, {{color|#000080|'''blue'''}}: <math>n \in T</math>.]]A set is called ''[[uncountable]]'' if it is not countable. That is, it is infinite and strictly larger than the set of natural numbers. The usual first example of this is the set of [[real numbers]] <math>(\R)</math>, which can be understood as the set of all numbers on the [[number line]]. One method of proving that the reals are uncountable is called [[Cantor's diagonal argument]], credited to Cantor for his 1891 proof,<ref name="Cantor.1891">{{cite journal |author=Georg Cantor |year=1891 |title=Ueber eine elementare Frage der Mannigfaltigkeitslehre |url=https://www.digizeitschriften.de/dms/img/?PID=GDZPPN002113910&physid=phys84#navi |journal=[[Jahresbericht der Deutschen Mathematiker-Vereinigung]] |volume=1 |pages=75–78}} English translation: {{cite book |title=From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Volume 2 |publisher=Oxford University Press |year=1996 |isbn=0-19-850536-1 |editor-last=Ewald |editor-first=William B. |pages=920–922}}</ref> though his differs from the more common presentation.

It begins by assuming, [[Proof by contradiction|by contradiction]], that there is some one-to-one mapping between the natural numbers and the set of real numbers between 0 and 1 (the interval <math>[0,1]</math>). Then, take the [[decimal expansion]]s of each real number, which looks like <math>0.d_1d_2d_3...</math> Considering these real numbers in a column, create a new number such that the first digit of the new number is different from that of the first number in the column, the second digit is different from the second number in the column and so on. We also need to make sure that the number we create has a unique decimal representation, that is, it cannot end in [[0.999...|repeating nines]] or repeating zeros. For example, if the digit isn't a 7, make the digit of the new number a 7, and if it was a seven, make it a 3.<ref>{{Cite book |last=Bloch |first=Ethan D. |url=https://link.springer.com/book/10.1007/978-1-4419-7127-2 |title=Proofs and Fundamentals |date=2011 |publisher=Springer Science+Business Media |series=Undergraduate Texts in Mathematics |pages=242–243 |language=en |doi=10.1007/978-1-4419-7127-2 |isbn=978-1-4419-7126-5 |issn=0172-6056 |archive-url=https://archive.org/details/proofsfundamenta0002bloc/ |archive-date=2022-01-22}}</ref> Then, this new number will be different from each of the numbers in the list by at least one digit, and therefore must not be in the list. This shows that the real numbers cannot be put into a one-to-one correspondence with the naturals, and thus must be strictly larger.<ref>{{Cite book|last1=Ashlock |first1=Daniel |last2=Lee |first2=Colin |date=2020 |title=An Introduction to Proofs with Set Theory |url=https://link.springer.com/book/10.1007/978-3-031-02426-9 |publisher=Springer Cham |series=Synthesis Lectures on Mathematics & Statistics |language=en |pages=181–182 |doi=10.1007/978-3-031-02426-9 |isbn=978-3-031-01298-3 |issn=1938-1743}}</ref>

Another classical example of an uncountable set, established using a related reasoning, is the [[power set]] of the natural numbers, denoted <math>\mathcal{P}(\N)</math>. This is the set of all [[subsets]] of <math>\N</math>, including the [[empty set]] and <math>\N</math> itself. The method is much closer to Cantor's original diagonal argument. Consider any function <math>f: \N \to \mathcal{P}(\N)</math>. One may define a subset <math>T \subseteq \N</math> which cannot be in the image of <math>f</math> by: if <math>1 \in f(1)</math>, then <math>1 \notin T</math>, and if <math>2 \notin f(2) </math>, then <math>2 \in T</math>, and in general, for each natural number <math>n</math>, <math>n \in T</math> if and only if <math>n \notin f(n) </math>. Then if the subset <math>T = f (t)</math> was in the image of <math>f</math>, then <math>t \in f (t) \iff t \notin f (t)</math>, a contradiction. So <math>f</math> cannot be surjective. Therefore no bijection can exist between <math>\N</math> and <math>\mathcal{P}(\N)</math>. Thus <math>\mathcal{P}(\N)</math> must not be countable. The two sets, <math>\R</math> and <math>\mathcal{P}(\N)</math> can be shown to have the same cardinality (by, for example, assigning each subset to a decimal expansion). Whether there exists a set <math>A</math> with cardinality between these two sets <math>|\N| < |A| < |\R|</math> is known as the [[continuum hypothesis]].

[[Cantor's theorem]] generalizes the second theorem above, showing that every set is strictly smaller than its powerset. The proof roughly goes as follows: Given a set <math>A</math>, if <math>f</math> is a function from <math>A</math> to <math>\mathcal{P}(A)</math>, let the subset <math>T \subseteq A</math> be given by <math>T = \{ a \in A : a \notin f(a) \}</math>. If <math>T = f (t)</math>, then <math>t \in f (t) \iff t \notin f (t)</math> a contradiction. So <math>f</math> cannot be surjective and thus cannot be a bijection. So <math>|A| < |\mathcal{P}(A)|</math>. (Notice that a trivial injection exists -- map <math>a</math> to <math>\{ a \}</math>.) Further, since <math>\mathcal{P}(A)</math> is itself a set, the argument can be repeated to show <math>|A| < |\mathcal{P}(A)| < |\mathcal{P}(\mathcal{P}(A))|</math>. Taking <math>A = \N</math>, this shows that <math>\mathcal{P}(\mathcal{P}(\N))</math> is even larger than <math>\mathcal{P}(\N) </math>, which was already shown to be uncountable. Repeating this argument shows that there are infinitely many "sizes" of infinity.

==Cardinal numbers==
{{main|Cardinal number}}In the above section, "cardinality" of a set was defined relationally. In other words, while it was closely tied to the concept of number, the meaning of "number of elements" has not yet been defined. This can be formalized from basic set-theoretic principles, relying on some number-like structures. For finite sets, this is simply the [[natural number]] found by counting the elements. This number is called the ''cardinal number'' of that set, or simply ''the cardinality'' of that set. The cardinal number of a set <math>A</math> is generally denoted by <math>|A|,</math> with a [[vertical bar]] on each side,<ref name=":7">{{Cite web |title=Cardinality {{!}} Brilliant Math & Science Wiki |url=https://brilliant.org/wiki/cardinality/ |access-date=2020-08-23 |website=brilliant.org |language=en-us}}</ref> though it may also be denoted by <math>n(A),</math> <span style="border-top: 3px double;"><math>A</math></span>, <math>\operatorname{card}(A),</math> or <math>\#A.</math> 

For infinite sets, "cardinal number" is somewhat more difficult to define formally. However, cardinal numbers are not usually thought of in terms of their formal definition, but immaterially in terms of their arithmetic/algebraic properties. For example, defining <math>|\N| = \aleph_0</math>, and <math>A \sim B</math> if and only if <math>|A| = |B| </math>. Then <math>2^{\aleph_0} = |\mathcal{P}(\N)| \sim |\R|. </math>

=== Finite sets ===
{{main|Finite set}}
[[File:Bijection.svg|thumb|200x200px|A [[bijective function]], ''f'': ''X'' → ''Y'', from set ''X'' to set ''Y''  demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4.]]Given a basic sense of [[natural numbers]], a set is said to have cardinality <math>n</math> if it can be put in one-to-one correspondence with the set <math>\{1,\,2,\, \dots, \, n\}.</math> For example, the set <math>S = \{ A,B,C,D \} </math> has a natural correspondence with the set <math>\{1,2,3,4\},</math> and therefore is said to have cardinality 4. Other terminologies include "Its cardinality is 4" or "Its cardinal number is 4". While this definition uses a basic sense of natural numbers, it may be that cardinality is used to define the natural numbers, in which case, a simple construction of objects satisfying the [[Peano axioms]] can be used as a substitute. Most commonly, the [[Von Neumann ordinal]]s.

Showing that such a correspondence exists is not always trivial, which is the subject matter of [[combinatorics]].

==== Uniqueness ====
An intuitive property of finite sets is that, for example, if a set has cardinality 4, then it cannot also have cardinality 5. Intuitively meaning that a set cannot have both exactly 4 elements and exactly 5 elements. However, it is not so obviously proven. The following proof is adapted from ''Analysis I'' by [[Terence Tao]].{{Sfn|Tao|2022|p=59}}
[[File:Lemma function.png|thumb|Intuitive depiction of the function <math>g</math> in the lemma, for the case <math>|X| = 7.</math>]]
Lemma: If a set <math>X</math> has cardinality <math>n \geq 1,</math> and <math>x_0 \in X,</math> then the set <math>X - \{x_0\} </math> (i.e. <math>X</math> with the element <math>x_0</math> removed) has cardinality <math>n-1.</math>

Proof: Given <math>X</math> as above, since <math>X</math> has cardinality <math>n,</math> there is a bijection <math>f</math> from <math>X</math> to <math>\{1,\,2,\, \dots, \, n\}.</math> Then, since <math>x_0 \in X,</math> there must be some number <math>f(x_0)</math> in <math>\{1,\,2,\, \dots, \, n\}.</math> We need to find a bijection from <math>X - \{x_0\} </math> to <math>\{1, \dots n-1\}</math> (which may be empty). Define a function <math>g</math> such that <math>g(x) = f(x)</math> if <math>f(x) < f(x_0),</math> and <math>g(x) = f(x)-1</math> if <math>f(x) > f(x_0).</math> Then <math>g</math> is a bijection from <math>X - \{x_0\} </math> to <math>\{1, \dots n-1\}.</math>

Theorem: If a set <math>X</math> has cardinality <math>n,</math> then it cannot have any other cardinality. That is, <math>X</math> cannot also have cardinality <math>m \neq n.</math>

Proof: If <math>X</math> is empty (has cardinality 0), then there cannot exist a bijection from <math>X</math> to any nonempty set <math>Y,</math> since nothing mapped to <math>y_0 \in Y.</math> Assume, by [[Mathematical induction|induction]] that the result has been proven up to some cardinality <math>n.</math> If <math>X,</math> has cardinality <math>n+1,</math> assume it also has cardinality <math>m.</math> We want to show that <math>m = n+1.</math> By the lemma above, <math>X - \{x_0\} </math> must have cardinality <math>n</math> and <math>m-1.</math> Since, by induction, cardinality is unique for sets with cardinality <math>n,</math> it must be that <math>m-1 = n,</math> and thus <math>m = n+1.</math>

==== Combinatorics ====
{{Main|Combinatorial principles}}
[[File:Inclusion-exclusion.svg|thumb|[[Inclusion–exclusion]] illustrated for three sets.]]
[[Combinatorics]] is the area of mathematics primarily concerned with [[counting]], both as a means and as an end to obtaining results, and certain properties of finite structures. The notion cardinality of finite sets is closely tied to many basic [[combinatorial principles]], and provides a set-theoretic foundation to prove them. The above shows uniqueness of finite cardinal numbers, and therefore, <math>A \sim B</math> if and only if <math>|A| = |B|</math>, formalizing the notion of a [[bijective proof]]. 

The [[addition principle]] asserts that given [[Disjoint sets|disjoint]] sets <math>A</math> and <math>B</math>, <math>|A \cup B| = |A| + |B|</math>, intuitively meaning that the sum of parts is equal to the sum of the whole. The [[multiplication principle]] asserts that given two sets <math>A</math> and <math>B</math>, <math>|A \times B| = |A| \cdot |B|</math>, intuitively meaning that there are <math>|A| \cdot |B|</math> ways to pair objects from these sets. Both of these can be proven by a bijective proof, together with induction. The more general result is the [[inclusion–exclusion principle]], which defines how to count the number of elements in overlaping sets.

=== Aleph numbers ===
{{Main|Aleph number}}
[[File:Aleph0.svg|right|thumb|169x169px|[[Aleph-nought]], aleph-zero, or aleph-null: the smallest infinite cardinal number, and the cardinal number of the set of natural numbers. ]]
The [[aleph numbers]] are a sequence of cardinal numbers that denote the size of [[infinite sets]], denoted with an [[aleph]] <math>\aleph,</math> the first letter of the [[Hebrew alphabet]]. The first aleph number is <math>\aleph_0,</math> called "aleph-nought", "aleph-zero", or "aleph-null", which represents the cardinality of the set of all [[natural numbers]]: <math>\aleph_0 = |\N| = |\{0,1,2,3,\cdots\}| .</math>  Then, <math>\aleph_1</math> represents the next largest cardinality. The most common way this is formalized in set theory is through [[Von Neumann ordinal]]s, known as [[Von Neumann cardinal assignment]].

[[Ordinal number]]s generalize the notion of ''order'' to infinite sets. For example, 2 comes after 1, denoted <math>1 < 2,</math> and 3 comes after both, denoted <math>1 < 2 < 3.</math> Then, one defines a new number, <math>\omega,</math> which comes after every natural number, denoted <math>1 < 2 < 3 < \cdots < \omega.</math> Further <math>\omega < \omega+1 ,</math> and so on. More formally, these ordinal numbers can be defined as follows:

<math>0 := \{\},</math> the [[empty set]], <math>1 := \{0\} ,</math> <math>2 := \{0,1\},</math> <math>3 := \{0,1,2\},</math> and so on. Then one can define <math>m < n \text{, if } \, m \in n,</math> for examlpe, <math>2 \in \{0,1,2\} = 3,</math> therefore <math>2 < 3.</math>  Defining <math>\omega := \{0,1,2,3,\cdots\}</math> (a [[limit ordinal]]) gives <math>\omega</math> the desired property of being the smallest ordinal greater than all finite ordinal numbers. Further, <math>\omega+1 := \{1,2,\cdots,\omega\}</math>, and so on.

Since <math>\omega \sim \N</math> by the natural correspondence, one may define <math>\aleph_0</math> as the set of all finite ordinals. That is, <math>\aleph_0 := \omega.</math> Then, <math>\aleph_1</math> is the set of all countable ordinals (all ordinals <math>\kappa</math> with cardinality <math>|\kappa| \leq \aleph_0</math>), the [[first uncountable ordinal]]. Since a set cannot contain itself, <math>\aleph_1</math> must have a strictly larger cardinality: <math>\aleph_0 < \aleph_1.</math> Furthermore, <math>\aleph_2</math> is the set of all ordinals with cardinality <math>\aleph_1,</math> and so on. By the [[well-ordering theorem]], there cannot exist any set with cardinality between <math>\aleph_0</math> and <math>\aleph_1,</math> and every infinite set has some cardinality corresponding to some aleph <math>\aleph_\alpha,</math> for some ordinal <math>\alpha.</math>

===Cardinality of the continuum===
{{main|Cardinality of the continuum|Continuum hypothesis}}
[[File:Number line.png|thumb|The [[number line]], containing all points in its continuum.|314x314px]]
The [[number line]] is a geometric construct of the intuitive notions of "[[space]]" and "[[distance]]" wherein each point corresponds to a distinct quantity or position along a continuous path. The terms "continuum" and "continuous" refer to the totality of this line, having some space (other points) between any two points on the line ([[Dense order|dense]] and [[Archimedean property|archimedian]]) and the absence of any gaps ([[Completeness of the real numbers|completeness]]), This intuitive construct is formalized by the set of [[real numbers]] <math>(\R)</math> which model the continuum as a complete, densely ordered, uncountable set. 
[[File:Cantor set binary tree.svg|thumb|262x262px|First five itterations approaching the Cantor set]]
The [[Cardinality of the continuum|cardinality of the]] [[Cardinality of the continuum|continuum]], denoted by "<math>\mathfrak c</math>" (a lowercase [[fraktur (script)|fraktur script]] "c"), remains invariant under various transformations and mappings, many considered surprising. For example, all intervals on the real line e.g. <math>[0,1]</math>, and <math>[0,2]</math>, have the same cardinality as the entire set <math>\R</math>. First, <math>f(x) = 2x</math> is a bijection from <math>[0,1]</math> to <math>[0,2]</math>. Then, the [[tangent function]] is a bijection from the interval <math display="inline">\left( \frac{-\pi}{2} \, ,  \frac{\pi}{2} \right)</math> to the whole real line. A more surprising example is the [[Cantor set]], which is defined as follows: take the interval <math>[0,1]</math> and remove the middle third <math display="inline">\left( \frac{1}{3}, \frac{2}{3} \right)</math>, then remove the middle third of each of the two remaining segments, and continue removing middle thirds (see image). The Cantor set is the set of points that survive this process. This set that remains is all of the points whose decimal expansion can be written in [[Ternary numeral system|ternary]] without a 1. Reinterpreting these decimal expansions as [[Binary number|binary]] (e.g. by replacing the 2s with 1s) gives a bijection between the Cantor set and the interval <math>[0,1]</math>.
[[File:Peanocurve.svg|thumb|339x339px|Three iterations of a [[Peano curve]] construction, whose [[Limit of a sequence|limit]] is a [[space-filling curve]].]]
[[Space-filling curves]] are continuous surjective maps from the [[unit interval]] <math>[0,1]</math> onto the [[unit square]] on <math>\R^2</math>, with classical examples such as the [[Peano curve]] and [[Hilbert curve|Hilbert]] [[Hilbert curve|curve]]. Although such maps are not injective, they are indeed surjective, and thus suffice to demonstrate cardinal equivalence. They can be reused at each dimension to show that <math>|\R| = |\R^n| = \mathfrak{c}</math> for any dimension <math>n \geq 1.</math> The infinite [[cartesian product]] <math>\R^\infty</math>, can also be shown to have cardinality <math>\mathfrak c</math>. This can be established by cardinal exponentiation: <math>|\R^\infty| = \mathfrak{c}^{\aleph_0} = \left(2^{\aleph_0} \right)^{\aleph_0} = 2^{(\aleph_0 \cdot \aleph_0)} = 2^{\aleph_0} = \mathfrak{c} = |\R|</math>.  Thus, the real numbers, all finite-dimensional real spaces, and the countable cartesian product share the same cardinality.

As shown in {{slink||Unountable sets}}, the set of real numbers is strictly larger than the set of natural numbers. Specifically, <math>|\R| = |\mathcal{P}(\N)|  </math>. The [[Continuum Hypothesis]] (CH) asserts that the real numbers have the next largest cardinality after the natural numbers, that is <math>|\R| = \aleph_1</math>. As shown by [[Kurt Gödel|Gödel]] and [[Paul Cohen|Cohen]], the continuum hypothesis is [[independence (mathematical logic)|independent]] of [[Zermelo–Fraenkel set theory with the axiom of choice|ZFC]], a standard axiomatization of set theory; that is, it is impossible to prove the continuum hypothesis or its negation from ZFC—provided that ZFC is [[Consistency|consistent]].<ref>{{Cite journal |last=Cohen |first=Paul J. |date=December 15, 1963 |title=The Independence of the Continuum Hypothesis |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=50 |issue=6 |pages=1143–1148 |bibcode=1963PNAS...50.1143C |doi=10.1073/pnas.50.6.1143 |jstor=71858 |pmc=221287 |pmid=16578557 |doi-access=free}}</ref><ref>{{Cite journal |last=Cohen |first=Paul J. |date=January 15, 1964 |title=The Independence of the Continuum Hypothesis, II |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=51 |issue=1 |pages=105–110 |bibcode=1964PNAS...51..105C |doi=10.1073/pnas.51.1.105 |jstor=72252 |pmc=300611 |pmid=16591132 |doi-access=free}}</ref><ref>{{Citation |last=Penrose |first=R |title=The Road to Reality: A Complete Guide to the Laws of the Universe |year=2005 |publisher=Vintage Books |isbn=0-09-944068-7 |author-link=Roger Penrose}}</ref> The [[Generalized Continuum Hypothesis]] (GCH) extends this to all infinite cardinals, stating that <math>2^{\aleph_\alpha} = \aleph_{\alpha+1}</math> for every ordinal <math>\alpha</math>. Without GHC, the cardinality of <math>\R</math> cannot be written in terms of alephs. The [[Beth numbers]] provide a concise notation for powersets of the real numbers starting from <math>\beth_1 = |\R|</math>, then <math>\beth_2 = |\mathcal{P}(\R)| = 2^{\beth_1}</math>, and in general <math>\beth_{n+1} = 2^{\beth_n}</math>.

== Paradoxes ==
During the rise of set theory, several [[paradoxes]] (see: ''[[Paradoxes of set theory]]''). These can be divided into two kinds: ''real paradoxes'' and ''apparent paradoxes''. Apparent paradoxes are those which follow a series of reasonable steps and arrive at a conclusion which seems impossible or incorrect according to one's [[intuition]], but aren't necessarily logically impossible. Two historical examples have been given, ''Galileo's Paradox'' and ''Aristotle's Wheel'', in {{Section link|2=History|nopage=y}}. Real paradoxes are those which, through reasonable steps, prove a [[Contradiction|logical contradiction]]. The real paradoxes here apply to [[naive set theory]] or otherwise informal statements, and have been resolved by restating the problem in terms of a [[Set theory#Formalized set theory|formalized set theory]], such as [[Zermelo–Fraenkel set theory]].

=== Apparent paradoxes ===

==== Hilbert's hotel ====
{{Main|Hilbert's paradox of the Grand Hotel}}
[[File:Hilbert's Hotel.png|thumb|259x259px|Visual representation of Hilbert's hotel]]
[[Hilbert's Hotel]] is a [[thought experiment]] devised by the German mathematician [[David Hilbert]] to illustrate a counterintuitive property of infinite sets (assuming the axiom of choice), allowing them to have the same cardinality as a [[proper subset]] of themselves. The scenario begins by imagining a hotel with an infinite number of rooms, all of which are occupied. But then a new guest walks in asking for a room. The hotel accommodates by moving the occupant of room 1 to room 2, the occupant of room 2 to room 3, room three to room 4, and in general, room n to room n+1. Then every guest still has a room, but room 1 opens up for the new guest.<ref name=":5">{{Cite book |last=Gamov |first=George |title=One two three... infinity |title-link=One Two Three... Infinity |publisher=Viking Press |year=1947 |language=English |lccn=62-24541}} [https://archive.org/details/OneTwoThreeInfinity_158/ Archived] on 2016-01-06</ref>

Then, the scenario continues by imagining an infinite bus of new guests seeking a room. The hotel accommodates by moving the person in room 1 to room 2, room 2 to room 4, and in general, room n to room 2n. Thus, all the even-numbered rooms are occupied, but all the odd-numbered rooms are vacant, leaving room for the infinite bus of new guests. The scenario continues by assuming an infinite number of these infinite buses arrive at the hotel, and showing that the hotel is still able to accommodate. Finally, an infinite bus which has a seat for every [[real number]] arrives, and the hotel is no longer able to accommodate.<ref name=":5" />

==== Skolem's paradox ====
{{Main|Skolem's paradox}}
[[File:Lowenheim-skolem.svg|thumb|Illustration of the [[Löwenheim–Skolem theorem]], where <math>\mathcal{M}</math> and <math>\mathcal{N}</math> are models of set theory, and <math>\kappa</math> is an arbitrary infinite cardinal number.]]
In [[model theory]], a [[Model (mathematical logic)|model]] corresponds to a specific interpretation of a [[formal language]] or [[Theory (mathematical logic)|theory]]. It consists of a [[Domain of discourse|domain]] (a set of objects) and an [[Interpretation (logic)|interpretation]] of the symbols and formulas in the language, such that the axioms of the theory are satisfied within this structure. The [[Löwenheim–Skolem theorem]] shows that any model of set theory in [[first-order logic]], if it is [[consistent]], has an equivalent [[Structure (mathematical logic)|model]] which is countable. This appears contradictory, because [[Georg Cantor]] proved that there exist sets which are not countable. Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets, [[Satisfiability|satisfies]] the first-order sentence that intuitively states "there are uncountable sets".<ref name=":6">{{Citation |last=Bays |first=Timothy |title=Skolem's Paradox |date=2025 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/paradox-skolem/ |access-date=2025-04-13 |edition=Spring 2025 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref>

A mathematical explanation of the paradox, showing that it is not a true contradiction in mathematics, was first given in 1922 by [[Thoralf Skolem]]. He explained that the countability of a set is not absolute, but relative to the model in which the cardinality is measured. Skolem's work was harshly received by [[Ernst Zermelo]], who argued against the limitations of first-order logic and Skolem's notion of "relativity", but the result quickly came to be accepted by the mathematical community.<ref>{{cite journal |last1=van Dalen |first1=Dirk |author-link1=Dirk van Dalen |last2=Ebbinghaus |first2=Heinz-Dieter |author2-link=Heinz-Dieter Ebbinghaus |date=Jun 2000 |title=Zermelo and the Skolem Paradox |url=https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=9d51449b161b9e1d352e103af28fe07f5e15cb4b |journal=The Bulletin of Symbolic Logic |volume=6 |pages=145–161 |citeseerx=10.1.1.137.3354 |doi=10.2307/421203 |jstor=421203 |s2cid=8530810 |number=2 |hdl=1874/27769}}</ref><ref name=":6" />

=== Real paradoxes ===

==== Cantor's paradox ====
{{Main|Cantor's paradox}}
[[Cantor's theorem]] state's that, for any set <math>A,</math> possibly infinite, its [[powerset]] <math>\mathcal{P}(A)</math> has a strictly greater cardinality. For example, this means there is no bijection from <math>\N</math> to <math>\mathcal{P}(\N) \sim \R.</math> [[Cantor's paradox]] is a paradox in [[naive set theory]], which shows that there cannot exist a "set of all sets" or "[[Universe (mathematics)|universe set]]". It starts by assuming there is some set of all sets, <math>U := \{x \; | \; x \,\text{ is a set} \},</math> then it must be that <math>U</math> is strictly smaller than <math>\mathcal{P}(U),</math> thus <math>|U| \leq |\mathcal{P}(U)| .</math> But since <math>U</math> contains all sets, we must have that <math>\mathcal{P}(U) \subseteq U,</math> and thus <math>|\mathcal{P}(U)| \leq |U|.</math> Therefore <math>|\mathcal{P}(U)| = |U|,</math> contradicting Cantor's theorem. This was one of the original paradoxes that added to the need for a formalized set theory to avoid these paradoxes. This paradox is usually resolved in formal set theories by disallowing [[unrestricted comprehension]] and the existence of a universe set.

==== Set of all cardinal numbers ====
Similar to Cantor's paradox, the paradox of the set of all cardinal numbers is a result due to unrestricted comprehension. It often uses the definition of cardinal numbers as ordinal numbers for representatives. It is related to the [[Burali-Forti paradox]]. It begins by assuming there is some set <math>S := \{ X \, | X \text{ is a cardinal number}\}.</math> Then, if there is some largest element <math>\aleph \in S ,</math> then the powerset <math>\mathcal{P}(\aleph)</math> is strictly greater, and thus not in <math>S.</math> Conversly, if there is no largest element, then the [[Union (set theory)#Arbitrary union|union]] <math>\bigcup S</math> contains the elements of all elements of <math>S,</math> and is therefore greater than or equal to each element. Since there is no largest element in <math>S,</math> for any element <math>x \in S,</math> there is another element <math>y \in S</math> such that <math>|x| < |y|</math> and <math>|y| \leq \Bigl| \bigcup S \Bigr|.</math> Thus, for any <math>x \in S,</math> <math>|x| < \Bigl| \bigcup S \Bigr|,</math> and so <math>\Bigl| \bigcup S \Bigr| \notin S.</math>

==See also==
{{Commons category}}
{{Wikidata property|P1164|P2820}}
* [[Cardinal function]]
* [[Inaccessible cardinal]]
* [[Infinitary combinatorics]]
* [[Large cardinal]]
** [[List of large cardinal properties]]
* [[Regular cardinal]]
* [[Scott's trick]]
* [[The Higher Infinite|''The Higher Infinite'']]
* [[Von Neumann universe]]
* [[Zero sharp]]

==References==

=== Citations ===
{{reflist}}
{{notelist}}

=== Bibliography ===

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* {{Cite book |last=Bourbaki |first=Nicholas |author-link=Nicolas Bourbaki |url=https://archive.org/details/theoryofsets0000bour/ |title=Theory of Sets |date=1968 |publisher=[[Éditions Hermann]] |series=[[Éléments de mathématique]] |location=Paris |lccn=68-25177}}
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{{Mathematical logic}}
{{Set theory}}
{{Authority control}}

[[Category:Cardinal numbers| ]]
[[Category:Basic concepts in infinite set theory]]