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# Infinity

At the end of 19th century, Georg Cantor introduced and studied infinite sets and infinite numbers, which are now an essential part of the foundation of mathematics.[1] For example, in modern mathematics, a line is commonly viewed as the set of all its points, and their infinite number (the cardinality of the line) is larger than the number of integers.[2] Thus the mathematical concept of infinity refines and extends the old philosophical concept. It is used everywhere in mathematics, even in areas such as combinatorics and number theory that may seem to have nothing to do with it. For example, Wiles's proof of Fermat's Last Theorem uses the existence of very large infinite sets.

The concept of infinity is also used in physics and the other sciences.

## History

The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.

### Early Greek

The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus. He used the word apeiron, which means infinite or limitless.[3] However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea (born c. 490 BCE), a pre-Socratic Greek philosopher of southern Italy and member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic.[4][5] He is best known for his paradoxes,[4] described by Bertrand Russell as "immeasurably subtle and profound".[6]

In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the actual infinity; for example, instead of saying that there are an infinity of primes, Euclid prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers.[7]

### Early Indian

The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:[8]

• Enumerable: lowest, intermediate, and highest

• Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable

• Infinite: nearly infinite, truly infinite, infinitely infinite

In this work, two basic types of infinite numbers are distinguished.

On both physical and ontological grounds, a distinction was made between asaṃkhyāta*]]("countless, innumerable") and* ananta"endless, unlimited"), between rigidly bounded and loosely bounded infinities.[9]

### 17th century

In 1699 Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.[12]

## Mathematics

Hermann Weyl opened a mathematico-philosophic address given in 1930 with:[13]

Mathematics is the science of the infinite.

### Infinity symbol

It was introduced in 1655 by John Wallis,[14][15] and, since its introduction, has also been used outside mathematics in modern mysticism[16] and literary symbology.[17]

### Calculus

Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.[18][19]

#### Real analysis

• means that f (t) does not bound a finite area from to

• means that the area under f (t) is infinite.

• means that the total area under f (t) is finite, and equals

Infinity is also used to describe infinite series:

• means that the sum of the infinite series converges to some real value

• means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.

### Nonstandard analysis

The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the twentieth century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in Keisler (1986).

### Set theory

A different form of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of collections, or sets.

Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive integers which are squares and all positive integers are the same size). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".

Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers.

Ordinal numbers may be identified with well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity.[25] Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.

#### Cardinality of the continuum

Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.

The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square.[28]

### Geometry and topology

Infinite-dimensional spaces are widely used in geometry and topology, particularly as classifying spaces, such as Eilenberg−MacLane spaces. Common examples are the infinite-dimensional complex projective space K(Z,2) and the infinite-dimensional real projective space K(Z/2Z,1).

### Fractals

The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters—some with infinite, and others with finite surface areas. One such fractal curve with an infinite perimeter and finite surface area is the Koch snowflake.

### Mathematics without infinity

Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of constructivism and intuitionism.[29]

## Physics

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value, for instance by taking an infinite value in an extended real number system, or by requiring the counting of an infinite number of events. It is, for example, presumed impossible for any type of body to have infinite mass or infinite energy. Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.[30]

### Cosmology

The first published proposal that the universe is infinite came from Thomas Digges in 1576.[31] Giordano Bruno]] proposed an unbounded universe in numerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."[32]

Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.[33]

The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. As to date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.[34][35][36]

However, the universe could be finite, even if its curvature is flat.

An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other.

The topology of such games is toroidal and the geometry is flat.

Many possible bounded, flat possibilities also exist for three-dimensional space.[37]

The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes.[38]

## Logic

In logic an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."[39]

## Computing

The IEEE floating-point standard (IEEE 754) specifies the positive and negative infinity values (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.

Some programming languages, such as Java[40] and J,[41] allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least elements, as they compare (respectively) greater than or less than all other values. They have uses as sentinel values in algorithms involving sorting, searching, or windowing.

In languages that do not have greatest and least elements, but do allow overloading of relational operators, it is possible for a programmer to create the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floating-point data type, the infinity values may still be accessible and usable as the result of certain operations.

In programming, an infinite loop is a loop whose exit condition is never satisfied, thus theoretically executing indefinitely.

Infinite sequences can be represented in the finite memory of a computer as a composite data structure consisting of a few first members of the sequence, and a recursive routine for computing the nth element from the preceding ones. Several techniques can be used for avoiding computing several times the same element of the sequence. One is lazy evaluation. Another one, available in Maple, consists of having routines with a remember option. This option consists of keeping in memory the results of the function that have been computed, and, at each call of the routine, looking if this particular result has been computed, for avoiding to compute it again. For example, the standard definition of the Fibonacci sequence is

Fib(n) == ifn= 1 orn= 2 then 1 else Fib(n
• Fib(
n
• 2).

With a standard implementation, an exponential number of function calls is needed for computing Fib(n), while, with the remember option, only n - 1 function calls are needed.

## Arts, games, and cognitive sciences

Perspective artwork utilizes the concept of vanishing points, roughly corresponding to mathematical points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms.[42] Artist M.C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.

Variations of chess played on an unbounded board are called infinite chess.[43][44]

Cognitive scientist George Lakoff considers the concept of infinity in mathematics and the sciences as a metaphor. This perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1,2,3,...>.

The symbol is often used romantically to represent eternal love.

Several types of jewelry are fashioned into the infinity shape for this purpose.

• 0.999...

• Aleph number

• Exponentiation

• Indeterminate form

• Infinite monkey theorem

• Infinite set

• Infinitesimal

• Surreal number

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