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The Lemniscate, ∞, in several typefaces.

Infinity (symbolically represented by ∞) is a concept in mathematics and philosophy that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity. In mathematics, infinity is defined in the context of set theory. The word comes from the Latin infinitas or "unboundedness."

In mathematics, "infinity" is often used in contexts where it is treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers. The German mathematician Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. He also discovered that there are different "kinds" or "measures" of infinity, a concept called cardinality. For example, the set of integers is countably infinite. However, the set of real numbers is uncountably infinite.

A set of elements can be defined as infinite if the set has a seemingly paradoxical quality: a subset of elements in an infinite set can be matched up, one-to-one, to all of the elements in a set.[1] The paradoxical nature of infinity is illustrated by the idea of a grand hotel, with infinitely many rooms—all of which are occupied by guests—but can nevertheless manage to accommodate a new guest by moving each existing guest over, one by one, to other rooms.

History

Main article: Infinity (philosophy)

Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks, unable to codify infinity in terms of a formalized mathematical system approached infinity as a philosophical concept.

Early Indian

The Isha Upanishad of the Yajurveda (c. 4th to 3rd century BC) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".

The Indian mathematical text Surya Prajnapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:

Enumerable: lowest, intermediate, and highest
Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
Infinite: nearly infinite, truly infinite, infinitely infinite

In the Indian work on the theory of sets, 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.

Buddhism

In some Buddhist imagery including Tibetan Buddhist thangka & vajrayana meditation deities such as Chenrezig the deity is often pictured holding a mala twisted in the middle to form a figure of 8. This represents the endless (infinite) cycle of existence, of birth, death & rebirth, i.e. the [infinity] of samsara.

Early Greek

In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the actual infinite; 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 (Elements, Book IX, Proposition 20).

However, recent readings of the Archimedes Palimpsest have hinted that at least Archimedes had an intuition about actual infinite quantities.

Mathematics

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.[2][3]^{[verification needed]}

Real analysis

In real analysis, the symbol $\infty$ , called "infinity", denotes an unbounded limit. $x \rightarrow \infty$ means that x grows without bound, and $x \to -\infty$ means the value of x is decreasing without bound. If f(t) ≥ 0 for every t, then

$\int_{a}^{b} \, f(t)\ dt \ = \infty$ means that f(t) does not bound a finite area from a to b
$\int_{-\infty}^{\infty} \, f(t)\ dt \ = \infty$ means that the area under f(t) is infinite.
$\int_{-\infty}^{\infty} \, f(t)\ dt \ = n$ means that the total area under f(t) is finite, and equals n

Infinity is also used to describe infinite series:

$\sum_{i=0}^{\infty} \, f(i) = a$ means that the sum of the infinite series converges to some real value a.
$\sum_{i=0}^{\infty} \, f(i) = \infty$ means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.

Infinity is often used not only to define a limit but as a value in the affinely extended real number system. Points labeled $+\infty$ and $-\infty$ can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat $\infty$ and $-\infty$ as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions.

Complex analysis

As in real analysis, in complex analysis the symbol $\infty$ , called "infinity", denotes an unsigned infinite limit. $x \rightarrow \infty$ means that the magnitude $| x |$ of x grows beyond any assigned value. A point labeled $\infty$ can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given below for the extended real numbers can also be defined, though there is no distinction in the signs (therefore one exception is that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely $z/0 = \infty$ for any complex number z except for zero. In this context it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of $\infty$ at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations.

Nonstandard analysis

Main article: Nonstandard analysis

The original formulation of infinitesimal calculus by Newton and 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 whole 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 H. Jerome Keisler's book (see below).

Set theory

Main articles: Cardinality and Ordinal number

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 $(\aleph_0)$ , 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. 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.

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. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.

Cardinality of the continuum

Main article: Cardinality of the continuum

One of Cantor's most important results was that the cardinality of the continuum $\mathbf c$ is greater than that of the natural numbers ${\aleph_0}$ ; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that $\mathbf{c} = 2^{\aleph_0} > {\aleph_0}$ (see Cantor's diagonal argument).

The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, $\mathbf{c} = \aleph_1 = \beth_1$ (see Beth one). However, this hypothesis can neither be proved nor disproved within the widely accepted Zermelo-Fraenkel set theory, even assuming the Axiom of Choice.

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 that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.

The first three steps of a fractal construction whose limit is a space-filling curve, showing that there are as many points in a one-dimensional line segment as in a two-dimensional square.

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