Infinity

The lemniscate, ∞, in several typefaces.

Infinity (sometimes symbolically represented by ∞) is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity. The word comes from the Latin infinitas or "unboundedness".

In mathematics, "infinity" is often 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. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e. a number greater than any real number. 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" of infinite sets, a concept called cardinality. For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite.

History

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 Greek

The earliest attestable acounts of mathematical infinity come from Zeno of Elea (ca. 490 BC? – ca. 430 BC?), 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. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".

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 (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.

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 and vajrayana meditation deities such as Chenrezig, the deity is 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 and rebirth, i.e. the [infinity of] samsara.

Mathematics

The infinity symbol

John Wallis introduced the infinity symbol to mathematical literature.

The infinity symbol $\infty$ is sometimes called the lemniscate, from the Latin lemniscus, meaning "ribbon". John Wallis is credited with introducing the symbol in 1655 in his De sectionibus conicis.^[1] ^[2] One conjecture about why he chose this symbol is that he derived it from a Roman numeral for 1000 that was in turn derived from the Etruscan numeral for 1000, which looked somewhat like CIƆ and was sometimes used to mean "many." Another conjecture is that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet.^[3] Also, before typesetting machines were invented, ∞ was easily made in printing by typesetting an 8 type on its side.

The infinity symbol is available in standard HTML as ∞ and in LaTeX as \infty. In Unicode, it is the character at code point U+221E (∞), or 8734 in decimal notation.

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.^[4]^[5]

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 nonzero complex number z. 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

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 Howard 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

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.

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 in the side of a square and those in the square.

Geometry and topology

Infinite-dimensional spaces are widely used in geometry and topology, particularly as classifying spaces, notably 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 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of the philosophical and mathematical schools of constructivism and Intuitionism.^[6]

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 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.

Theoretical applications of physical infinity

The practice of refusing infinite values for measurable quantities does not come from a priori or ideological motivations, but rather from more methodological and pragmatic motivations. One of the needs of any physical and scientific theory is to give usable formulas that correspond to or at least approximate reality. As an example if any object of infinite gravitational mass were to exist, any usage of the formula to calculate the gravitational force would lead to an infinite result, which would be of no benefit since the result would be always the same regardless of the position and the mass of the other object. The formula would be useful neither to compute the force between two objects of finite mass nor to compute their motions. If an infinite mass object were to exist, any object of finite mass would be attracted with infinite force (and hence acceleration) by the infinite mass object, which is not what we can observe in reality. Sometimes infinite result of a physical quantity may mean that the theory being used to compute the result may be approaching the point where it fails. This may help to indicate the limitations of a theory.

This point of view does not mean that infinity cannot be used in physics. For convenience's sake, calculations, equations, theories and approximations often use infinite series, unbounded functions, etc., and may involve infinite quantities. Physicists however require that the end result be physically meaningful. In quantum field theory infinities arise which need to be interpreted in such a way as to lead to a physically meaningful result, a process called renormalization.

However, there are some theoretical circumstances where the end result is infinity. One example is the singularity in the description of black holes. Some solutions of the equations of the general theory of relativity allow for finite mass distributions of zero size, and thus infinite density. This is an example of what is called a mathematical singularity, or a point where a physical theory breaks down. This does not necessarily mean that physical infinities exist; it may mean simply that the theory is incapable of describing the situation properly. Two other examples occur in inverse-square force laws of the gravitational force equation of Newtonian gravity and Coulomb's law of electrostatics. At r=0 these equations evaluate to infinities.

Cosmology

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. Note that 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 one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology; if one travelled in a straight line through the universe perhaps one would eventually revisit one's starting point.

If, on the other hand, the universe were not curved like a sphere but had a flat topology, it could be both unbounded and infinite. 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. The Planck spacecraft launched in 2009 is expected to record the cosmic background radiation with 10 times higher precision, and will give more insight into the question whether the universe is infinite or not.

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."^[7]

Computing

The IEEE floating-point standard specifies positive and negative infinity values; these can be the result of arithmetic overflow, division by zero, or other exceptional operations.

Some programming languages (for example, J and UNITY) specify greatest and least elements, i.e. values that compare (respectively) greater than or less than all other values. These may also be termed top and bottom, or plus infinity and minus infinity; they are useful 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 to create greatest and least elements.

The arts and cognitive sciences

Perspective artwork utilizes the concept of imaginary vanishing points, or points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms.^[8] Artist M. C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.

From the perspective of cognitive scientists George Lakoff, concepts of infinity in mathematics and the sciences are metaphors, based on what they term the Basic Metaphor of Infinity (BMI), namely the ever-increasing sequence <1,2,3,...>.

References

↑ Scott, Joseph Frederick (1981), The mathematical work of John Wallis, D.D., F.R.S., (1616-1703) (2 ed.), AMS Bookstore, p. 24, ISBN 0-828-40314-7, http://books.google.com/books?id=XX9PKytw8g8C , Chapter 1, page 24
↑ COLOG-88: International Conference on Computer Logic Tallinn, USSR, December 12-16, 1988: proceedings, Springer, 1990, p. 147, ISBN 3-540-52335-9, http://books.google.com/books?id=nfnGohZvXDQC , page 147
↑ The History of Mathematical Symbols, By Douglas Weaver, Mathematics Coordinator, Taperoo High School with the assistance of Anthony D. Smith, Computing Studies teacher, Taperoo High School.
↑ Continuity and Infinitesimals entry by John Lane Bell in the Stanford Encyclopedia of Philosophy
↑ Jesseph, Douglas Michael (1998). "Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes". Perspectives on Science 6 (1&2): 6–40. ISSN 1063-6145. OCLC 42413222. Archived from the original on 16 February 2010. http://www.webcitation.org/5nZWht6FE. Retrieved 16 February 2010.
↑ Kline, Morris (1972). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press. pp. 1197–1198. ISBN 77-170263.
↑ Cambridge Dictionary of Philosophy, Second Edition, p. 429
↑ Kline, Morris (1985). Mathematics for the nonmathematician. Courier Dover Publications. p. 229. ISBN 0-486-24823-2. http://books.google.com/books?id=f-e0bro-0FUC&pg=PA229. , Section 10-7, p. 229

Other references

Amir D. Aczel (2001). The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity. New York: Pocket Books. ISBN 0-7434-2299-6.
D. P. Agrawal (2000). Ancient Jaina Mathematics: an Introduction, Infinity Foundation.
Bell, J. L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.
L. C. Jain (1982). Exact Sciences from Jaina Sources.
L. C. Jain (1973). "Set theory in the Jaina school of mathematics", Indian Journal of History of Science.
George G. Joseph (2000). The Crest of the Peacock: Non-European Roots of Mathematics (2nd edition ed.). Penguin Books. ISBN 0-14-027778-1.
H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
Eli Maor (1991). To Infinity and Beyond. Princeton University Press. ISBN 0-691-02511-8.
John J. O'Connor and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor', MacTutor History of Mathematics archive.
John J. O'Connor and Edmund F. Robertson (2000). 'Jaina mathematics', MacTutor History of Mathematics archive.
Ian Pearce (2002). 'Jainism', MacTutor History of Mathematics archive.
Rudy Rucker (1995). Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton University Press. ISBN 0-691-00172-3.
Navjyoti Singh (1988). Jaina Theory of Actual Infinity and Transfinite Numbers. 30.
David Foster Wallace (2004). Everything and More: A Compact History of Infinity. Norton, W. W. & Company, Inc.. ISBN 0-393-32629-2.

External links

A Crash Course in the Mathematics of Infinite Sets, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1-59. The stand-alone appendix to Infinite Reflections, below. A concise introduction to Cantor's mathematics of infinite sets.
Infinite Reflections, by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1-59.
Infinity, Principia Cybernetica
Hotel Infinity
Source page on medieval and modern writing on Infinity
The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity
Dictionary of the Infinite (compilation of articles about infinity in physics, mathematics, and philosophy)

Large numbers

Subarticles

Names of large numbers · History of large numbers

Examples in numerical order

million · googol · googolplex · Skewes' number · Moser's number · Graham's number · Transfinite numbers · Infinity

Expression methods

Notations	Knuth's up-arrow notation · Conway chained arrow notation · Steinhaus–Moser notation

Operators	Hyper operators (Tetration) · Ackermann function

Number systems · Number names · Orders of magnitude · List of numbers · Indefinite and fictitious numbers · Extended real number line