Set size

In mathematics, the cardinality of a finite set is the number of its elements, and is therefore a measure of size of the set. Since the discovery by Georg Cantor, in the late 19th century, of different sizes of infinite sets, the term ''cardinality'' was coined for generalizing to infinite sets the concept of the number of elements.

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

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

The great discovery of Cantor is that infinite sets of apparently different size may have the same cardinality, and, nevertheless, there are infinitely many different cardinalities. For example, the even numbers, the prime numbers and the polynomials whose coefficients are rational number have the same cardinality as the natural numbers. The set of the real numbers has a greater cardinality than the natural numbers, and has the same cardinality as the interval and as every Euclidean space of any dimension. For every set, its power set (the set of all its subsets) has a greater cardinality.

Cardinalities are represented with cardinal numbers, which are specific sets of a given cardinality, which have been chosen once for all. Some infinite cardinalities have been given a specific name, such as for the cardinality of the natural numbers and , the cardinality of the continuum, for the cardinality of the real numbers.

Etymology

In English, the term ''cardinality'' originates from the post-classical Latin ''cardinalis'', meaning "principal" or "chief", which derives from ''cardo'', a noun meaning "hinge". In Latin, ''cardo'' referred to something central or pivotal, both literally and metaphorically. This concept of centrality passed into medieval Latin and then into English, where ''cardinal'' came to describe things considered to be, in some sense, fundamental, such as ''cardinal virtues'', '' cardinal sins'', ''cardinal directions'', and (in the grammatical sense) ''cardinal numbers.'' The last of which referred to numbers used for counting (e.g., one, two, three), as opposed to ''ordinal numbers'', which express order (e.g., first, second, third), and ''nominal numbers'' used for labeling without meaning (e.g., jersey numbers and serial numbers).

In mathematics, the notion of cardinality was first introduced by Georg Cantor in the late 19th century, wherein he used the used the term ''Mächtigkeit'', which may be translated as "magnitude" or "power", though Cantor credited the term to a work by Jakob Steiner on projective geometry. The terms ''cardinality'' and ''cardinal number'' were eventually adopted from the grammatical sense, and later translations would use these terms.

History

Prehistory

A crude sense of cardinality, an awareness that groups of things or events compare with other groups by containing more, fewer, or the same number of instances, is observed in a variety of present-day animal species, suggesting an origin millions of years ago. Human expression of cardinality is seen as early as years ago, with equating the size of a group with a group of recorded notches, or a representative collection of other things, such as sticks and shells. The abstraction of cardinality as a number is evident by 3000 BCE, in Sumerian mathematics and the manipulation of numbers without reference to a specific group of things or events.

Ancient history

From the 6th century BCE, the writings of Greek philosophers, such as Anaximander, show hints of comparing infinite sets or shapes. While the Greeks considered notions of infinity, they viewed it as paradoxical and imperfect (cf. ''Zeno's paradoxes''), often associating good and evil with finite and infinite. Aristotle distinguished between the notions of actual infinity and potential infinity, arguing that Greek mathematicians understood the difference, and that they "do not need the ctualinfinite and do not use it." The greek notion of number (''αριθμός'', ''arithmos'') was used exclusively for a definite number of definite objects (i.e. finite numbers). This would be codified in Euclid's ''Elements'', where the 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.

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.

One of the earliest explicit uses of a one-to-one correspondence is recorded in Aristotle's ''Mechanics'' (), known as Aristotle's wheel paradox. The paradox can be briefly described as follows: A wheel is depicted as two concentric circles. The larger, outer circle is tangent to a horizontal line (e.g. a road that it rolls on), while the smaller, inner circle is rigidly affixed to the larger. Assuming the larger circle rolls along the line without slipping (or skidding) for one full revolution, the distances moved by both circles are the same: the circumference of the larger circle. Further, the lines traced by the bottom-most point of each is the same length. Since the smaller wheel does not skip any points, and no point on the smaller wheel is used more than once, there is a one-to-one correspondence between the two circles.

Pre-Cantorian set theory

Galileo Galilei presented what was later coined Galileo's paradox in his book ''Two New Sciences'' (1638), where he attempts to show that infinite quantities cannot be called greater or less than one another. He presents the paradox roughly as follows: a square number is one which is the product of another number with itself, such as 4 and 9, which are the squares of 2 and 3, respectively. Then the square root of a square number is that multiplicand. He then notes that there are as many square numbers as there are square roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square. But there are as many square roots as there are numbers, since every number is the square root of some square. He, 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.

Bernard Bolzano's ''Paradoxes of the Infinite'' (''Paradoxien des Unendlichen'', 1851) is often considered the first systematic attempt to introduce the concept of sets into mathematical analysis. In this work, Bolzano defended the notion of actual infinity, examined various properties of infinite collections, including an early formulation of what would later be recognized as one-to-one correspondence between infinite sets, and proposed to base mathematics on a notion similar to sets. He discussed examples such as the pairing between the intervals ,5/math> and ,12/math> by the relation $5y = 12x.$ Bolzano also revisited and extended Galileo's paradox. However, he too resisted saying that these sets were, in that sense, the same size. Thus, while ''Paradoxes of the Infinite'' anticipated several ideas central to later set theory, the work had little influence on contemporary mathematics, in part due to its posthumous publication and limited circulation.

Other, more minor contributions include David Hume in ''A Treatise of Human Nature'' (1739), who said ''"When two numbers are so combined, as that the one has always a unit answering to every unit of the other, we pronounce them equal",'' now called ''Hume's principle'', which was used extensively by Gottlob Frege later during the rise of set theory. Jakob Steiner, whom Georg Cantor credits the original term, ''Mächtigkeit'', for cardinality (1867). Peter Gustav Lejeune Dirichlet is commonly credited for being the first to explicitly formulate the pigeonhole principle in 1834, though it was used at least two centuries earlier by Jean Leurechon in 1624.

Early set theory

Georg Cantor

The concept of cardinality, as a formal measure of the size of a set, emerged nearly fully formed in the work of Georg Cantor during the 1870s and 1880s, in the context of mathematical analysis. In a series of papers beginning with '' On a Property of the Collection of All Real Algebraic Numbers'' (1874), Cantor introduced the idea of comparing the sizes of infinite sets, through the notion of one-to-one correspondence. He showed that the set of real numbers was, in this sense, strictly larger than the set of natural numbers using a nested intervals argument. This result was later refined into the more widely known diagonal argument of 1891, published in ''Über eine elementare Frage der Mannigfaltigkeitslehre,'' where he also proved the more general result (now called Cantor's Theorem) that the power set of any set is strictly larger than the set itself.

Cantor introduced the notion cardinal numbers in terms of ordinal numbers. He viewed cardinal numbers as an abstraction of sets, introduced the notations, where, for a given set $M$, the order type of that set was written $\overline$, and the cardinal number was $M$, a double abstraction. 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'' (18951897). In these works, Cantor developed an arithmetic of cardinal numbers, defining addition, multiplication, and exponentiation of cardinal numbers based on set-theoretic constructions. This led to the formulation of the Continuum Hypothesis (CH), the proposition that no set has cardinality strictly between $\aleph_0$ and the cardinality of the continuum, that is $, \R, = \aleph_1$. Cantor was unable to resolve CH and left it as an open problem.

Other contributors

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

After Cantor's 1883 proof that all finite-dimensional spaces $(\R^n)$ have the same cardinality, in 1890, Giuseppe Peano introduced the Peano curve, which was a more visual proof that the unit interval ,1/math> has the same cardinality as the unit square on $\R^2.$ This created a new area of mathematical analysis studying what is now called space-filling curves.

German logician 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). 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 Alfred Whitehead in ''Principia Mathematica'' (19101913, vol. II) using a theory of types. Though Russell initially had difficulties understanding Cantor's and Frege’s intuitions of cardinality. This definition of cardinal numbers is now referred to as the ''Frege-Russell'' definition.

Axiomatic set theory

In 1908, 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). Zermelo's system would later be extended by Abraham Fraenkel and 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, albeit non- categorical, foundation through which infinite cardinals could be systematically studied while avoiding the paradoxes of naive set theory.

In 1940, Kurt Gödel showed that CH cannot be ''disproved'' from ZF set theory (ZFC without Choice), even if the axiom of choice is adopted, i.e. from ZFC. Gödel's proof shows that both CH and AC hold in the constructible universe, an inner model of ZF set theory, assuming only the axioms of ZF. The existence of an inner model of ZF in which additional axioms hold shows that the additional axioms are (relatively) consistent with ZF, provided ZF itself is consistent. In 1963, Paul Cohen showed that CH cannot be ''proven'' from the ZFC axioms, which showed that CH was independent from ZFC. To prove his result, Cohen developed the method of forcing, which has become a standard tool in set theory. Essentially, this method begins with a model of ZFC in which CH holds and constructs another model which contains more sets than the original in a way that CH does not hold in the new model. Cohen was awarded the Fields Medal in 1966 for his proof.

Comparing sets

Introduction

The basic notions of sets and functions are used to develop the concept of cardinality, and technical terms therein are used throughout this article. A set can be understood as any collection of objects, usually represented with curly braces. For example, $S = \$ specifies a set, called $S$, which contains the numbers 1, 2, and 3. The symbol $\in$ represents set membership, for example $1 \in S$ says "1 is a member of the set $S$" which is true by the definition of $S$ above.

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

Equinumerosity

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

Two sets $A$ and $B$ are said to have the ''same cardinality'' if their elements can be paired one-to-one. That is, if there exists a function $f:A \mapsto B$ which is bijective. This is written as $A \sim B,$ $A \approx B,$ $A \equiv B,$ and eventually $, A, = , B, ,$ once $, A,$ has been defined. Alternatively, these sets, $A$ and $B,$ may be said to be ''similar'', ''equinumerous'', or ''equipotent''. For example, the set $E = \$ of non-negative even numbers has the same cardinality as the set $\N = \$ of natural numbers, since the function $f(n) = 2n$ is a bijection from to (see picture above).

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

Equivalence

A fundamental result necessary in developing a theory of cardinality is relating it to an equivalence relation. A binary relation is an equivalence relation if it satisfies the three basic properties of equality: reflexivity, symmetry, and transitivity.

* Reflexivity: For any set $A$, $A \sim A.$
** Given any set $A,$ there is a bijection from $A$ to itself by the identity function, therefore equinumerosity is reflexive.
* Symmetry: If $A \sim B$ then $B \sim A.$
** Given any sets $A$ and $B,$ such that there is a bijection $f$ from $A$ to $B,$ then there is an inverse function $f^$ from $B$ to $A,$ which is also bijective, therefore equinumerosity is symmetric.
* Transitivity: If $A \sim B$ and $B \sim C$ then $A \sim B.$
** Given any sets $A,$ $B,$ and $C$ such that there is a bijection $f$ from $A$ to $B,$ and $g$ from $B$ to $C,$ then their composition $g \circ f$ is a bijection from $A$ to $C,$ and so cardinality is transitive.

Since equinumerosity satisfies these three properties, it forms an equivalence relation. This means that cardinality, in some sense, partitions sets into equivalence classes, and one may assign a representative to denote this class. This motivates the notion of a cardinal number. Somewhat more formally, a relation must be a certain set of ordered pairs. Since there is no set of all sets in standard set theory (see: '), equinumerosity is not a relation in the usual sense, but a predicate or a relation over classes.

Inequality

A set $A$ is not larger than a set $B$ if it can be mapped into $B$ without overlap. That is, the cardinality of $A$ is less than or equal to the cardinality of $B$ if there is an injective function from $A$ to $B$. This is written $A \preceq B,$ $A \lesssim B$ and eventually $, A, \leq , B, .$ If $A \preceq B,$ but there is no injection from $B$ to $A,$ then $A$ is said to be ''strictly'' smaller than $B,$ written without the underline as $A \prec B$ or $, A, < , B, .$ For example, if $A$ has four elements and $B$ has five, then the following are true $A \preceq A,$ $A \preceq B,$ and $A \prec B.$

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

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

The above shows that cardinal inequality is a partial order. A total order has the additional property that, for any $a$ and $b$, either $a \leq b$ or $b \leq a.$ This can be established by the well-ordering theorem. Every well-ordered set is isomorphic to a unique ordinal number, called the order type of the set. Then, by comparing their order types, one can show that $A \preceq B$ or $B \preceq A$. This result is equivalent to the axiom of choice.

Countable sets

A set is called ''countable'' if it is finite or has a bijection with the set of natural numbers $(\N),$ in which case it is called ''countably infinite''. The term '' denumerable'' is also sometimes used for countably infinite sets. For example, the set of all even natural numbers is countable, and therefore has the same cardinality as the whole set of natural numbers, even though it is a proper subset. Similarly, the set of square numbers is countable, which was considered paradoxical for hundreds of years before modern set theory (see: '). However, several other examples have historically been considered surprising or initially unintuitive since the rise of set theory.

The rational numbers $(\Q)$ are those which can be expressed as the quotient or fraction of two integers. 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 $\Z\times\Z,$ which can be visualized as the set of all integer points on a grid. Then, an intuitive function can be described by drawing a line in a repeating pattern, or spiral, which eventually goes through each point in the grid. For example, going through each diagonal on the grid for positive fractions, or through a lattice spiral for all integer pairs. These technically over cover the rationals, since, for example, the rational number $\frac$ gets mapped to by all the fractions $\frac,\, \frac, \, \frac, \, \dots ,$ as the grid method treats these all as distinct ordered pairs. So this function shows $, \Q, \leq , \N,$ not $, \Q, = , \N, .$ This can be corrected by "skipping over" these numbers in the grid, or by designing a function which does this naturally, for example using the Calkin–Wilf tree.

A number is called algebraic if it is a solution of some polynomial equation (with integer coefficients). For example, the square root of two $\sqrt2$ is a solution to $x^2 - 2 = 0,$ and the rational number $p/q$ is the solution to $qx - p = 0.$ Conversely, a number which cannot be the root of any polynomial is called transcendental. Two examples include Euler's number (') and pi (). In general, proving a number is transcendental is considered to be very difficult, and only a few classes of transcendental numbers are known. However, it can be shown that the set of algebraic numbers is countable (for example, see '). Since the set of algebraic numbers is countable while the real numbers are uncountable (shown in the following section), the transcendental numbers must form the vast majority of real numbers, even though they are individually much harder to identify. That is to say, almost all real numbers are transcendental.

Uncountable sets

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 $(\R)$, which can be understood as the set of all numbers on the number line. One method of proving that the reals are uncountable is called Cantor's diagonal argument, credited to Cantor for his 1891 proof, English translation: though his differs from the more common presentation.

It begins by assuming, 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 ,1/math>). Then, take the decimal expansions of each real number, which looks like $0.d_1d_2d_3...$ Considering these real numbers in a column, create a new number such that the first digit of the new number is different from that of the first number in the column, the second digit is different from the second number in the column and so on. We also need to make sure that the number we create has a unique decimal representation, that is, it cannot end in repeating nines or repeating zeros. For example, if the digit isn't a 7, make the digit of the new number a 7, and if it was a seven, make it a 3. 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.

Another classical example of an uncountable set, established using a related reasoning, is the power set of the natural numbers, denoted $\mathcal(\N)$. This is the set of all subsets of $\N$, including the empty set and $\N$ itself. The method is much closer to Cantor's original diagonal argument. Again, assume by contradiction that there exists a one-to-one correspondence $f$ between $\N$ and $\mathcal(\N)$, so that every subset of $\N$ is assigned to some natural number. These subsets are then placed in a column, in the order defined by $f$ (see image). Now, one may define a subset $T$ of $\N$ which is not in the list by taking the negation of the "diagonal" of this column as follows:

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

These two sets, $\R$ and $\mathcal(\N)$ can be shown to have the same cardinality (by, for example, assigning each subset to a decimal expansion). Whether there exists a set $A$ with cardinality between these two sets $, \N, < , A, < , \R,$ is known as the continuum hypothesis.

Cantor's theorem generalizes the second theorem above, showing that every set is strictly smaller than its powerset. The proof roughly goes as follows: Given a set $A$, assume by contradiction that there is a bijection $f$ from $A$ to $\mathcal(A)$. Then, the subset $T \subseteq A$ given by taking the negation of the "diagonal", formally, $T = \$, cannot be in the list. Therefore, every function is missing at least one element, and so $, A, < , \mathcal(A),$. Further, since $\mathcal(A)$ is itself a set, the argument can be repeated to show $, A, < , \mathcal(A), < , \mathcal(\mathcal(A)),$. Taking $A = \N$, this shows that $\mathcal(\mathcal(\N))$ is even larger than $\mathcal(\N)$, which was already shown to be uncountable. Repeating this argument shows that there are infinitely many "sizes" of infinity.

Cardinal numbers

In the above section, "cardinality" of a set was defined relationally. In other words, while it was closely tied to the concept of number, the meaning of "number of elements" has not yet been defined. This can be formalized from basic set-theoretic principles, relying on some number-like structures. For finite sets, this is simply the natural number found by counting the elements. This number is called the ''cardinal number'' of that set, or simply ''the cardinality'' of that set. The cardinal number of a set $A$ is generally denoted by $, A, ,$ with a vertical bar on each side, though it may also be denoted by $n(A),$ $A$, $\operatorname(A),$ or $\#A.$

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

Finite sets

Given a basic sense of natural numbers, a set is said to have cardinality $n$ if it can be put in one-to-one correspondence with the set $\.$ For example, the set $S = \$ has a natural correspondence with the set $\,$ and therefore is said to have cardinality 4. Other terminologies include "Its cardinality is 4" or "Its cardinal number is 4". While this definition uses a basic sense of natural numbers, it may be that cardinality is used to define the natural numbers, in which case, a simple construction of objects satisfying the Peano axioms can be used as a substitute. Most commonly, the Von Neumann ordinals.

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

Uniqueness

An intuitive property of finite sets is that, for example, if a set has cardinality 4, then it cannot also have cardinality 5. Intuitively meaning that a set cannot have both exactly 4 elements and exactly 5 elements. However, it is not so obviously proven. The following proof is adapted from ''Analysis I'' by Terence Tao.

Lemma: If a set $X$ has cardinality $n \geq 1,$ and $x_0 \in X,$ then the set $X - \$ (i.e. $X$ with the element $x_0$ removed) has cardinality $n-1.$

Proof: Given $X$ as above, since $X$ has cardinality $n,$ there is a bijection $f$ from $X$ to $\.$ Then, since $x_0 \in X,$ there must be some number $f(x_0)$ in $\.$ We need to find a bijection from $X - \$ to $\$ (which may be empty). Define a function $g$ such that $g(x) = f(x)$ for all $x \neq n$ and $g(n) = f(x_0)$. Then $g$ is a bijection from $X - \$ to $\.$

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

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

Combinatorics

Combinatorics is the area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. The notion cardinality of finite sets is closely tied to many basic combinatorial principles, and provides a set-theoretic foundation to prove them. The above shows uniqueness of finite cardinal numbers, and therefore, $A \sim B$ if and only if $, A, = , B,$, formalizing the notion of a bijective proof.

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

Aleph numbers

The aleph numbers are a sequence of cardinal numbers that denote the size of infinite sets, denoted with an aleph $\aleph,$ the first letter of the Hebrew alphabet. The first aleph number is $\aleph_0,$ called "aleph-nought", "aleph-zero", or "aleph-null", which represents the cardinality of the set of all natural numbers: $\aleph_0 = , \N, = , \, .$ Then, $\aleph_1$ represents the next largest cardinality. The most common way this is formalized in set theory is through Von Neumann ordinals, known as Von Neumann cardinal assignment.

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

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

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

Cardinality of the continuum

The number line is a geometric construct of the intuitive notions of "space" and "distance" wherein each point corresponds to a distinct quantity or position along a continuous path. The terms "continuum" and "continuous" refer to the totality of this line, having some space (other points) between any two points on the line (dense and archimedian) and the absence of any gaps ( completeness), This intuitive construct is formalized by the set of real numbers $(\R)$ which model the continuum as a complete, densely ordered, uncountable set.

The cardinality of the continuum, denoted by "$\mathfrak c$" (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. ,1/math>, and ,2/math>, have the same cardinality as the entire set $\R$. First, $f(x) = 2x$ is a bijection from ,1/math> to ,2/math>. Then, the tangent function is a bijection from the interval $\left( \frac \, , \frac \right)$ to the whole real line. A more surprising example is the Cantor set, which is defined as follows: take the interval ,1/math> and remove the middle third $\left( \frac, \frac \right)$, 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 without a 1. Reinterpreting these decimal expansions as binary (e.g. by replacing the 2s with 1s) gives a bijection between the Cantor set and the interval ,1/math>.

Space-filling curves are continuous surjective maps from the unit interval ,1/math> onto the unit square on $\R^2$, 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 $, \R, = , \R^n, = \mathfrak$ for any dimension $n \geq 1.$ The infinite cartesian product $\R^\infty$, can also be shown to have cardinality $\mathfrak c$. This can be established by cardinal exponentiation: $, \R^\infty, = \mathfrak^ = \left(2^ \right)^ = 2^ = 2^ = \mathfrak = , \R,$. Thus, the real numbers, all finite-dimensional real spaces, and the countable cartesian product share the same cardinality.

As shown in , the set of real numbers is strictly larger than the set of natural numbers. Specifically, $, \R, = , \mathcal(\N),$. The Continuum Hypothesis (CH) asserts that the real numbers have the next largest cardinality after the natural numbers, that is $, \R, = \aleph_1$. As shown by Gödel and Cohen, the continuum hypothesis is 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 consistent. The Generalized Continuum Hypothesis (GCH) extends this to all infinite cardinals, stating that $2^ = \aleph_$ for every ordinal $\alpha$. Without GHC, the cardinality of $\R$ cannot be written in terms of alephs. The Beth numbers provide a concise notation for powersets of the real numbers starting from $\beth_1 = , \R,$, then $\beth_2 = , \mathcal(\R), = 2^$, and in general $\beth_ = 2^$.

Paradoxes

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

Apparent paradoxes

Hilbert's hotel

Hilbert's Hotel is a thought experiment devised by the German mathematician 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, all of which are occupied. But then a new guest walks in asking for a room. The hotel accommodates by moving the occupant of room 1 to room 2, the occupant of room 2 to room 3, room three to room 4, and in general, room n to room n+1. Then every guest still has a room, but room 1 opens up for the new guest.Archived
on 2016-01-06

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

Skolem's paradox

In model theory, a model corresponds to a specific interpretation of a formal language or theory. It consists of a domain (a set of objects) and an interpretation of the symbols and formulas in the language, such that the axioms of the theory are satisfied within this structure. The Löwenheim–Skolem theorem shows that any model of set theory in first-order logic, if it is consistent, has an equivalent model which is countable. This appears contradictory, because Georg Cantor proved that there exist sets which are not countable. Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets, satisfies the first-order sentence that intuitively states "there are uncountable sets".

A mathematical explanation of the paradox, showing that it is not a true contradiction in mathematics, was first given in 1922 by Thoralf Skolem. He explained that the countability of a set is not absolute, but relative to the model in which the cardinality is measured. Skolem's work was harshly received by Ernst Zermelo, who argued against the limitations of first-order logic and Skolem's notion of "relativity", but the result quickly came to be accepted by the mathematical community.

Real paradoxes

Cantor's paradox

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

Set of all cardinal numbers

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

See also

* Cardinal function
* Inaccessible cardinal
* Infinitary combinatorics
* Large cardinal
** List of large cardinal properties
* Numerosity (mathematics)
* Regular cardinal
* Scott's trick
* ''The Higher Infinite''
* Von Neumann universe
* Zero sharp

References

Citations

Bibliography

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Basic concepts in infinite set theory