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Cardinal number

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Size of a possibly infinite set

This article is about the mathematical concept. For number words indicating quantity ("three" apples, "four" birds, etc.), seeCardinal numeral.

History

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The notion of cardinality, as now understood, was formulated byGeorg Cantor, the originator ofset theory, in 1874–1884. Cardinality can be used to compare an aspect of finite sets. For example, the sets {1,2,3} and {4,5,6} are notequal, but have thesame cardinality, namely three. This is established by the existence of abijection (i.e., a one-to-one correspondence) between the two sets, such as the correspondence {1→4, 2→5, 3→6}.

Cantor applied his concept of bijection to infinite sets^[1] (for example the set of natural numbersN = {0, 1, 2, 3, ...}). Thus, he called all sets having a bijection withNdenumerable (countably infinite) sets, which all share the same cardinal number. This cardinal number is called $\aleph _{0}$ ,aleph-null. He called the cardinal numbers of infinite setstransfinite cardinal numbers.

Cantor proved that anyunbounded subset ofN has the same cardinality asN, even though this might appear to run contrary to intuition. He also proved that the set of allordered pairs of natural numbers is denumerable; this implies that the set of allrational numbers is also denumerable, since every rational can be represented by a pair of integers. He later proved that the set of all realalgebraic numbers is also denumerable. Each real algebraic numberz may be encoded as a finite sequence of integers, which are the coefficients in the polynomial equation of which it is a solution, i.e. the ordered n-tuple (a₀,a₁, ...,a_n),a_i ∈Z together with a pair of rationals (b₀,b₁) such thatz is the unique root of the polynomial with coefficients (a₀,a₁, ...,a_n) that lies in the interval (b₀,b₁).

In his 1874 paper "On a Property of the Collection of All Real Algebraic Numbers", Cantor proved that there exist higher-order cardinal numbers, by showing that the set of real numbers has cardinality greater than that ofN. His proof used an argument withnested intervals, but in an 1891 paper, he proved the same result using his ingenious and much simplerdiagonal argument. The new cardinal number of the set of real numbers is called thecardinality of the continuum and Cantor used the symbol ${\mathfrak {c}}$ for it.

Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number ( $\aleph _{0}$ , aleph-null), and that for every cardinal number there is a next-larger cardinal

(\aleph _{1},\aleph _{2},\aleph _{3},\ldots ).

Hiscontinuum hypothesis is the proposition that the cardinality ${\mathfrak {c}}$ of the set of real numbers is the same as $\aleph _{1}$ . This hypothesis is independent of the standard axioms of mathematical set theory, that is, it can neither be proved nor disproved from them. This was shown in 1963 byPaul Cohen, complementing earlier work byKurt Gödel in 1940.

Motivation

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In informal use, a cardinal number is what is normally referred to as acounting number, provided that 0 is included: 0, 1, 2, .... They may be identified with thenatural numbers beginning with 0. The counting numbers are exactly what can be defined formally as thefinite cardinal numbers. Infinite cardinals only occur in higher-level mathematics andlogic.

More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. For finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can construct a set that has exactly the right size. For example, 3 describes the position of 'c' in the sequence <'a','b','c','d',...>, and we can construct the set {a,b,c}, which has 3 elements.

However, when dealing withinfinite sets, it is essential to distinguish between the two, since the two notions are in fact different for infinite sets. Considering the position aspect leads toordinal numbers, while the size aspect is generalized by the cardinal numbers described here.

The intuition behind the formal definition of cardinal is the construction of a notion of the relative size or "bigness" of a set, without reference to the kind of members which it has. For finite sets this is easy; one simply counts the number of elements a set has. In order to compare the sizes of larger sets, it is necessary to appeal to more refined notions.

A setY is at least as big as a setX if there is aninjective mapping from the elements ofX to the elements ofY. An injective mapping identifies each element of the setX with a unique element of the setY. This is most easily understood by an example; suppose we have the setsX = {1,2,3} andY = {a,b,c,d}, then using this notion of size, we would observe that there is a mapping:

1 → a

2 → b

3 → c

which is injective, and hence conclude thatY has cardinality greater than or equal toX. The element d has no element mapping to it, but this is permitted as we only require an injective mapping, and not necessarily abijective mapping. The advantage of this notion is that it can be extended to infinite sets.

We can then extend this to an equality-style relation. TwosetsX andY are said to have the samecardinality if there exists abijection betweenX andY. By theSchroeder–Bernstein theorem, this is equivalent to there beingboth an injective mapping fromX toY,and an injective mapping fromY toX. We then write |X| = |Y|. The cardinal number ofX itself is often defined as the least ordinala with |a| = |X|.^[2] This is called thevon Neumann cardinal assignment; for this definition to make sense, it must be proved that every set has the same cardinality assome ordinal; this statement is thewell-ordering principle. It is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.

The classic example used is that of the infinite hotel paradox, also calledHilbert's paradox of the Grand Hotel. Supposing there is an innkeeper at a hotel with an infinite number of rooms. The hotel is full, and then a new guest arrives. It is possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant. We can explicitly write a segment of this mapping:

1 → 2

2 → 3

3 → 4

...

n →n + 1

...

With this assignment, we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...}, since a bijection between the first and the second has been shown. This motivates the definition of an infinite set being any set that has a proper subset of the same cardinality (i.e., aDedekind-infinite set); in this case {2,3,4,...} is a proper subset of {1,2,3,...}.

When considering these large objects, one might also want to see if the notion of counting order coincides with that of cardinal defined above for these infinite sets. It happens that it does not; by considering the above example we can see that if some object "one greater than infinity" exists, then it must have the same cardinality as the infinite set we started out with. It is possible to use a different formal notion for number, calledordinals, based on the ideas of counting and considering each number in turn, and we discover that the notions of cardinality and ordinality are divergent once we move out of the finite numbers.

It can be proved that the cardinality of thereal numbers is greater than that of the natural numbers just described. This can be visualized usingCantor's diagonal argument; classic questions of cardinality (for instance thecontinuum hypothesis) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals. In more recent times, mathematicians have been describing the properties of larger and larger cardinals.

Since cardinality is such a common concept in mathematics, a variety of names are in use. Sameness of cardinality is sometimes referred to asequipotence,equipollence, orequinumerosity. It is thus said that two sets with the same cardinality are, respectively,equipotent,equipollent, orequinumerous.

Cardinality function

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Main article:Cardinal function

The cardinality function is acardinal function which takes in a set $A {\displaystyle A}$ and returns its cardinal number: $A\mapsto |A|$ . For finite sets, this is simply thenatural number found by counting the elements. This function applied to a set $A {\displaystyle A}$ is generally denoted by $|A|,$ with avertical bar on each side,^[3] though it may also be denoted by $A {\displaystyle A}$ , $\operatorname {card} (A),$ or $\#A.$ ^[17]

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.^[18] Thus, The assumption that there issomecardinal function $A\mapsto |A|$ which satisfies $A\sim B\iff |A|=|B|$ , sometimes called theaxiom of cardinality^[19] orHume's principle,^[20] is sufficient for deriving most properties of cardinal numbers.^[21] It will be shown later that such a function can be constructed without the need to define it axiomatically.

Constructive definition

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Formally, assuming theaxiom of choice, the cardinality of a setX is the leastordinal number α such that there is a bijection betweenX and α. This definition is known as thevon Neumann cardinal assignment. If the axiom of choice is not assumed, then a different approach is needed. The oldest definition of the cardinality of a setX (implicit in Cantor and explicit in Frege andPrincipia Mathematica) is as the class [X] of all sets that are equinumerous withX. This does not work inZFC or other related systems ofaxiomatic set theory because ifX is non-empty, this collection is too large to be a set. In fact, forX ≠ ∅ there is an injection from the universe into [X] by mapping a setm to {m} ×X, and so by theaxiom of limitation of size, [X] is a proper class. The definition does work however intype theory and inNew Foundations and related systems. However, if we restrict from this class to those equinumerous withX that have the leastrank, then it will work (this is a trick due toDana Scott:^[22] it works because the collection of objects with any given rank is a set).

Von Neumann cardinal assignment implies that the cardinal number of a finite set is the common ordinal number of all possible well-orderings of that set, and cardinal and ordinal arithmetic (addition, multiplication, power, proper subtraction) then give the same answers for finite numbers. However, they differ for infinite numbers. For example, $2^{\omega }=\omega <\omega ^{2}$ in ordinal arithmetic while $2^{\aleph _{0}}>\aleph _{0}=\aleph _{0}^{2}$ in cardinal arithmetic, although the von Neumann assignment puts $\aleph _{0}=\omega$ . On the other hand, Scott's trick implies that the cardinal number 0 is $\{\emptyset \}$ , which is also the ordinal number 1, and this may be confusing. A possible compromise (to take advantage of the alignment in finite arithmetic while avoiding reliance on the axiom of choice and confusion in infinite arithmetic) is to apply von Neumann assignment to the cardinal numbers of finite sets (those which can be well ordered and are not equipotent to proper subsets) and to use Scott's trick for the cardinal numbers of other sets.

Formally, the order among cardinal numbers is defined as follows: |X| ≤ |Y| means that there exists aninjective function fromX toY. TheCantor–Bernstein–Schroeder theorem states that if |X| ≤ |Y| and |Y| ≤ |X| then |X| = |Y|. The axiom of choice is equivalent to the statement that given two setsX andY, either |X| ≤ |Y| or |Y| ≤ |X|.^[23]^[24]

A setX isDedekind-infinite if there exists aproper subsetY ofX with |X| = |Y|, andDedekind-finite if such a subset does not exist. Thefinite cardinals are just thenatural numbers, in the sense that a setX is finite if and only if |X| = |n| =n for some natural numbern. Any other set isinfinite.

Assuming the axiom of choice, it can be proved that the Dedekind notions correspond to the standard ones. It can also be proved that the cardinal $\aleph _{0}$ (aleph null or aleph-0, where aleph is the first letter in theHebrew alphabet, represented $\aleph$ ) of the set of natural numbers is the smallest infinite cardinal (i.e., any infinite set has a subset of cardinality $\aleph _{0}$ ). The next larger cardinal is denoted by $\aleph _{1}$ , and so on. For every ordinal α, there is a cardinal number $\aleph _{\alpha },$ and this list exhausts all infinite cardinal numbers.

Cardinal arithmetic

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We can definearithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers. It can be shown that for finite cardinals, these operations coincide with the usual operations for natural numbers. Furthermore, these operations share many properties with ordinary arithmetic.

Successor cardinal

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Further information:Successor cardinal

If the axiom of choice holds, then every cardinal κ has a successor, denoted κ⁺, where κ⁺ > κ and there are no cardinals between κ and its successor. (Without the axiom of choice, usingHartogs' theorem, it can be shown that for any cardinal number κ, there is a minimal cardinal κ⁺ such that $\kappa ^{+}\nleq \kappa .$ ) For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from thesuccessor ordinal.

Cardinal addition

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IfX andY aredisjoint, addition is given by theunion ofX andY. If the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replaceX byX×{0} andY byY×{1}).

|X|+|Y|=|X\cup Y|.

^[25]

Zero is an additive identityκ + 0 = 0 +κ =κ.

Addition isassociative (κ +μ) +ν =κ + (μ +ν).

Addition iscommutativeκ +μ =μ +κ.

Addition is non-decreasing in both arguments:

(\kappa \leq \mu )\rightarrow ((\kappa +\nu \leq \mu +\nu ){\mbox{ and }}(\nu +\kappa \leq \nu +\mu )).

Assuming the axiom of choice, addition of infinite cardinal numbers is easy. If eitherκ orμ is infinite, then

\kappa +\mu =\max\{\kappa ,\mu \}\,.

Subtraction

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Assuming the axiom of choice and, given an infinite cardinalσ and a cardinalμ, there exists a cardinalκ such thatμ +κ =σ if and only ifμ ≤σ. It will be unique (and equal toσ) if and only ifμ <σ.

Cardinal multiplication

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The product of cardinals comes from theCartesian product.

|X|\cdot |Y|=|X\times Y|

^[25]

Zero is a multiplicativeabsorbing element:κ·0 = 0·κ = 0.

There are no nontrivialzero divisors:κ·μ = 0 → (κ = 0 orμ = 0).

One is a multiplicative identity:κ·1 = 1·κ =κ.

Multiplication is associative: (κ·μ)·ν =κ·(μ·ν).

Multiplication iscommutative:κ·μ =μ·κ.

Multiplication is non-decreasing in both arguments:κ ≤μ → (κ·ν ≤μ·ν andν·κ ≤ν·μ).

Multiplicationdistributes over addition:κ·(μ +ν) =κ·μ +κ·ν and(μ +ν)·κ =μ·κ +ν·κ.

Assuming the axiom of choice, multiplication of infinite cardinal numbers is also easy. If eitherκ orμ is infinite and both are non-zero, then

\kappa \cdot \mu =\max\{\kappa ,\mu \}.

Thus the product of two infinite cardinal numbers is equal to their sum.

Division

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Assuming the axiom of choice and given an infinite cardinalπ and a non-zero cardinalμ, there exists a cardinalκ such thatμ ·κ =π if and only ifμ ≤π. It will be unique (and equal toπ) if and only ifμ <π.

Cardinal exponentiation

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Exponentiation is given by

|X|^{|Y|}=\left|X^{Y}\right|,

whereX^Y is the set of allfunctions fromY toX.^[25] It is easy to check that the right-hand side depends only on ${|X|}$ and ${|Y|}$ .

κ⁰ = 1 (in particular 0⁰ = 1), seeempty function.

Ifμ ≥ 1, then 0^μ = 0.

1^μ = 1.

κ¹ =κ.

κ^{μ +ν} =κ^μ·κ^ν.

κ^{μ ·ν} = (κ^μ)^ν.

(κ·μ)^ν =κ^ν·μ^ν.

Exponentiation is non-decreasing in both arguments:

(1 ≤ν andκ ≤μ) → (ν^κ ≤ν^μ) and

(κ ≤μ) → (κ^ν ≤μ^ν).

2^|X| is the cardinality of thepower set of the setX andCantor's diagonal argument shows that 2^|X| > |X| for any setX. This proves that no largest cardinal exists (because for any cardinalκ, we can always find a larger cardinal 2^κ). In fact, theclass of cardinals is aproper class. (This proof fails in some set theories, notablyNew Foundations.)

All the remaining propositions in this section assume the axiom of choice:

Ifκ andμ are both finite and greater than 1, andν is infinite, thenκ^ν =μ^ν.

Ifκ is infinite andμ is finite and non-zero, thenκ^μ =κ.

If 2 ≤κ and 1 ≤μ and at least one of them is infinite, then:

Max (κ, 2^μ) ≤κ^μ ≤ Max (2^κ, 2^μ).

UsingKönig's theorem, one can proveκ <κ^cf(κ) andκ < cf(2^κ) for any infinite cardinalκ, where cf(κ) is thecofinality ofκ.

Roots

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Assuming the axiom of choice and, given an infinite cardinalκ and a finite cardinalμ greater than 0, the cardinalν satisfying $\nu ^{\mu }=\kappa$ will be $\kappa$ .

Logarithms

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Assuming the axiom of choice and, given an infinite cardinalκ and a finite cardinalμ greater than 1, there may or may not be a cardinalλ satisfying $\mu ^{\lambda }=\kappa$ . However, if such a cardinal exists, it is infinite and less thanκ, and any finite cardinalityν greater than 1 will also satisfy $\nu ^{\lambda }=\kappa$ .

The logarithm of an infinite cardinal numberκ is defined as the least cardinal numberμ such thatκ ≤ 2^μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study ofcardinal invariants oftopological spaces, though they lack some of the properties that logarithms of positive real numbers possess.^[26]^[27]^[28]

The continuum hypothesis

[edit]

Thecontinuum hypothesis (CH) states that there are no cardinals strictly between $\aleph _{0}$ and $2^{\aleph _{0}}.$ The latter cardinal number is also often denoted by ${\mathfrak {c}}$ ; it is thecardinality of the continuum (the set ofreal numbers). In this case $2^{\aleph _{0}}=\aleph _{1}.$

Similarly, thegeneralized continuum hypothesis (GCH) states that for every infinite cardinal $\kappa$ , there are no cardinals strictly between $\kappa$ and $2^{\kappa }$ . Both the continuum hypothesis and the generalized continuum hypothesis have been proved to beindependent of the usual axioms of set theory, the Zermelo–Fraenkel axioms together with the axiom of choice (ZFC).

Indeed,Easton's theorem shows that, forregular cardinals $\kappa$ , the only restrictions ZFC places on the cardinality of $2^{\kappa }$ are that $\kappa <\operatorname {cf} (2^{\kappa })$ , and that the exponential function is non-decreasing.

References

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Notes

^Dauben 1990, pg. 54
^Weisstein, Eric W."Cardinal Number".mathworld.wolfram.com. Retrieved2020-09-06.
^Hrbáček & Jech 2017, p. 65
^Kleene 1952, p. 9.
^Krivine 1971, p. 23.
^Kuratowski 1968, p. 174.
^Stoll 1963, p. 80.
^Suppes 1972, p. 109.
^Takeuti & Zaring 1982, p. 85.
^Bourbaki 1968, p. 158.
^Enderton 1977, p. 136.
^Halmos 1998, p. 94.
^Schumacher 1996, p. 103.
^Halmos 1998, p. 53.
^Pinter 2014, Page 3 of Chaper 8.
^Tao 2022, p. 60.
^ $A {\displaystyle A}$ ^[4]^[5]^[6]^[7]^[8]^[9]
$\operatorname {card} (A)$ ^[10]^[11]^[12]^[13]
$\#A$ ^[14]^[15]^[16]
^Kleene 1952, p. 9
^Pinter 2014, Page 2 of Chapter 8
^Potter, Michael (2004-01-15).Set Theory and its Philosophy: A Critical Introduction. Clarendon Press.ISBN 978-0-19-155643-2.
^Enderton 1977, p. 136
^Deiser, Oliver (May 2010). "On the Development of the Notion of a Cardinal Number".History and Philosophy of Logic.31 (2):123–143.doi:10.1080/01445340903545904.S2CID 171037224.
^Enderton, Herbert. "Elements of Set Theory", Academic Press Inc., 1977.ISBN 0-12-238440-7
^Friedrich M. Hartogs (1915),Felix Klein;Walther von Dyck;David Hilbert;Otto Blumenthal (eds.),"Über das Problem der Wohlordnung",Math. Ann., Bd. 76 (4), Leipzig: B. G. Teubner:438–443,doi:10.1007/bf01458215,ISSN 0025-5831,S2CID 121598654,archived from the original on 2016-04-16, retrieved2014-02-02
^^a ^b ^cSchindler 2014, pg. 34
^Robert A. McCoy and Ibula Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathematics 1315,Springer-Verlag.
^Eduard Čech, Topological Spaces, revised by Zdenek Frolík and Miroslav Katetov, John Wiley & Sons, 1966.
^D. A. Vladimirov, Boolean Algebras in Analysis, Mathematics and Its Applications, Kluwer Academic Publishers.

Bibliography

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Hahn, Hans,Infinity, Part IX, Chapter 2, Volume 3 ofThe World of Mathematics. New York: Simon and Schuster, 1956.
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Hrbáček, Karel;Jech, Thomas (2017) [1999].Introduction to Set Theory (3rd, Revised and Expanded ed.). New York:CRC Press.doi:10.1201/9781315274096.ISBN 978-0-82477915-3.LCCN 99-15458.
Jech, Thomas (2002) [1997].Set Theory. Springer Monographs in Mathematics (3rd ed.). Berlin:Springer-Verlag.doi:10.1007/3-540-44761-X.ISBN 978-3-540-44085-7.ISSN 1439-7382.
Kanamori, Akihiro (2003) [1994].The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Springer Monographs in Mathematics (2nd ed.). Berlin:Springer-Verlag.doi:10.1007/978-3-540-88867-3.ISBN 978-3-540-88866-6.ISSN 1439-7382.LCCN 2008940025.
Kleene, Stephen Cole (1952).Introduction To Metamathematics. New York:D. Van Nostrand Company.
Krivine, Jean-Louis (1971).Introduction to Axiomatic Set Theory. Synthese Library. New York:D. Reidel Publishing Company.doi:10.1007/978-94-010-3144-8.ISBN 9780391001794.ISSN 0166-6991.LCCN 71-146965. Archived fromthe original on 2014-08-06.Alt URL
Koellner, Peter (2011),Zalta, Edward N. (ed.),"Independence and Large Cardinals",The Stanford Encyclopedia of Philosophy (Summer 2011 ed.), Metaphysics Research Lab, Stanford University, retrieved2025-07-08
Koellner, Peter (2014),Zalta, Edward N. (ed.),"Large Cardinals and Determinacy",The Stanford Encyclopedia of Philosophy (Spring 2014 ed.), Metaphysics Research Lab, Stanford University, retrieved2025-07-08
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Lévy, Azriel (1979).Basic Set Theory. Perspectives in Mathematical Logic. Berlin:Springer-Verlag.ISBN 3-540-08417-7.LCCN 78-1917.
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Schumacher, Carol (1996).Chapter Zero: Fundamental Notions of Abstract Mathematics. Reading:Addison-Wesley.ISBN 9780201826531.LCCN 95-22675.
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External links

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"Cardinal number",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

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