View source for Cardinality

In [[Mathematics|mathematics]], '''cardinality''' is an inherent property of [[Set (mathematics)|sets]], roughly meaning the number of individual [[Philosophy:Mathematical object|objects]] they contain, which may be [[Infinity|infinite]]. The concept is understood through [[One-to-one correspondence|one-to-one correspondence]]s between sets. That is, if their objects can be paired such that each object has a pair, and no object is paired more than once.

The basic concepts of cardinality go back as early as the 6th century BCE, and there are several close encounters with it throughout history, however, the results were generally dismissed as paradoxical. It is considered to have been first introduced formally to mathematics by [[Biography:Georg Cantor|Georg Cantor]] at the turn of the 20th century. Cantor's theory of cardinality was then formalized, popularized, and explored by many influential mathematicians of the time, and has since become a fundamental concept of mathematics.

Two sets are said to be '''''equinumerous''''' or ''have the same cardinality'' if there exists a one-to-one correspondence between them. Otherwise, one is said to be ''strictly larger'' or ''strictly smaller'' than the other. A set is [[Countable set|countably infinite]] if it can be placed in one-to-one correspondence with the set of natural numbers <math>\{1,2,3,4,\cdots\}.</math> For example, the set of even numbers <math>\{2,4,6,..\}</math> and the set of rational numbers are countable. [[Uncountable set]]s are those strictly larger than the set of natural numbers—for example, the set of all real numbers or the [[Company:Powerset|powerset]] of the set of natural numbers.

For finite sets, cardinality coincides with the natural number found by [[Counting|counting]] their elements. However, it is more often difficult to ascribe "sizes" to infinite sets. Thus, a system of cardinal numbers can be developed to extend the [[Cardinal numeral|role of natural numbers in answering "how many"]]. The cardinal number corresponding to a set <math>A</math> is written as <math>|A|</math> between two vertical bars. Most commonly, the Aleph numbers <math>\aleph_0, \aleph_1, \aleph_2, ... \aleph_\omega, \aleph_{\omega+1}...</math> are used, since it can be shown every infinite set has cardinality equivalent to some Aleph.

The set of natural numbers has cardinality <math>\aleph_0</math>. The question of whether the real numbers have cardinality <math>\aleph_1</math> is known as the [[Continuum hypothesis|continuum hypothesis]], which has been shown to be unprovable in standard set theories such as [[Zermelo–Fraenkel set theory]]. Alternative set theories and additional axioms give rise to different properties and have often strange or unintuitive consequences. However, every theory of cardinality using standard logical foundations of mathematics admits [[Skolem's paradox]], which roughly asserts that basic properties of cardinality are not [[Philosophy:Absoluteness (logic)|absolute]], but relative to the [[Structure (mathematical logic)|model]] in which the cardinality is measured.

== Introduction ==

=== Definition ===
Cardinality is an inherent property of [[Set (mathematics)|sets]] which defines their size, roughly meaning the number of individual [[Philosophy:Mathematical object|objects]] they contain.<ref>{{Cite book |last=Hazewinkel |first=Michiel |title=Encyclopaedia of Mathematics: C An updated and annotated translation of the Soviet 'Mathematical Encyclopaedia' |publisher=Kluwer Academic Publishers |year=2013 |isbn=978-94-009-6000-8 |volume=2 |location=Dordrecht |pages=24 |doi=10.1007/978-94-009-6000-8 |lccn=87-26437 |orig-year=1998|quote=CARDINALITY, cardinal number, of a set A - That property of A which is inherent in any set B equivalent to A. Here two sets are called equivalent (or equipotent or of the same cardinality) if it is possible to construct a one-to-one correspondence between them.}}</ref> Fundamentally however, it is different from the concepts of [[Number|number]] or [[Counting|counting]] as the cardinalities of two sets can be compared without referring to their number of elements, or defining number at all. For example, in the image [[Cardinality#top|above]], a set of apples is compared to a set of oranges such that every fruit is used exactly once which shows these two sets have the same cardinality, even if one doesn't know how many of each there are.<ref>{{Cite book |last=Clegg |first=Brian |author-link=Brian Clegg (writer) |url=http://archive.org/details/briefhistoryofin0000cleg |title=A Brief History of Infinity |publisher=Constable & Robinson |year=2003 |isbn=978-1-84119-650-3 |location=London |page=150}}</ref><ref>{{Harvard citation no brackets|Enderton|1977|pp=128–129}}</ref> Thus, cardinality is measured by putting sets in [[One-to-one correspondence|one-to-one correspondence]]. If it is possible, the sets are said to have the ''same cardinality'', and if not, one set is said to be ''strictly larger'' or ''strictly smaller'' than the other.<ref>{{Harvard citation no brackets|Bourbaki|1968|p=157}} {{Break}}{{Harvard citation no brackets|Takeuti|Zaring|1982|p=83}} {{Break}}{{Harvard citation no brackets|Tao|2022|pp=57–58}}</ref>

=== Note on notation and terminology ===
{| class="wikitable floatright" style="text-align:center"
!
!not surjective
!surjective
|-
!not 
injective
|[[File:Total_function_2.svg|alt=Total function|frameless|150x150px|class=skin-invert-image]]
'''general''' '''function'''
|[[File:Surjection.svg|frameless|150x150px|class=skin-invert-image]]
'''surjective''' '''only'''
|-
!injective
|[[File:Injection.svg|frameless|150x150px|class=skin-invert-image]]
'''injective''' '''only'''
|[[File:Bijection.svg|frameless|150x150px|class=skin-invert-image]]
'''bijective'''
|}
The basic concepts of cardinality are developed in terms of [[Set (mathematics)|sets]] and [[Function (mathematics)|functions]], which are somewhat more abstract than their counterparts outside of mathematics. Informally, a 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 example <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. Here <math>S</math> is [[Philosophy:Finiteness|finite]], but that is not a requirement in general. The only requirement for a set is that it is well-defined. That is, for any object, <math>x</math>, one can determine whether <math>x</math> belongs to that set <math>(x \in S)</math>, or <math>x</math> does not belong to that set <math>(x \notin S)</math>. One example of an infinite set is the set of all natural numbers <math>\{ 1,2,3,... \}</math>.{{Efn|The actually-infinite set of natural numbers is guaranteed to exist by the Axiom of Infinity.}}

A [[Function (mathematics)|function]], or correspondence, maps members of one set to the members of another, often represented with an arrow diagram. For example, the adjacent table depicts several functions which map sets of natural numbers to sets of letters. If a function does not map two members to the same place, it is called injective. If a function covers every member in the output set, it is called surjective. If a function is both injective and surjective, it is called bijective or a one-to-one correspondence. Functions are not limited to those one can draw an arrow diagram for, so long as the function is well-defined. That is, for each possible input, one can determine the output. For example, one may define a function from the natural numbers to the natural numbers by multiplying by two: <math>1 \mapsto 2, \; </math><math>2\mapsto 4, \, </math><math>3\mapsto 6 ... \,</math> <math>n \mapsto 2n  ...</math>

=== Etymology and related terms ===
The term ''cardinality'' originates from the post-classical Latin ''cardo'' ("to hinge"), which referred to something central or pivotal, both literally and metaphorically. This passed into [[Social:Medieval Latin|medieval Latin]] and then into English, where ''cardinal'' came to describe things considered to be, in some sense, fundamental, such as, ''cardinal sins'', ''cardinal directions'', and (in linguistics) ''cardinal numbers''.<ref>Oxford English Dictionary, "cardinal (adj.), Etymology," March 2025, https://doi.org/10.1093/OED/1490074521.</ref><ref>Harper, Douglas, "[https://www.etymonline.com/word/cardinal Origin and history of ''cardinal'']", Online Etymology Dictionary, accessed April 20, 2025.</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'', 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 number]]s'' 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 German mathematician [[Biography:Georg Cantor|Georg Cantor]] in the late 19th century, who used the term ''Mächtigkeit'', which may be translated as "magnitude" or "power", though Cantor credited the term to a work by [[Biography:Jakob Steiner|Jakob Steiner]] on [[Projective geometry|projective geometry]].<ref>{{Harvard citation no brackets|Ferreirós|2007|p=24}}</ref><ref name=":3">{{Cite book |last=Cantor |first=Georg |url=https://link.springer.com/book/10.1007/978-3-662-00274-2 |title=Gesammelte Abhandlungen |date=1932 |publisher=Springer |isbn=978-3-662-00254-4 |editor-last=Zermelo |editor-first=Ernst |location=Berlin |pages=151 |doi=10.1007/978-3-662-00274-2 |orig-date=1932 |url-access=subscription}}</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=Teubner |others=Ghent University |location=Leipzig |via=Internet Archive}}</ref> Around 1930, the terms ''cardinality'' and ''cardinal number'' were adopted from the grammatical sense, and later translations would use these terms.<ref name=":15">Harper, Douglas, "[https://www.etymonline.com/word/cardinality Origin and history of ''cardinality'']", Online Etymology Dictionary, accessed April 20, 2025.</ref><ref>Oxford English Dictionary, "cardinality (n.2), Etymology," March 2025, https://doi.org/10.1093/OED/5444748676.</ref>{{Efn|Etymonline gives the year 1935,<ref name=":15" /> however Oxford English Dictionary cites an earlier example from 1926 by [[Biography:Cooper Harold Langford|C. H. Langford]].<ref>Oxford English Dictionary, “cardinality (n.2), sense 2,” March 2025, https://doi.org/10.1093/OED/8293133300.</ref><ref>{{Cite journal |last=Langford |first=C. H. |author-link=Cooper Harold Langford |date=1926 |title=Analytic Completeness of Sets of Postulates |url=https://academic.oup.com/plms/article-abstract/s2-25/1/115/1519960 |journal=[[Organization:Proceedings of the London Mathematical Society|Proceedings of the London Mathematical Society]] |volume=25 |issue=1 |pages=116, 118 |doi=10.1112/plms/s2-25.1.115 |via=Oxford Academic}}</ref>}}

==Comparing sets==

=== Equinumerosity ===
{{redirect-distinguish|Equipollent|Equipollent (geometry)}}

[[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|even number]]s. Although '''E''' is a proper subset of '''N''', both sets have the same cardinality.|class=skin-invert-image]]
An intuitive relationship between two sets having the "same size" is that their objects can be paired one-to-one. That is, if each object in one set can be assigned a dedicated "pair" in the other, and no object from either set left unpaired, it seems reasonable to say there are the same number of objects in each set. 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, at least intuitively, equivalent.<ref>{{Harvard citation no brackets|Enderton|1977|pp=128–129}}  {{Break}}{{Harvard citation no brackets|Kleene|1952|p=3}} {{Break}}{{Harvard citation no brackets|Suppes|1972|p=91}} {{Break}}{{Harvard citation no brackets|Tao|2022|pp=57–58}}

</ref> Thus, the following definition is given:

Two sets are said to have the ''same cardinality'' or be ''equinumerous'' if their members can be paired one-to-one. That is, if there exists a function between them which is bijective. This is often written as <math>A \sim B,</math> or<math>A \approx B.</math>{{refn|<math>A \sim B</math>{{Sfn|Halmos|1998|p=52}}{{Sfn|Kleene|1952|p=9}}{{Sfn|Kuratowski|1968|p=169}}{{Sfn|Stoll|1963|p=79}}{{Break}}<math>A \approx B</math>{{Sfn|Enderton|1977|p=129}}{{Sfn|Lévy|1979|p=76}}{{Sfn|Pinter|2014|loc=Page 2 of Chapter 7}}{{Sfn|Suppes|1972|p=91}}}} Alternatively, these sets may be said to be ''equivalent'', ''similar'', ''equipotent'', or ''equipollent''.{{refn|Equivalent{{Sfn|Halmos|1998|p=52}}{{Sfn|Kleene|1952|p=9}}{{Sfn|Takeuti|Zaring|1982|p=83}} {{Break}}Similar{{Sfn|Russell|Whitehead|1925|p=419}}{{Sfn|Stoll|1963|p=79}}{{Break}}Equinumerous{{Sfn|Enderton|1977|p=129}}{{Sfn|Lévy|1979|p=76}}{{Sfn|Stoll|1963|p=79}} {{Break}}Equipotent{{Sfn|Bourbaki|1968|p=157}}{{Sfn|Hrbáček|Jech|2017|p=65}}{{Sfn|Pinter|2014|loc=Page 2 of Chapter 7}} {{Break}}Equipollent{{Sfn|Krivine|1971|p=23}}{{Sfn|Kuratowski|1968|p=169}}{{Sfn|Suppes|1972|p=91}}}} For example, the set <math>E = \{0, 2, 4, 6, \text{...}\}</math> of [[Even number|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}}.

The property for finite sets that "the whole is greater than the part" is no longer true for infinite sets, and the existence of surjections or injections that don't work does not prove that there is no bijection. For example, the function {{tmath|g}} from {{tmath|\N}} to {{tmath|E}}, defined by multiplying by 4, <math>g(n) = 4n</math>, is injective, but not surjective (since 2, for instance, is not mapped to). Further, the function {{tmath|h}} from {{tmath|\N}} to {{tmath|E}}, defined by rounding down to the nearest even number, <math>h(n) = 2 \operatorname{floor}(n/2)</math> (cf. ''[[Floor function|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}}.<ref>{{Harvard citation no brackets|Halmos|1998|p=52}} {{Break}}{{Harvard citation no brackets|Pinter|2014|loc=page 2 of chapter 7}}{{Break}}{{Harvard citation no brackets|Schumacher|1996|pp=93–94}}{{Break}}{{Harvard citation no brackets|Suppes|1972|p=92}}</ref>

==== Equivalence ====
[[File:Example for a composition of two functions.svg|thumb|Example of the composition of two functions.|300x300px|class=skin-invert-image]]
A fundamental result necessary in developing a theory of cardinality is relating it to an [[Equivalence relation|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]].<ref name=":10">{{Harvard citation no brackets|Bourbaki|1968|p=157}} {{Break}}{{Harvard citation no brackets|Suppes|1972|p=92}} {{Break}}{{Harvard citation no brackets|Hrbáček|Jech|2017|p=66}}</ref>

* Reflexivity: For any set <math>A</math>, <math>A \sim A.</math><!--
-->
** Given any set <math>A,</math> there is a bijection from <math>A</math> to itself by the [[Identity function|identity function]], therefore equinumerosity is reflexive.<ref name=":10" />
* Symmetry: If <math>A \sim B</math> then <math>B \sim A.</math><!--
-->
** 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|inverse function]] <math>f^{-1}</math> from <math>B</math> to <math>A,</math> which is also bijective, therefore equinumerosity is symmetric.<ref name=":10" />
* Transitivity: If <math>A \sim B</math> and <math>B \sim C</math> then <math>A \sim C.</math><!--
-->
** 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> is a bijection from <math>A</math> to <math>C,</math> and so cardinality is transitive.<ref name=":10" />

Since equinumerosity satisfies these three properties, it forms an equivalence relation. This means that cardinality, in some sense, [[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 cardinal number.<ref>{{Harvard citation no brackets|Abbott|2015|p=36}}</ref> 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, equinumerosity is not a relation in the usual sense, but a predicate, defined formally as:<ref>{{Harvard citation no brackets|Takeuti|Zaring|1982|p=83}}</ref> <math display="block">A \sim B \iff \exists (f: A \to B) (\forall b \in B \, (\exists ! a \in A \, (f(a) = b))</math>

=== Inequality ===
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|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>{{Refn|
* <math>A \preceq B</math>{{Sfn|Enderton|1977|p=145}}{{Sfn|Lévy|1979|p=84}}{{Sfn|Suppes|1972|p=94}}
* <math>A \lesssim B</math>{{Sfn|Halmos|1998|p=87}}{{Sfn|Stoll|1963|p=81}}
* <math>|A| \leq |B|</math>{{Sfn|Hrbáček|Jech|2017|p=66}}{{Sfn|Schumacher|1996|p=100}}}} and read as <math>A</math> is ''not greater than'' '''''<math>B,</math>''''' or <math>A</math> is ''dominated by '''<math>B.</math>'''''<ref>{{Harvard citation no brackets|Enderton|1977|p=145}} {{br}}{{harvnb|Halmos|1998|p=87}}{{br}}{{Harvard citation no brackets|Stoll|1963|p=81}}</ref> 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><ref>{{Harvard citation no brackets|Halmos|1998|p=90}}{{br}}{{Harvard citation no brackets|Stoll|1963|p=82}}{{br}}{{Harvard citation no brackets|Suppes|1972|p=97}}</ref> 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|identity function]] is injective, and is transitive by function composition.<ref>{{Harvard citation no brackets|Enderton|1977|pp=146–147}}{{br}}{{Harvard citation no brackets|Halmos|1998|p=87}}{{br}}{{Harvard citation no brackets|Hrbáček|Jech|2017|p=66}}

</ref> Antisymmetry is established by the [[Schröder–Bernstein theorem]].<ref name=":6">{{Harvard citation no brackets|Aigner|Ziegler|2018|pp=134–135}}{{br}}{{Harvard citation no brackets|Enderton|1977|pp=147–148}}{{br}}{{Harvard citation no brackets|Schumacher|1996|pp=104–105}}{{br}}{{Harvard citation no brackets|Stoll|1963|pp=81–82}}</ref>

The above shows that cardinal inequality is a partial order. A [[Total order|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|well-ordering theorem]]. Every well-ordered set is isomorphic to a unique [[Ordinal number|ordinal number]], called the [[Order type|order type]] of the 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|axiom of choice]].<ref>{{Harvard citation no brackets|Enderton|1977|p=|pp=151–153}}</ref><ref>{{citation | author=Friedrich M. Hartogs |  editor=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>

== Countability ==

=== Countable sets ===
A set is called ''countable'' if it is [[Finite set|finite]] or has a bijection with the set of [[Natural number|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.<ref>{{Harvard citation no brackets|Halmos|1998|p=91}} {{br}}{{Harvard citation no brackets|Kuratowski|1968|p=174}} {{br}}{{Harvard citation no brackets|Stoll|1963|p=87}}{{br}}{{Harvard citation no brackets|Tao|2022|p=159}}</ref> 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 (cf. ''{{section link||Pre-Cantorian set theory}}''). However, several other examples have historically been considered surprising or initially unintuitive since the rise of set theory.<ref>{{Harvard citation no brackets|Schumacher|1996|pp=93–99}} {{br}}{{Harvard citation no brackets|Kleene|1952|pp=3–4}}</ref>

{{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> for each fraction <math>p/q.</math>
}}

The rational numbers <math>(\Q)</math> are those which can be expressed as the [[Quotient|quotient]] or [[Fraction (mathematics)|fraction]] {{tmath|\tfrac p q}} of two [[Integer|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.<ref>{{Harvard citation no brackets|Aigner|Ziegler|2018|p=128}}{{br}}{{Harvard citation no brackets|Kleene|1952|pp=4–5}} {{br}}{{Harvard citation no brackets|Pinter|2014|loc=Page 2 of chapter 7}} {{br}}{{Harvard citation no brackets|Stoll|1963|p=88}}</ref> 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, using the Schröder–Bernstein theorem, or by designing a function which does this naturally, for example using the [[Calkin–Wilf tree]].<ref>{{Harvard citation no brackets|Aigner|Ziegler|2018|pp=129–131}}</ref>
[[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|polynomial]] equation (with integer [[Coefficient|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 by ordering the polynomials lexicographically (for example, see ''{{section link|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|almost all]] real numbers are transcendental.<ref>{{Harvard citation no brackets|Enderton|1977|pp=160–161}}{{br}} {{Harvard citation no brackets|Kleene|1952|pp=5–8}}{{br}}{{Harvard citation no brackets|Kuratowski|1968|pp=177–178}}{{br}}{{Harvard citation no brackets|Stoll|1963|pp=88–89}}</ref>

==== Hilbert's hotel ====
[[File:Hilbert's Hotel.png|thumb|291x291px|Visual representation of [[Hilbert's hotel]]. Each guest goes to the room with a number that is double their room number, leaving the odd-numbered rooms vacant.|class=skin-invert-image]]
[[Hilbert's paradox of the Grand Hotel]] is a popular [[Thought experiment|thought experiment]] devised by the German mathematician [[Biography:David Hilbert|David Hilbert]] to illustrate a counterintuitive property of countably infinite sets, 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, one for each natural number, 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 3 to room 4, and in general, room n to room n+1. Then every guest still has a room, but room 1 is open 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}} Archived on 2016-01-06</ref><ref name=":11">{{Harvard citation no brackets|Schumacher|1996|p=96}}{{br}}{{Harvard citation no brackets|Aigner|Ziegler|2018|pp=128–129}}</ref>

Then, the scenario continues by imagining an infinitely long 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 further 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|real number]] arrives, and the hotel is no longer able to accommodate.<ref name=":5" /><ref name=":11" />

=== Uncountable sets ===
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|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 method differs from the more common presentation.<ref>{{Harvard citation no brackets|Abbott|2015|pp=32–34}}</ref>

{{Quote box
| quote = <math>\begin{align}
1 \to & \; 0.{ \color{red} \textbf{6} }18033... \\
2 \to & \; 0.1{ \color{red}\textbf{2} }3456... \\ 
3 \to & \; 0.60{ \color{red}\textbf{7} }927... \\
4 \to & \; 0.222{ \color{red}\textbf{2} }22... \\
\vdots \\
& \; 0.{ \color{red}\textbf{2323}22}...
\end{align}</math>
| border = 0px
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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 representation|decimal representation]] of each real number, for example, <math>0.5772...</math> with a leading zero followed by any sequence of digits. The number 1 is included in this set since {{nowrap|1 {{=}} [[0.999...|0.999...]]}} Considering these real numbers in a column, it is always possible to 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. The new number must also have a unique decimal representation, that is, it cannot end in [[0.999...|repeating nines]] or repeating zeros. For example, if the digit isn't 2, make the digit of the new number 2, and if it was 2, make it 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://web.archive.org/web/20220122000000/https://link.springer.com/book/10.1007/978-1-4419-7127-2 |archive-date=2022-01-22}} [https://archive.org/details/proofsfundamenta0002bloc/ Alt URL]</ref><ref>{{Harvard citation no brackets|Abbott|2015|p=33}} {{br}}{{Harvard citation no brackets|Kleene|1952|p=|pp=6–7}}{{br}}{{Harvard citation no brackets|Pinter|2014|loc=Page 3 of Chapter 7}} {{br}}{{Harvard citation no brackets|Schumacher|1996|p=101}}</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><ref>{{Harvard citation no brackets|Aigner|Ziegler|2018|p=128}}{{br}}{{Harvard citation no brackets|Hrbáček|Jech|2017|pp=90–91}}</ref>

[[File:Diagonal argument powerset svg.svg|thumb|255x255px|<math>\N</math> does not have the same cardinality as its [[Power set|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>.|class=skin-invert-image]]Another classical example of an uncountable set, established using a related reasoning, is the [[Power set|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|empty set]] and <math>\N</math> itself. The method is much closer to Cantor's original diagonal argument. Again, assume by contradiction that there exists a one-to-one correspondence <math>f</math> between <math>\N</math> and <math>\mathcal{P}(\N)</math>, so that every subset of <math>\N</math> is assigned to some natural number. These subsets are then placed in a column, in the order defined by <math>f</math> (see image). Now, one may define a subset <math>T</math> of <math>\N</math> which is not in the list by taking the negation of the "diagonal" of this column as follows:<ref name=":12">{{Harvard citation no brackets|Halmos|1998|p=93}} {{br}}{{Harvard citation no brackets|Hrbáček|Jech|2017|p=91}}</ref>

If <math>1 \in f(1)</math>, then <math>1 \notin T</math>, that is, if 1 is in the first subset of the list, then 1 is ''not'' in the subset <math>T</math>. Further, if <math>2 \notin f(2) </math>, then <math>2 \in T</math>, that is if the number 2 is not in the second subset of the column, then 2 ''is'' in the subset <math>T</math>. Then in general, for each natural number <math>n</math>, <math>n \in T</math> if and only if <math>n \notin f(n) </math>, meaning <math>n</math> is put in the subset <math>T</math> only if the nth subset in the column does not contain the number <math>n</math>. Then, for each natural number <math>n</math>, <math>T \neq f(n)</math>, meaning, <math>T</math> is not the nth subset in the list, for any number <math>n</math>, and so it cannot appear anywhere in the list defined by <math>f</math>. Since <math>f</math> was chosen arbitrarily, this shows that every function from <math>\N</math> to <math>\mathcal{P}(\N)</math> must be missing at least one element, therefore no such bijection can exist, and so <math>\mathcal{P}(\N)</math> must be not be countable.<ref name=":12" />

These 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) called the [[Cardinality#Cardinality of the continuum|cardinality of the continuum]].<ref>{{Harvard citation no brackets|Hrbáček|Jech|2017|p=91}}{{br}}{{Harvard citation no brackets|Enderton|1977|p=149}}</ref>

[[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>, assume by contradiction that there is a bijection <math>f</math> from <math>A</math> to <math>\mathcal{P}(A)</math>. Then, the subset <math>T \subseteq A</math> given by taking the negation of the "diagonal", formally, <math>T = \{ a \in A : a \notin f(a) \}</math>, cannot be in the list. Therefore, every function is missing at least one element, and so <math>|A| < |\mathcal{P}(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.<ref>{{Harvard citation no brackets|Abbott|2015|pp=34–35}}{{br}}{{Harvard citation no brackets|Lévy|1979|p=87}} {{br}}{{Harvard citation no brackets|Stoll|1963|p=86}} {{br}}{{Harvard citation no brackets|Tao|2022|p=171}}</ref>

==Cardinal numbers==
In the above sections, "the cardinality of a set" was described relationally. In other words, one set could be compared to another, intuitively comparing their "size". [[Cardinal number]]s are a means of measuring this "size" more explicitly. For finite sets, this is simply the [[Natural number|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|vertical bar]] on each side,<ref>{{Harvard citation no brackets|Hrbáček|Jech|2017|p=65}} {{br}}{{Harvard citation no brackets|Lévy|1979|p=83}}{{br}}{{Harvard citation no brackets|Jech|2003|p=27}}</ref> though it may also be denoted by  <span style="border-top: 3px double;"><math>A</math></span>, <math>\operatorname{card}(A),</math> or <math>\#A.</math>{{refn|1= <span style="border-top: 3px double;"><math>A</math></span>{{sfn|Kleene|1952|p=9}}{{Sfn|Krivine|1971|p=23}}{{Sfn|Kuratowski|1968|p=174}}{{Sfn|Stoll|1963|p=80}}{{Sfn|Suppes|1972|p=109}}{{Sfn|Takeuti|Zaring|1982|p=85}}{{br}}<math>\operatorname{card}(A)</math>{{Sfn|Bourbaki|1968|p=158}}{{Sfn|Enderton|1977|p=136}}{{Sfn|Halmos|1998|p=94}}{{Sfn|Schumacher|1996|p=103}} {{br}}<math>\#A</math>{{Sfn|Halmos|1998|p=53}}{{Sfn|Pinter|2014|loc=Page 3 of Chaper 8}}{{Sfn|Tao|2022|p=60}}}}

For infinite sets, "cardinal number" is somewhat more difficult to define formally. Cardinal numbers are not usually thought of in terms of their formal definition, but immaterially in terms of their arithmetic/algebraic properties.<ref>{{Harvard citation no brackets|Kleene|1952|p=9}}{{br}}{{Harvard citation no brackets|Stoll|1963|p=80}}</ref> The assumption that there is ''some'' [[Cardinal function|cardinal function]] <math>A \mapsto |A|</math> which satisfies <math>A \sim B \iff |A| = |B|</math>, sometimes called the ''axiom of cardinality''<ref>{{Harvard citation no brackets|Pinter|2014|loc=Page 2 of Chapter 8}} {{br}}{{Harvard citation no brackets|Suppes|1972|p=111}}</ref> or ''[[Hume's principle]]'',<ref>{{Cite book |last=Potter |first=Michael |url=https://books.google.com/books?id=FxRoPuPbGgUC |title=Set Theory and its Philosophy: A Critical Introduction |date=2004-01-15 |publisher=Clarendon Press |isbn=978-0-19-155643-2 |language=en}}</ref> is sufficient for deriving most properties of cardinal numbers.<ref>{{Harvard citation no brackets|Enderton|1977|p=136}}{{br}}{{Harvard citation no brackets|Halmos|1998|p=94}}{{br}}{{Harvard citation no brackets|Hrbáček|Jech|2017|p=68}}</ref>

Commonly in mathematics, if a relation satisfies the properties of an [[Equivalence relation|equivalence relation]], the objects used to materialize this relation are equivalence classes, which groups all the objects equivalent to one another. These called the ''Frege{{En dash}}Russell'' cardinal numbers.<ref name="Kleene 1952 44">{{Harvard citation no brackets|Kleene|1952|p=44}} {{Break}}{{Harvard citation no brackets|Lévy|1979|p=84}} {{Break}}{{Harvard citation no brackets|Suppes|1972|p=109}}</ref> However, this would mean that cardinal numbers are too large to form sets (apart from the cardinal number <math>\bold 0</math> whose only element is the [[Empty set|empty set]]), since, for example, the cardinal number <math>\bold 1</math> would be the set of all sets with one element, then <math>\{ \bold 1 \} \in \bold 1</math>, and would therefore contain itself, violating [[Axiom of regularity|regularity]].<ref>{{Harvard citation no brackets|Enderton|1977|p=137}}</ref> Thus, due to [[Biography:John von Neumann|John von Neumann]], it is more common to assign representatives of these classes.<ref>{{Harvard citation no brackets|Lévy|1979|p=84}}{{br}}{{Harvard citation no brackets|Stoll|1963|p=80}}</ref>

=== Finite sets ===
{{main|Finite set|Combinatorial principles}}

[[File:Bijection.svg|thumb|224x224px|A [[bijective function]], <math>f: X \to Y</math> from the set {{nowrap|X {{=}} {1,2,3,4}}} to the set Y demonstrates that Y has cardinality 4.|class=skin-invert-image]]
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> analogous to [[Counting|counting]] its elements.<ref name=":02">{{Harvard citation no brackets|Tao|2022|p=58}}</ref><ref name=":13">{{Cite book |last1=Hashisaki |first1=Joseph |url=https://archive.org/details/theoryofarithmet0000john |title=Theory of Arithmetic |last2=Peterson |first2=John |publisher=[[Company:John Wiley & Sons|John Wiley & Sons]] |year=1963 |location=New York |lccn=63-11445}}</ref> 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". In formal contexts, the natural numbers can be understood as some construction of objects satisfying the [[Peano axioms]].<ref>{{Harvard citation no brackets|Halmos|1998|pp=46–49}} {{br}}{{Harvard citation no brackets|Hrbáček|Jech|2017|p=39}}{{br}}{{Harvard citation no brackets|Tao|2022|p=12}}</ref>

Showing that such a correspondence exists is not always trivial. [[Combinatorics]] is the area of mathematics primarily concerned with [[Counting|counting]], both as a means and as an end to obtaining results, and certain properties of finite structures.<ref>{{Harvard citation no brackets|Brualdi|2004|pp=1–4}}</ref> The notion cardinality of finite sets is closely tied to many basic [[Combinatorial principles|combinatorial principles]], and provides a set-theoretic foundation to prove them. It can be shown by induction on the possible sizes of sets that finite cardinality corresponds uniquely with natural numbers (cf. ''{{section link|Finite set|Uniqueness of cardinality}}'').<ref>{{Harvard citation no brackets|Halmos|1998|pp=52–53}}{{br}}{{Harvard citation no brackets|Hrbáček|Jech|2017|p=70}} {{br}}{{Harvard citation no brackets|Tao|2022|p=59}}</ref> This is related{{em dash}}though not equivalent{{em dash}}to the [[Pigeonhole principle|pigeonhole principle]].<ref>{{Harvard citation no brackets|Enderton|1977|pp=134–136}} {{br}}{{Harvard citation no brackets|Kuratowski|1968|p=102}}</ref>

The [[Addition principle|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 the parts is equal to the whole.<ref>{{Harvard citation no brackets|Brualdi|2004|p=45}}</ref> 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.<ref>{{Harvard citation no brackets|Brualdi|2004|p=46}}</ref> Both of these can be proven by a [[Bijective proof|bijective proof]], together with induction.<ref>{{Harvard citation no brackets|Hrbáček|Jech|2017|pp=71–72}}</ref> The more general result is the [[Inclusion–exclusion principle|inclusion–exclusion principle]], which defines how to count the number of elements in overlapping sets.{{Sfn|Brualdi|2004|p=160}}

Naturally, a set is defined to be finite if it is [[Empty set|empty]] or can be put in correspondence with the set <math>\{1,\,2,\, \dots, \, n\},</math> for some natural number <math>n.</math><ref name=":02" /><ref name=":13" /> However, there exist [[Finite set#Necessary and sufficient conditions for finiteness|other definitions of "finite"]] which don't rely on a definition of "number." For example, a set is called Dedekind-finite if it cannot be put in one-to-one correspondence with a proper subset of itself.<ref>{{Harvard citation no brackets|Kuratowski|1968|p=103}} {{br}}{{Harvard citation no brackets|Lévy|1979|pp=79–80}} {{br}}{{Harvard citation no brackets|Suppes|1972|p=99}}</ref>

=== Aleph numbers ===
[[File:Aleph0.svg|right|thumb|208x208px|[[Aleph-nought]], aleph-zero, or aleph-null: the smallest infinite cardinal number, and the cardinal number of the set of natural numbers.<ref name=":7" />]]
The aleph numbers are a sequence of cardinal numbers that represent the sizes of infinite sets, denoted with an [[Aleph|aleph]] <math>\aleph,</math> the first letter of the [[Social:Hebrew alphabet|Hebrew alphabet]].<ref name=":7">{{Harvard citation no brackets|Enderton|1977|p=137}}{{br}}{{Harvard citation no brackets|Halmos|1998|p=101}}{{br}}{{Harvard citation no brackets|Suppes|1972|p=156}}</ref> 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, then <math>\aleph_2</math>, and so on.<ref>{{Harvard citation no brackets|Enderton|1977|pp=212–213}}{{br}}{{Harvard citation no brackets|Jech|2003|pp=29–30}}</ref> The most common way this is formalized in set theory is through Von Neumann ordinals, known as [[Von Neumann cardinal assignment]].<ref>{{Cite book |last=Moschovakis |first=Yiannis N. |url=http://archive.org/details/notesonsettheory0000mosc |title=Notes on Set Theory |date=1994 |publisher=[[Physics:Springer-Verlag|Springer-Verlag]] |isbn=978-0-387-94180-6 |series=[[Undergraduate Texts in Mathematics]] |location=New York |lccn=93-35825}}</ref>

[[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.<ref>{{Harvard citation no brackets|Enderton|1977|pp=8–9}}{{br}}{{Harvard citation no brackets|Halmos|1998|pp=74–76}}</ref> More formally, these ordinal numbers can be defined as follows:

<math>0 := \{\},</math> the [[Empty set|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 example, <math>2 \in \{0,1,2\} = 3,</math> therefore <math>2 < 3.</math>  Defining <math>\omega := \{0,1,2,3,\cdots\}</math> 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.<ref>{{Harvard citation no brackets|Halmos|1998|pp=74–80}}{{br}}{{Harvard citation no brackets|Lévy|1979|p=52}}</ref>

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>\alpha</math> with cardinality <math>|\alpha| \leq \aleph_0</math>), the [[First uncountable ordinal|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><ref>{{Harvard citation no brackets|Halmos|1998|pp=100–102}}</ref>  Furthermore, <math>\aleph_2</math> is the set of all ordinals with cardinality less than or equal to <math>\aleph_1,</math> and in general the [[Successor cardinal|successor cardinal]] <math>\kappa^+</math>is the set of all ordinals with cardinality up to <math>\kappa</math>. Put another way for infinite cardinals, <math>\kappa^+</math> is the number of possible [[Well-order|well-order]]ings on <math>\kappa</math> up to [[Order isomorphism|order isomorphism]].<ref>{{Harvard citation no brackets|Hrbáček|Jech|2017|pp=130–131}}{{br}}{{Harvard citation no brackets|Kuratowski|1968|p=277}}{{br}}{{Harvard citation no brackets|Suppes|1972|pp=128–129}}</ref> Proving that such a set always exists is known as Hartogs' theorem, wherein the smallest ordinal not less than or equal to than a set <math>\kappa</math> is called the [[Hartogs number]] of <math>\kappa</math>.<ref>{{Harvard citation no brackets|Hrbáček|Jech|2017|p=130}}{{br}}{{Harvard citation no brackets|Lévy|1979|p=89}}{{br}}{{Harvard citation no brackets|Suppes|1972|p=226}}</ref> Then, <math>\aleph_\lambda </math> for a limit ordinal <math>\lambda</math> is the [[Union (set theory)|union]] of all lesser alephs.<ref>{{Harvard citation no brackets|Enderton|1977|p=214}}{{br}}{{Harvard citation no brackets|Jech|2003|pp=29–30}}</ref>

The importance of ordinal numbers here is to generalize the notion of counting to infinite sets. When counting, one implicitly assigns an order to their set of objects, but no matter what order one assigns the final result of the count is always the same, which demonstrates the connection between cardinal and ordinal numbers.<ref>{{Harvard citation no brackets|Enderton|1977|p=|pp=136,197}}{{br}}{{Harvard citation no brackets|Halmos|1998|pp=75,86}}</ref> Further, by the [[Well-ordering theorem|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 uniquely to some aleph <math>\aleph_\alpha,</math> for some ordinal <math>\alpha.</math><ref>{{Harvard citation no brackets|Enderton|1977|p=197}}{{br}}{{Harvard citation no brackets|Hrbáček|Jech|2017|p=170}}{{br}}{{Harvard citation no brackets|Suppes|1972|p=236}}</ref> This allows one to use a constructive definition of the cardinality function, by assigning each set to its equinumerous aleph.<ref>{{Harvard citation no brackets|Lévy|1979|pp=|p=83}}{{br}}{{Harvard citation no brackets|Enderton|1977|pp=195–198}}</ref>

=== Cardinal arithmetic ===
[[File:AdditionShapes.svg|right|thumb|210x210px|The addition of the objects from the two disjoint sets is an instance of <math> 3 + 2 = 5 </math>.]]
Basic arithmetic can be done on cardinal numbers in a very natural way, by extending the theorems for finite combinatorial principles above. The intuitive principle that is <math>A</math> and <math>B</math> are disjoint then addition of these sets is simply taking their [[Union (set theory)|union]], written as <math>|A \cup B| = |A| + |B|</math>.<ref>{{Harvard citation no brackets|Brualdi|2004|p=28}}{{br}}{{Harvard citation no brackets|Enderton|1977|p=139}}{{br}}{{Harvard citation no brackets|Halmos|1998|p=95}}{{br}}{{Harvard citation no brackets|Hrbáček|Jech|2017|p=93}}</ref> Thus if <math>A</math> and <math>B</math> are infinite, cardinal addition is defined as <math>|A| + |B| := |A \sqcup B|</math> where <math>\sqcup</math> denotes [[Disjoint union|disjoint union]]. Similarly, the multiplication of two sets is intuitively the number of ways to pair their elements (as in the multiplication principle), therefore cardinal multiplication is defined as <math>|A| \cdot |B| := |A \times B| </math>, where <math>\times</math> denotes the [[Cartesian product]].<ref>{{Harvard citation no brackets|Brualdi|2004|pp=28–30}}{{br}}{{Harvard citation no brackets|Enderton|1977|p=139}}{{br}}{{Harvard citation no brackets|Halmos|1998|p=95}}{{br}}{{Harvard citation no brackets|Hrbáček|Jech|2017|p=94}}</ref> These definitions can be shown to satisfy the basic properties of standard arithmetic:<ref>{{Harvard citation no brackets|Bourbaki|1968|p=162}}{{br}}{{Harvard citation no brackets|Hrbáček|Jech|2017|p=94}}{{br}}{{Harvard citation no brackets|Krivine|1971|p=26}}{{br}}{{Harvard citation no brackets|Kuratowski|1968|pp=183–184, 198–202}}</ref>

* [[Associativity]]: <math>|A| + |B \sqcup C| = |A \sqcup B| + |C| </math>, and <math>|A| \cdot |B \times C| = |A \times B| \cdot |C|</math>
* [[Commutativity]]: <math>|A| + |B| = |B| + |A|</math>, and <math>|A| \cdot |B| = |B| \cdot |A|</math>
* [[Distributivity]]: <math>|A| \cdot |B \sqcup C| = |A \times B| + |A \times C|</math>

Cardinal exponentiation <math>|A|^{|B|}</math> is defined via set exponentiation, the set of all functions <math>f:B \mapsto A</math>, that is, <math>|A|^{|B|} := |A^B|.</math><ref>{{Harvard citation no brackets|Hrbáček|Jech|2017|p=95}}{{br}}{{Harvard citation no brackets|Lévy|1979|p=98}}</ref> For finite sets this can be shown to coincide with standard [[Exponentiation#Integer exponents|natural number exponentiation]], but includes as a corollary that [[Zero to the power of zero|zero to the power of zero]] is one <math>(0^0 = 1)</math> since there is exactly one function from the [[Empty set|empty set]] to itself: the empty function.<ref>{{harvnb|Bourbaki|1968|p=164}}{{br}}{{harvnb|Enderton|1977|p=141}}{{br}}{{Harvard citation no brackets|Hrbáček|Jech|2017|p=97}}{{br}}{{Harvard citation no brackets|Lévy|1979|p=98}}</ref> A combinatorial argument can be used to show <math>2^{|A|} = |\mathcal{P}(A)|</math>, by considering the [[Indicator function|indicator function]] of each subset. In general, cardinal exponentiation is not as well-behaved as cardinal addition and multiplication. For example, even though it can be proven that the expression <math>2^{\aleph_0}</math>does indeed correspond to some aleph, it is unprovable from standard set theories which aleph it corresponds to.<ref>{{Harvard citation no brackets|Hrbáček|Jech|2017|p=101}}{{br}}{{Harvard citation no brackets|Lévy|1979|p=99}}</ref> 
{| class="wikitable"
|+Basic results
!Exponentiation
!Inequality and Monotonicity
![[Identity element]]s
![[Absorption law]]s (for infinite A,B)
|-
|<math>\big(|A|^{|B|} \big)^{|C|} = |A|^{|B \times C|}</math> {{Efn|cf. ''[[Currying]]''}}<ref>{{harvnb|Enderton|1977|p=142}}{{br}}{{Harvard citation no brackets|Hrbáček|Jech|2017|p=95}}{{br}}{{Harvard citation no brackets|Kuratowski|1968|p=185}}</ref>
|<math>|A| \leq |B|</math> implies  <math>|A|+|C| \leq |B| + |C|</math><ref>{{Harvard citation no brackets|Kuratowski|1968|p=187}}{{br}}{{harvnb|Lévy|1979|p=93}}</ref>
|<math>|A| + \bold 0 = |A|</math><ref>{{Harvard citation no brackets|Lévy|1979|p=93}}{{br}}{{harvnb|Enderton|1977|p=143}}</ref>
|<math>|A|+|B| = \operatorname{max}(|A|,|B|)</math><ref name="Bourbaki 1968 188">{{harvnb|Bourbaki|1968|p=188}}{{br}}{{harvnb|Enderton|1977|p=164}}</ref>
|-
|<math>|A|^{|B| + |C|} = |A|^{|B|} \cdot |A|^{|C|}</math><ref name="Hrbáček 2017 95">{{Harvard citation no brackets|Hrbáček|Jech|2017|p=95}}{{br}}{{Harvard citation no brackets|Krivine|1971|p=27}}{{br}}{{Harvard citation no brackets|Kuratowski|1968|p=185}}</ref>
|<math>|A| \leq |B|</math> implies  <math>|A| \cdot |C| \leq |B| \cdot |C|</math><ref>{{Harvard citation no brackets|Kuratowski|1968|p=187}}{{br}}{{harvnb|Lévy|1979|p=95}}</ref>
|<math>|A| \cdot \bold 1 = |A|</math><ref>{{Harvard citation no brackets|Hrbáček|Jech|2017|p=94}}{{br}}{{Harvard citation no brackets|Krivine|1971|p=26}}</ref>
|<math>|A|\cdot|B| = \operatorname{max}(|A|,|B|)</math><ref name="Bourbaki 1968 188"/>
|-
|<math>|A \times B|^{|C|} = |A|^{|C|} \cdot |B|^{|C|}</math><ref name="Hrbáček 2017 95"/>
|<math>|A| \leq |B|</math> implies  <math>|A|^{|C|} \leq |B|^{|C|}</math><ref name="Enderton 1977 149">{{harvnb|Enderton|1977|p=149}}{{br}}{{Harvard citation no brackets|Kuratowski|1968|p=187}}</ref>
|<math>|A|^{\bold 1} = |A|</math><ref>{{Harvard citation no brackets|Hrbáček|Jech|2017|p=97}}{{br}}{{Harvard citation no brackets|Kuratowski|1968|p=185}}</ref>
|<math>\bold 0 \cdot |A|  = \bold 0</math> ([[Absorbing element|annihilator]])<ref>{{harvnb|Enderton|1977|p=143}}{{br}}{{Harvard citation no brackets|Lévy|1979|p=95}}</ref>
|-
|<math>|A|^\bold 0 = \bold 1</math><ref>{{harvnb|Bourbaki|1968|p=164}}{{br}}{{harvnb|Enderton|1977|p=143}}</ref>
|<math>|A| \leq |B|</math> and <math>\bold 0 < |C|</math> implies <math>|C|^{|A|} \leq |C|^{|B|}</math><ref name="Enderton 1977 149"/>
|
|<math>\bold 1^{|A|} = \bold 1</math><ref>{{Harvard citation no brackets|Kuratowski|1968|p=185}}{{br}}{{Harvard citation no brackets|Lévy|1979|p=98}}</ref>
|}

=== Set of all cardinal numbers ===
The ''set of all cardinal numbers'' refers to a hypothetical set which contains all cardinal numbers. Such a set cannot exist, which has been considered paradoxical, and related to the [[Burali-Forti paradox]]. Using the definition of cardinal numbers as representatives of their cardinalities.  It begins by assuming there is some set <math>S := \{ X \, | X \text{ is a cardinal number}\}.</math> Then, if there is some largest cardinal <math>\kappa \in S ,</math> then the powerset <math>2^\kappa</math> is strictly greater, and thus not in <math>S.</math> Conversely, 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> Thus, the collection of all cardinal numbers is too large to form a set, and is a [[Cardinality#Proper classes|proper class]].<ref>{{harvnb|Bourbaki|1968|pp=165,328}}{{br}}{{Harvard citation no brackets|Hrbáček|Jech|2017|p=97}}{{br}}{{Harvard citation no brackets|Kuratowski|1968|pp=179–181}}</ref>

== Cardinality of the continuum ==
{{main|Cardinality of the continuum|Continuum hypothesis}}

[[File:Number line.png|thumb|The [[Number line|number line]], containing all points in its continuum.|314x314px|class=skin-invert-image]]
The [[Number line|number line]] is a geometric construct of the intuitive notions of "[[Space|space]]" and "[[Distance|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 iterations approaching the Cantor set]]
The [[Cardinality of the continuum|cardinality of the continuum]], denoted by "<math>\mathfrak c</math>" (a lowercase 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 curve]].|class=skin-invert-image]]
Space-filling curves are continuous surjective maps from the [[Unit interval|unit interval]] <math>[0,1]</math> onto the [[Unit square|unit square]] on <math>\R^2</math>, with classical examples such as the [[Peano curve]] and [[Hilbert 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|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 {{section link||Uncountable 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 [[Biography:Kurt Gödel|Gödel]] and [[Biography:Paul Cohen|Cohen]], the continuum hypothesis is [[Independence (mathematical logic)|independent]] of 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]], meaning it produces no contradictions.<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>. Research on CH and GCH continues independent of ZFC, especially in [[Descriptive set theory|descriptive set theory]] and through the exploration of [[Cardinality#Large cardinals|large cardinal axioms]].<ref>{{Harvard citation no brackets|Heller|Woodin|2011|pp=2–3}} {{br}}{{Harvard citation no brackets|Kanamori|2003|p=XV}}</ref> Without GHC, the cardinality of <math>\R</math> cannot be written in terms of specific alephs. The Beth numbers provide a concise notation for powersets of the real numbers starting from <math>\beth_0 = |\N|</math>, then <math>\beth_1 = 2^{\beth_0} =|\R|</math>, and <math>\beth_2 = |\mathcal{P}(\R)| = 2^{\beth_1}</math>, and in general <math>\beth_{n+1} = 2^{\beth_n}</math> and <math>\beth_{\lambda} = \bigcup_{\alpha<\lambda} \beth_\alpha</math> if <math>\lambda</math> is a [[Limit ordinal|limit ordinal]].

== 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.|267x267px]]
In [[Model theory|model theory]], a model corresponds to a specific interpretation of a [[Formal language|formal language]] or [[Theory (mathematical logic)|theory]]. It consists of a [[Philosophy:Domain of discourse|domain]] (a set of objects) and an [[Philosophy:Interpretation (logic)|interpretation]] of the symbols in the language, such that the axioms of the theory are satisfied within this structure. In [[First-order logic|first-order logic]], the [[Löwenheim–Skolem theorem]] states that if a theory has an infinite model, then it also has models of every other infinite cardinality. Applied to set theory, it asserts that [[Zermelo–Fraenkel set theory]], which proves the existence of uncountable sets such as <math>\mathcal P(\N)</math>, nevertheless has a countable model. Thus, [[Skolem's paradox]] was posed as follows: how can it be that there exists a domain of set theory which only contains countably many objects, but is capable of [[Satisfiability|satisfying]] the statement "there exists a set with uncountably many elements"?<ref name=":62">{{Citation |last=Bays |first=Timothy |title=Skolem's Paradox |date=2025 |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 |encyclopedia=The Stanford Encyclopedia of Philosophy}}</ref>

Skolem's paradox does not only apply to ZFC, but ''any'' first-order set theory, if it is consistent, has a model which is countable. A mathematical explanation of the paradox, showing that it is not a true contradiction in mathematics, was first given in 1922 by [[Biography:Thoralf Skolem|Thoralf Skolem]]. He explained that the countability or uncountability of a set is not [[Philosophy:Absoluteness (logic)|absolute]], but relative to the model in which the cardinality is measured. This is because, for example, if the set <math>X</math> is countable in a model of set theory then there is a bijection <math>f : \N \mapsto X.</math> But a submodel containing <math>X</math> which excludes all such functions would thus contain no bijection between <math>X</math> and <math>\N</math>, and therefore <math>X</math> would be uncountable. In [[Philosophy:Second-order logic|second-order]] and higher-order logics, the Löwenheim–Skolem theorem does not hold. This is due to the fact that second-order logic quantifies over all subsets of the domain. Skolem's work was harshly received by [[Biography:Ernst Zermelo|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 |last2=Ebbinghaus |first2=Heinz-Dieter |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 |hdl=1874/27769 |jstor=421203 |s2cid=8530810 |number=2}}</ref><ref name=":62" /><ref>{{harvnb|Bourbaki|1968|pp=340–341}}</ref>

== Alternative and additional axioms ==
Around the turn of the 20th century, set theory turned to an axiomatic approach to avoid rampant foundational issues related to its naive study (cf. ''{{section link||Axiomatic set theory}}''). The most common axiomatic set theory used today is [[Zermelo–Fraenkel set theory]] (ZFC).<ref name=":14">{{Harvard citation no brackets|Ferreirós|2007|loc=Chapter XI: Consolidation of Axiomatic Set Theory}}</ref> In this system, the relevant axioms include: the Axiom of Infinity, stating roughly "there is an infinite set", specifically, a set with cardinality of the natural numbers <math>\N</math>; the [[Axiom of power set]], which says that, for any set <math>A</math>, the powerset <math>\mathcal P(A)</math> also exists; and the Axiom of Choice, explained below. ZFC has been criticized both for being too strong, and too weak. Similarly, there are many "natural extensions" of ZFC studied by set theorists. Thus, there exist many alternative systems of axioms each of which has implications to the standard theory of cardinality discussed above.

=== Without the axiom of choice ===
[[File:Axiome_du_choix.png|thumb|270x270px|Illustration of the axiom of choice, with each set represented as a jar and its elements represented as marbles.|class=skin-invert-image]]
The Axiom of Choice (AC) is a fundamental principle in the foundations of mathematics which has seen much controversy in its history. Roughly, it states that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. In many cases, a set created by choosing elements can be made without invoking the axiom of choice. But it is not possible to do this in general, in which case, the axiom of choice must be invoked.

If the Axiom of Choice is assumed to be false in ZF, it has several implications:

* Cardinal inequality <math>(\preceq)</math> cannot be a total order on all sets. That is, given any two sets <math>A, B,</math> it may be that neither <math>A \preceq B</math> nor <math>B \preceq A</math> hold. Further, since Alephs can naturally be compared, there exist sets which do not correspond to any Aleph number.
* There may exist sets which are Dedekind-finite, meaning they cannot be put in bijection with a proper subset of itself, but whose elements cannot be counted (put in bijection with <math>\{ 1, 2, \cdots , n \}</math> for any n).
* It is impossible to prove there exists a [[Cardinal function|cardinal function]] <math>S \mapsto |S|</math> which satisfies the [[Cardinality#Cardinal numbers|cardinal number axiom]], and satisfies <math>|S| \sim S</math> for all <math>S.</math><ref>{{Harvard citation no brackets|Lévy|1979|p=84}}</ref> Thus, some authors reserve <math>\operatorname{card}(S)</math> for the cardinality of well-orderable sets. Still, the function can be defined using [[Scott's trick]], which uses the Axiom of Regularity to assign each set <math>S</math> to the least rank <math>V_{\alpha}</math> of the Von Neumann universe which contains a set equinumerous to <math>S.</math>
* One may define the "infinite product" of cardinal numbers as the cardinality of the set of all sequences of their elements. Asserting that this set is always nonempty is equivalent to the axiom of choice. This is why Bertrand Russell called AC the ''Multiplicative axiom''.
* Although the Generalized Continuum Hypothesis (GCH) is independent of ZFC, it can be shown that ZF+GCH is sufficient to prove AC. Thus, if AC is false, it follows that the GCH does not hold.

=== Proper classes ===
[[File:Frank-Rühl_sample-Tav.svg|thumb|212x212px|alt=ת|The [[Hebrew alphabet|Hebrew]] letter Tav, denoting the size of the "[[Philosophy:Absolute infinite|Absolute infinite]]"]]
Some set theories allow for proper classes, which are, roughly, collections "too large" to form sets. For example, the [[Universe (mathematics)|Universe of all sets]], the [[Cardinality#Set of all cardinal numbers|class of all cardinal numbers]], or the [[Burali-Forti paradox|class of all ordinal numbers]]. Such set theories include [[Von Neumann–Bernays–Gödel set theory]], and [[Morse–Kelley set theory]]. In such set theories, some authors find this definition more elegant than assigning representatives, as it more accurately describes the concept by "definition by abstraction".

Proper classes can, in a sense, be assigned cardinalities. Cantor originally called the cardinality of a proper class "[[Philosophy:Absolute infinite|Absolute infinite]]", denoted with tav '''{{large|{{script|Hebr|ת}}}}''' (the last letter of the [[Social:Hebrew alphabet|Hebrew alphabet]]), and associated the concept with [[Religion:God|God]].<ref>{{Harvard citation no brackets|Heller|Woodin|2011|pp=40–41}}</ref> The first to distinguish between sets and classes was [[Biography:John von Neumann|John von Neumann]], who formalized the notion of "too large to form sets". More accurately, he defined a class to be a proper class if and only if it was equinumerous with the whole Universe of sets (cf. ''[[Axiom of limitation of size]]''). Thus, all proper classes have the same "size". The axiom has several implications, mostly relating to the [[Limitation of size|limitation of size]] principles of early set theory. It implies the axiom of specification, the axiom of replacement, the [[Axiom of union|axiom of union]], and the [[Axiom of global choice|axiom of global choice]].

=== Large cardinals ===
[[Large cardinal|Large cardinal axioms]] assert the existence of cardinal numbers which, as the name suggests, are very large{{mdash}}so large that they cannot be proven to exist within ZFC. For example, an [[Inaccessible cardinal|inaccessible cardinal]] is, roughly, a cardinal that cannot be approached from below using basic set-theoretic operations such as unions, limits, and powersets (more formally, a [[Regular cardinal|regular]], [[Limit cardinal|limit cardinal]] greater than <math>\aleph_0</math>). Large cardinals are understood in terms of the Von Neumann hierarchy, denoted <math>V_\alpha</math> (for some ordinal <math>\alpha</math>), which can be roughly understood as the sets that can be obtained from the empty set, followed by recursively applying the powerset <math>\alpha</math> times. Specifically, <math>V_0 = \varnothing</math>, <math>V_{\alpha+1} = \mathcal P(V_\alpha)</math> and <math>V_\lambda = \bigcup_{\alpha < \lambda} V_\alpha</math> for a limit ordinal <math>\lambda</math>.<ref>{{Harvard citation no brackets|Heller|Woodin|2011|p=89}}</ref>

There exist [[List of large cardinal properties|many known properties defining large cardinals]], which come roughly in a linear hierarchy, in terms of consistency strength. As an analogy, in ZFC without the Axiom of Infinity can only prove the existence of finite sets. Therefore <math>V_\omega ( = \mathcal{P}(\varnothing) \cup \mathcal{P}(\mathcal{P}(\varnothing)) \cup \cdots )</math>, whose existence is provable in usual ZFC, can serve as a model of ZFC{{en dash}}Infinity, and thus if ZFC is consistent, ZFC{{en dash}}Infinity is consistent.<ref>{{Harvard citation no brackets|Heller|Woodin|2011|pp=90–94|loc=Chapter 4.2: The Realm of the Finite}}</ref> Analogously, ZFC{{math|+}}"There exists an inaccessible cardinal" implies the consistency of ZFC, since if <math>\kappa</math> is inaccessible, <math>V_\kappa</math> can serve as a model of ZFC (cf. ''[[Grothendieck universe]]''). Stronger and stronger large cardinal axioms assert the existence of larger and larger cardinals, each of which proves the consistency—or inconsistency—of the weaker systems.<ref>{{Harvard citation no brackets|Koellner|2011|loc=Section 3}} {{br}}{{Harvard citation no brackets|Heller|Woodin|2011}}</ref>

Large cardinals are at the forefront of set-theoretic research for both practical and philosophical reasons. In a practical sense, it is often the case that unproven or unprovable conjectures can be settled by sufficiently strong large cardinal axioms. For example, the existence of a [[Measurable cardinal|measurable cardinal]] is inconsistent with [[Axiom of constructibility|Gödel's constructibility axiom]]. Similarly, the existence of a [[Reinhardt cardinal]] is inconsistent with the axiom of choice, which makes it a much more controversial large cardinal axiom.

In a philosophical sense, according to a [[Philosophy:Mathematical Platonism|platonic]] view among set-theorists such as [[Biography:W. Hugh Woodin|W. Hugh Woodin]], these axioms simply extend the system to include sets that are "supposed" to be considered. That is, there is some fundamental [[Universe (mathematics)|universe of sets]], which these axioms grant further access to.<ref>{{Harvard citation no brackets|Heller|Woodin|2011|pp=3–4}}</ref> For this reason, large cardinal axioms are generally given preference compared to other possible axioms of set theory. This view is controversial among a competing philosophy, sometimes called [[Philosophy:Pluralism|pluralism]],<ref>{{Harvard citation no brackets|Koellner|2014}}</ref> which posits that set theory should be understood as a [[Multiverse (set theory)|multiverse]] of set theories, but no "absolute" or "true" model.<ref>{{Harvard citation no brackets|Heller|Woodin|2011|loc=Chapter 4.4: The Generic Multiverse of Sets}}</ref>

=== Constructability ===
The axiom of constructability was the axiom introduced by [[Biography:Kurt Gödel|Kurt Gödel]] to prove the relative consistency of the axiom of choice and generalized continuum hypothesis. It states that all sets are "constructable" (cf. ''{{section link|Constructible universe|What L is}}''). It has several consequences, naturally, including AC and GCH, but resolves [[Axiom of constructibility#Statements true in L|several other important questions in set theory]]. The axiom, however, is not well-accepted by set-theorists, most believing it to be "too restrictive". In part it is because the axiom is contradicted by sufficiently strong large cardinal axioms.

=== Determinacy ===
[[File:Banach-Tarski_Paradox.svg|right|thumb|350x350px|Illustration of the [[Banach–Tarski paradox]], which arises in ZFC, but is impossible in ZF+AD.]]
The Axiom of Determinacy (AD) asserts that [[Axiom of determinacy#Types of games that are determined|certain kinds]] of mathematical games on the natural numbers are [[Determinacy|determined]]; that is, one player will always have a guaranteed winning strategy.<ref>{{Harvard citation no brackets|Koellner|2014|loc=Section 3.1 Determinacy}}{{br}}{{Harvard citation no brackets|Kanamori|2003|pp=369–370}}</ref> The first serious study of the consequences of AD started during the 1960s in [[Descriptive set theory|descriptive set theory]]{{em dash}}which, roughly, studies the definable sets of the real numbers{{em dash}}after it was noticed that it leads to very nice regulatory properties of the real numbers.<ref>{{Harvard citation no brackets|Koellner|2014|loc=Section 2.1 Historical Overview}}{{br}}{{Harvard citation no brackets|Kanamori|2003|p=XXI}}</ref> Specifically, it implies the [[Perfect set property|perfect set property]], the [[Property of Baire|property of Baire]], and that all sets of real numbers are Lebesgue measurable.<ref>{{Harvard citation no brackets|Koellner|2014|loc=Section 2.2.2 Regularity Properties}}{{br}}{{Harvard citation no brackets|Kanamori|2003|p=XXI}}</ref> However, it was shown that this axiom is inconsistent with AC (cf. ''{{section link|Axiom of determinacy|Incompatibility with the axiom of choice}}''), and thus was never taken as a fundamental axiom of set theory.<ref>{{Harvard citation no brackets|Koellner|2014|loc=Section 2.1 Historical Overview}}</ref> However, its relationship to and consistency with large cardinals is still of heavy interest among set theorists such as [[Biography:Donald A. Martin|Donald A. Martin]], [[Biography:John R. Steel|John R. Steel]], and [[Biography:W. Hugh Woodin|W. Hugh Woodin]].<ref>{{Harvard citation no brackets|Kanamori|2003|p=XXII}}</ref>

Replacing the Axiom of Choice with the Axiom of Determinacy has several implications:

* The [[Banach–Tarski paradox]] is a paradox arising from AC which allows one to disassemble a ball into finitely many pieces, and re-arrange them into two balls identical to the original. This, and related paradoxes, become impossible under AD.<ref>{{Harvard citation no brackets|Koellner|2014|loc=Section 2.2.2 Regularity Properties}}</ref>
* It is possible to [[Partition of a set|partition]] the set of real numbers in such a way that there are ''strictly'' more nonempty sets of real numbers than there are real numbers. That is, if <math>P</math> denotes a general partition of <math>\R</math>, there is a partition such that <math>|P| > |\R|</math>.<ref>{{Cite journal |last1=Taylor |first1=Alan D. |last2=Wagon |first2=Stan |date=2019 |title=A Paradox Arising from the Elimination of a Paradox |url=https://www.jstor.org/stable/48661905 |journal=The American Mathematical Monthly |volume=126 |issue=4 |pages=306–318 |doi=10.1080/00029890.2019.1559416 |jstor=48661905 |issn=0002-9890}}</ref>

==History==

=== Ancient history ===
[[File:AristotlesWheelLabeledDiagram.svg|thumb|264x264px|Diagram of Aristotle's wheel as described in ''Mechanica''|class=skin-invert-image]]
From the 6th century BCE, the writings of Greek philosophers, such as [[Biography:Anaximander|Anaximander]], show hints of comparing infinite sets or shapes, however, it was generally viewed as paradoxical and imperfect (cf. ''[[Zeno's paradoxes]]'').<ref name="Allen22">{{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> [[Biography:Aristotle|Aristotle]] distinguished between the notions of [[Philosophy:Actual infinity|actual infinity]] and potential infinity, arguing that Greek mathematicians understood the difference, and that they "do not need the [actual] infinite and do not use it."<ref>{{cite book |last1=Allen |first1=Reginald E. |url=https://books.google.com/books?id=3bNw_OmGNwYC&pg=PA256 |title=Plato's Parmenides |publisher=Yale University Press |year=1998 |isbn=9780300138030 |series=The Dialogues of Plato |volume=4 |location=New Haven |page=256 |oclc=47008500}}</ref> The Greek notion of number (''αριθμός'', ''arithmos'') was used exclusively for a definite number of definite objects (i.e. finite numbers).<ref>{{Cite book |last=Klein |first=Jacob |url=http://archive.org/details/jacob-klein-greek-mathematical-thought-and-the-origin-of-algebra |title=Greek Mathematical Thought And The Origin Of Algebra |date=1992 |publisher=[[Social:Dover Publications|Dover Publications]] |isbn=0-486-27289-3 |location=New York |page=46 |translator-last=Brann |translator-first=Eva |lccn=92-20992 |orig-year=1934 |translator-link=Eva Brann}}</ref> This would be codified in [[Euclid's Elements|Euclid's ''Elements'']], where the [[Euclidean geometry#common notions|fifth common notion]] states "The whole is greater than the part", often called the ''Euclidean principle''. This principle would be the dominating philosophy in mathematics until the 19th century.<ref name="Allen22" /><ref>{{Cite book |last=Mayberry |first=John P. |url=http://archive.org/details/foundationsofmat0000john |title=Foundations of Mathematics in the Theory of Sets |date=2011 |publisher=Cambridge University Press |isbn=978-0-521-17271-4 |series=Encyclopedia of Mathematics and its Applications |issn=0953-4806}}</ref>

Around the 4th century BCE, Jaina mathematics would be the first to discuss different sizes of infinity. They defined three major classes of number: enumerable (finite numbers), unenumerable (''asamkhyata'', roughly, countably infinite), and infinite (''ananta''). Then they had five classes of infinite numbers: infinite in one direction, infinite in both directions, infinite in area, infinite everywhere, and infinite perpetually.<ref>{{Cite book |last=Joseph |first=George Gheverghese |url=https://press.princeton.edu/books/paperback/9780691135267/the-crest-of-the-peacock |title=The Crest of the Peacock: Non-European Roots of Mathematics |date=Oct 24, 2010 |publisher=Princeton University Press |isbn=978-0-691-13526-7 |edition=3rd |location=Princeton, New Jersey |pages=349–351 |archive-url=https://web.archive.org/web/20240805000000/https://press.princeton.edu/books/paperback/9780691135267/the-crest-of-the-peacock |archive-date=2024-08-05}} Alt URL</ref><ref>{{Cite web |last1=O'Connor |first1=John J. |last2=Robertson |first2=Edmund F. |author-link2=E. F. Robertson |date=2000 |title=MacTutor{{snd}}Jaina mathematics |url=https://mathshistory.st-andrews.ac.uk/HistTopics/Jaina_mathematics/ |access-date=2025-06-09 |website=[[MacTutor History of Mathematics Archive]] |language=en |via=[[Organization:University of St Andrews|University of St Andrews]], Scotland}}</ref>

One of the earliest explicit uses of a one-to-one correspondence is recorded in [[Engineering:Mechanics (Aristotle)|Aristotle's ''Mechanics'']] ({{Circa|350 BCE}}), 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|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.<ref>{{Cite book |last=Pickover |first=Clifford A. |url= |title=The Math Book: 250 Milestones in the History of Mathematics |title-link=The Math Book |date=2014 |publisher=[[Company:Barnes & Noble|Barnes & Noble]] |isbn=978-1-4351-4803-1 |location=New York |pages=54 |chapter=Aristotle's Wheel Paradox |chapter-url=https://archive.org/details/mathbook250miles0000pick/page/54/}}</ref><ref>{{Cite book |last=Darling |first=David |author-link=David J. Darling |url= |title=Universal Book of Mathematics |title-link=The Universal Book of Mathematics |date=2008 |publisher=Wiley & Sons |isbn=978-0-470-30788-5 |location=Hoboken |chapter=Aristotle's wheel |chapter-url=http://archive.org/details/isbn_9780470307885}}</ref><ref>{{Cite book |last=Farlow |first=Stanley J. |author-link=Stanley Farlow |url=http://archive.org/details/paradoxesinmathe0000farl |title=Paradoxes in Mathematics |date=2014 |publisher=[[Social:Dover Publications|Dover Publications]] |isbn=978-0-486-49716-7 |location=Mineola |pages=92–95}}</ref>

=== Pre-Cantorian set theory ===
{{Multiple image
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| total_width       = 370
| footer            = Portrait of [[Biography:Galileo Galilei|Galileo Galilei]], {{circa|1640}} (left).  Portrait of [[Biography:Bernard Bolzano|Bernard Bolzano]] 1781–1848 (right).
}}

[[Biography:Galileo Galilei|Galileo Galilei]] presented what was later coined [[Galileo's paradox]] in his book ''[[Physics:Two New Sciences|Two New Sciences]]'' (1638),<ref>{{Cite book |last=Galilei |first=Galileo |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> where he presents a seeming paradox in infinite sequences of numbers. It goes roughly as follows: for each [[Square number|perfect square]] <math>(n^2)</math> 1, 4, 9, 16, and so on, there is a unique [[Square root|square root]] <math display="inline">(\sqrt{n^2} = n)</math> 1, 2, 3, 4, and so on. Therefore, there are as many perfect squares as there are square roots. However, every number is a square root, since it can be [[Square (algebra)|squared]], but not every number is a perfect square. Moreover, the proportion of perfect squares as one passes larger values diminishes, and is eventually smaller than any given fraction. Galileo denied that this was fundamentally contradictory, however he concluded that this meant we could not compare the sizes of infinite sets, missing the opportunity to discover cardinality.<ref>{{Harvard citation no brackets|Bourbaki|1968|p=323}} {{Break}}{{Harvard citation no brackets|Enderton|1977|p=131}}. {{Break}}{{Harvard citation no brackets|Kleene|1952|p=3}} {{Break}}{{Harvard citation no brackets|Schumacher|1996|pp=92–93}}</ref>

In ''[[Philosophy:A Treatise of Human Nature|A Treatise of Human Nature]]'' (1739), [[Biography:David Hume|David Hume]] is quoted for saying ''"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 |title=A Treatise of Human Nature |date=1739–1740 |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 [[Biography:Gottlob Frege|Gottlob Frege]] later during the rise of set theory.<ref name=":1">{{cite book |last=Frege |first=Gottlob |title=Die Grundlagen der Arithmetik |date=1884 |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><ref name=":2">{{Cite book |last=Demopoulos |first=William |url=http://archive.org/details/fregesphilosophy0000unse |title=Frege's Philosophy of Mathematics |date=1997 |publisher=Harvard University Press |isbn=978-0-674-31943-1 |location=Cambridge |chapter=Introduction |lccn=94-34381}}</ref><ref name=":8">{{Cite book |last=Boolos |first=George |url=http://archive.org/details/philosophyofmath0000unse_c8j9 |title=The Philosophy of Mathematics |date=1996 |publisher=[[Organization:Oxford University Press|Oxford University Press]] |isbn=978-0-19-875119-9 |editor-last=Hart |editor-first=W. D. |location=New York |chapter=Chapter IX. The Consistency of Frege’s ''Foundations of Arithmetic'' |lccn=95-49208}}</ref>

[[Biography:Bernard Bolzano|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|mathematical analysis]]. In this work, Bolzano defended the notion of [[Philosophy:Actual infinity|actual infinity]], presented an early formulation of what would later be recognized as one-to-one correspondence between infinite 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> and revisited Galileo's paradox. However, he too resisted saying that these sets were, in that sense, the same size. 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 name=":9">{{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>

=== Early set theory ===

==== Georg Cantor ====
[[File:Georg_Cantor3.jpg|alt=refer to caption|thumb|339x339px|[[Biography:Georg Cantor|Georg Cantor]], {{spaces|4|hair}}{{circa}} 1870]]
The concept of cardinality emerged nearly fully formed in the work of Georg Cantor during the 1870s and 1880s, in the context of [[Mathematical analysis|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 |volume=2 |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 |series=Teubner-Archiv zur Mathematik |place=Vienna |publisher=Springer |language=de |doi=10.1007/978-3-7091-9516-1_2 |isbn=978-3-7091-9516-1 |url-access=subscription}}</ref> Cantor introduced the idea of comparing the sizes of infinite sets, through the notion of one-to-one correspondence.<ref>{{Harvard citation no brackets|Ferreirós|2007|p=171}}</ref> 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]].<ref>{{Harvard citation no brackets|Ferreirós|2007|p=177}}</ref> 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|power set]] of any set is strictly larger than the set itself.<ref>{{Harvard citation no brackets|Bourbaki|1968|pp=324–326}} {{Break}}{{Harvard citation no brackets|Ferreirós|2007|p=|pp=286–288}}</ref>

Cantor introduced the notion cardinal numbers together with ordinal numbers, which he viewed as abstractions of sets. For a given set <math display="inline">M</math>, he wrote <math display="inline">\overline{M}</math> to mean the abstraction of that set from its elements, while maintaining their order, and cardinal numbers were a double abstraction, written <span style="border-top: 3px double;"><math display="inline">M</math></span>.<ref>{{Harvard citation no brackets|Kleene|1952|p=9}} {{Break}}{{Harvard citation no brackets|Stoll|1963|p=80}}. {{Break}}{{Harvard citation no brackets|Takeuti|Zaring|1982|p=82}}.</ref> Specifically, his definition was "the general concept which, with the aid of our intelligence, results from a set when we abstract from the nature of its various elements and from the order of their being given."<ref>{{Harvard citation no brackets|Stoll|1963|p=80}}</ref> This definition was considered to be imprecise, unclear, and purely psychological, and it would be some time before the concept was put on more rigorous footing.<ref>Imprecise - {{Harvard citation no brackets|Stoll|1963|p=80}}. {{Break}}Psychological - {{Harvard citation no brackets|Takeuti|Zaring|1982|p=82}} {{Break}}Unclear - {{Cite book |last=Skolem |first=Thoralf |url=https://archive.org/details/abstractsettheor0008skol/ |title=Abstract Set Theory |date=1962 |publisher=University of Notre Dame Press |page=3}}

</ref> He also introduced the 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 arithmetic of cardinal numbers, defining addition, multiplication, and exponentiation of cardinal numbers. This led to the formulation of the Continuum Hypothesis: whether <math>2^{\aleph_0} = \aleph_1</math>. Cantor was unable to resolve CH and left it as an [[Unsolved:Open problem|open problem]].<ref>{{Harvard citation no brackets|Ferreirós|2007|pp=172, 177}}</ref>

==== Other contributors ====
Parallel to Cantor’s development, [[Biography:Richard Dedekind|Richard Dedekind]] independently formulated many advanced theorems of set theory, and helped establish set-theoretic foundations of algebra and arithmetic.<ref>{{Harvard citation no brackets|Bourbaki|1968|p=321}} {{Break}}{{Harvard citation no brackets|Ferreirós|2007|pp=81–82}}</ref> Dedekind's ''{{ill|The Nature and Meaning of Numbers|de|Was sind und was sollen die Zahlen?}}'' (1888)<ref>{{Cite book |last=Dedekind |first=Richard |url=https://link.springer.com/book/10.1007/978-3-663-02788-1 |title=Was sind und was sollen die Zahlen? |date=1961 |publisher=Vieweg+Teubner Verlag Wiesbaden |language=de |doi=10.1007/978-3-663-02788-1 |orig-date=1888}}</ref> 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.<ref>{{Harvard citation no brackets|Bourbaki|1968|pp=324–326}} {{Break}}{{Harvard citation no brackets|Ferreirós|2007|pp=172–176, 178–179}}</ref><ref name=":9" /><ref>{{Citation |last=Reck |first=Erich |title=Dedekind's Contributions to the Foundations of Mathematics |date=2023 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/win2023/entries/dedekind-foundations/ |access-date=2025-07-11 |edition=Winter 2023 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri |encyclopedia=The Stanford Encyclopedia of Philosophy}}</ref>

After Cantor's 1883 proof that all finite-dimensional spaces <math>(\R^n)</math> 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 |url-access=subscription}}</ref> in 1890, [[Biography:Giuseppe Peano|Giuseppe Peano]] introduced the [[Peano curve]], which was a more visual proof that the [[Unit interval|unit interval]] <math>[0,1]</math> has the same cardinality as the [[Unit square|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 |url-access=subscription |archive-url=https://web.archive.org/web/20180722000000/https://doi.org/10.1007/BF01199438 |archive-date=2018-07-22}} Alt URL</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 removed as it refers to the 2012 edition, not the 1963 edition cited here --></ref>

German logician [[Biography:Gottlob Frege|Gottlob Frege]] attempted to ground the concepts of number and arithmetic in logic using Cantor's theory of cardinality and [[Hume's principle]] in ''Die Grundlagen der Arithmetik'' (1884) and the subsequent ''Grundgesetze der Arithmetik'' (1893, 1903).<ref name=":1" /><ref name=":2" /><ref name=":8" /> Frege defined cardinal numbers as equivalence classes of sets under equinumerosity. However, Frege's approach to set theory was later shown to be flawed. His approach was eventually reformalized by Bertrand Russell and [[Biography:Alfred North Whitehead|Alfred Whitehead]] in ''[[Principia Mathematica]]'' (1910{{En dash}}1913, vol. II){{Sfn|Russell|Whitehead|5=|1973}} using a [[Type theory#History|theory of types]].<ref>{{Harvard citation no brackets|Bourbaki|1968|pp=331–332}} {{Break}}{{Harvard citation no brackets|Takeuti|Zaring|1982|pp=1–3}}</ref> Though Russell initially had difficulties understanding Cantor's and Frege’s intuitions of cardinality.<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 |url-access=subscription}}</ref><ref>{{Harvard citation no brackets|Anellis||pp=1–11|5=1984}}</ref> This definition of cardinal numbers is now referred to as the ''Frege{{En dash}}Russell'' definition.<ref name="Kleene 1952 44"/> This definition was eventually superseded by the convention established by [[Biography:John von Neumann|John von Neumann]] in 1928 which uses representatives to define cardinal numbers.<ref>{{Harvard citation no brackets|Stoll|1963|p=80}} {{Break}}{{Harvard citation no brackets|Kleene|1952|p=9}}</ref>

At the [[Place:Paris|Paris]] conference of the International Congress of Mathematicians in 1900, [[Biography:David Hilbert|David Hilbert]], one of the most influential mathematicians of the time, gave a speech wherein he presented ten unsolved problems (of a total of 23, later published, now called ''[[Hilbert's problems]]''). Of these, he placed "Cantor's problem" (now called the Continuum Hypothesis) as the first on the list. This list of problems would prove to be very influential in 20th century mathematics, and attracted a lot of attention from other mathematicians toward Cantor's theory of cardinality.<ref>{{Harvard citation no brackets|Bourbaki|1968|p=327}} {{Break}}{{Harvard citation no brackets|Ferreirós|2007|pp=iiiv, 301}}, 312</ref><ref name=":9" />

=== Axiomatic set theory ===
{{Multiple image
| image1            = Hausdorff 1913-1921.jpg
| image2            = Kurt gödel.jpg
| total_width       = 350
| footer            = [[Biography:Felix Hausdorff|Felix Hausdorff]] and [[Biography:Kurt Gödel|Kurt Gödel]]
}}

In 1908, [[Biography:Ernst Zermelo|Ernst Zermelo]] proposed the first axiomatization of set theory, now called [[Zermelo set theory]], primarily to support his earlier (1904) proof of the [[Well-ordering theorem]], which showed that all cardinal numbers could be represented as Alephs, though the proof required a controversial principle now known as the Axiom of Choice (AC).<ref>{{Harvard citation no brackets|Bourbaki|1968|p=325}}</ref> Zermelo's system would later be extended by [[Biography:Abraham Fraenkel|Abraham Fraenkel]] and [[Biography:Thoralf Skolem|Thoralf Skolem]] in the 1920s to create the standard foundation of set theory, called [[Zermelo–Fraenkel set theory]] (ZFC, "C" for the Axiom of Choice). ZFC provided a rigorous foundation through which infinite cardinals could be systematically studied while avoiding the [[Paradoxes of set theory|paradoxes of naive set theory]].<ref name=":9" /><ref name=":14" />

Ignoring possible foundational issues, during the early 1900s, [[Biography:Felix Hausdorff|Felix Hausdorff]] would begin studying "exorbitant numbers": roughly, very large cardinal numbers, or what are now called inaccessible cardinals. This work would be continued and popularized by several other influential set theorists such as [[Biography:Paul Mahlo|Paul Mahlo]]{{em dash}}who introduced [[Mahlo cardinal]]s{{em dash}}as well as [[Biography:Wacław Sierpiński|Wacław Sierpiński]], and [[Biography:Alfred Tarski|Alfred Tarski]]. Their work would eventually be known collectively as the study of [[Cardinality#Large cardinals|large cardinals]].<ref>{{Harvard citation no brackets|Ferreirós|2007|p=334}}{{br}}{{Harvard citation no brackets|Kanamori|2003|p=XVI}}</ref>

In 1940, [[Biography:Kurt Gödel|Kurt Gödel]] showed that the Continuum Hypothesis (CH) cannot be ''disproved'' from the axioms of ZFC by showing that both CH and AC hold in his constructible universe: an [[Inner model|inner model]] of ZFC. The existence of a model of ZFC in which additional axioms hold shows that the additional axioms are (relatively) consistent with ZFC.<ref>{{cite journal |last1=Gödel |first1=Kurt |date=1938 |title=The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis |journal=Proceedings of the National Academy of Sciences |volume=24 |issue=12 |pages=556–557 |bibcode=1938PNAS...24..556G |doi=10.1073/pnas.24.12.556 |pmc=1077160 |pmid=16577857 |doi-access=free}}</ref> In 1963, [[Biography:Paul Cohen|Paul Cohen]] showed that CH cannot be ''proven'' from the ZFC axioms, which showed that CH was [[Independence (mathematical logic)|independent]] from ZFC. To prove his result, Cohen developed the method of [[Forcing (mathematics)|forcing]], which has become a standard tool in set theory. Cohen was awarded the Fields Medal in 1966 for his proof.<ref>{{Cite journal |last=Cohen |first=P. J. |date=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 |issn=0027-8424 |pmc=221287 |pmid=16578557 |doi-access=free}}</ref><ref>{{Cite book |last=Cohen |first=Paul Joseph |title=Set theory and the continuum hypothesis |date=2008 |publisher=Dover Publications |isbn=978-0-486-46921-8 |location=Mineola, New York City |orig-date=1966}}</ref>

==See also==
{{Wikidata property|P1164|P2820}}
* ''[[Cardinal and Ordinal Numbers]]''
* [[Infinitary combinatorics]]
* Numerosity (mathematics)
* [[Ross–Littlewood paradox]]
* [[Zero sharp]]

==References==

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

=== Bibliography ===

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* {{Citation |last=Koellner |first=Peter |title=Independence and Large Cardinals |date=2011 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/sum2011/entries/independence-large-cardinals/ |access-date=2025-07-08 |edition=Summer 2011 |publisher=Metaphysics Research Lab, Stanford University |}}
* {{Citation |last=Koellner |first=Peter |title=Large Cardinals and Determinacy |date=2014 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/spr2014/entries/large-cardinals-determinacy/ |access-date=2025-07-08 |edition=Spring 2014 |publisher=Metaphysics Research Lab, Stanford University |}}
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{{Mathematical logic}}

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

{{Sourceattribution|Cardinality}}