Infinity (symbol: ) is a concept describing something without any bound, or something larger than any natural number. Since pre-Socratic times, philosophers have speculated about the nature of the infinite. For example, Zeno of Elea proposed many paradoxes involving infinity, and Eudoxus of Cnidus used the idea of indefinitely small quantities in his method of exhaustion, though the notion of passing to the limit—which lies at the basis of the infinitesimal calculus—was never formulated in ancient Greece..
At the end of 19th century, Georg Cantor introduced and studied infinite sets and infinite numbers, which are now an essential part of the foundations of mathematics [clarification needed] Zermelo–Fraenkel set theory. For example, in modern mathematics, a line is commonly viewed as the set of all of its points, and their infinite number (the cardinality of the line) is larger than the number of integers. Thus the mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes (called cardinalities) of infinite sets.
The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem is written with the implicit reliance on the existence of very large infinite sets. The concept of infinity is also used in physics and the other sciences.
- 1 History
- 2 Mathematics
- 3 Physics
- 4 Logic
- 5 Computing
- 6 Arts, games, and cognitive sciences
- 7 See also
- 8 Notes
- 9 References
- 10 Sources
- 11 External links
Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
The earliest recorded idea of infinity may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded" and perhaps can be translated "infinite".
In accordance with the traditional view of Aristotle, the Hellenistic Greeks had a "horror of the infinite" and generally preferred to distinguish the potential infinity from the actual infinity; for example, instead of saying that there are an infinity of primes, Euclid said that there are more prime numbers than contained in any given collection of prime numbers.
Zeno: Achilles and the tortoise
Zeno of Elea (c. 495 – c. 430 BC) did not advance any views concerning the infinite. Nevertheless his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions.The paradoxes were described by Bertrand Russell as "immeasurably subtle and profound".
Achilles races a tortoise, giving the latter a head start.
- Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
- Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
- Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
- Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
Apparently Achilles never overtakes the tortoise since, however many steps he completes, the tortoise remains ahead of him.
Zeno was not attempting to make a point about infinity. A member of the Eleatic school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millenia to find other weaknesses in the argument.
Finally, in 1821, Augustin-Louis Cauchy provided both a satisfactory definition of a limit and a proof that, for 0 < x < 1,
- a + ax + ax2 + ax3 + ax4 + ax5 + · · · = a/ .
Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100 meter head start. The duration of the chase fits Cauchy's pattern with a = 10 seconds and x = 0.01. Achilles does overtake the tortoise; it takes him
- 10 + 0.1 + 0.001 + 0.00001 + · · · = 10/ = 10/ = 10 10/ seconds.
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:
- Enumerable: lowest, intermediate, and highest
- Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
- Infinite: nearly infinite, truly infinite, infinitely infinite
European mathematicians started using infinite numbers and expressions in a systematic fashion in the 17th century. In 1655 John Wallis first used the notation for such a number in his De sectionibus conicis and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of  But in Arithmetica infinitorum (1655 also) he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c." For example, "1, 6, 12, 18, 24, &c."
In 1699 Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.
Mathematics is the science of the infinite.
The infinity symbol (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ INFINITY (HTML
∞) and in LaTeX as
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.
In real analysis, the symbol , called "infinity", is used to denote an unbounded limit. The notation means that x grows without bound, and means that x decreases without bound. If f(t) ≥ 0 for every t, then
- 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.
Infinity can be used not only to define a limit but also as a value in the extended real number system. Points labeled and 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 and as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space, and a hyperplane at infinity for general dimensions, each consisting of points at infinity.
In complex analysis the symbol , called "infinity", denotes an unsigned infinite limit. means that the magnitude of x grows beyond any assigned value. A point labeled 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 above 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 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 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.
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).
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.[page needed] 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 is greater than that of the natural numbers ; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that (see Cantor's diagonal argument or Cantor's first uncountability proof).
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, (see Beth one). 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 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.
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).
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.
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.
The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable 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."
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.
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.
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.
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."
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 and J, 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.
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) == if n = 1 or n = 2 then 1 else Fib(n - 1) + 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. Artist M.C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.
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.
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- "The Infinite". Internet Encyclopedia of Philosophy.
- Infinity on In Our Time at the BBC
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- 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.
- Grime, James. "Infinity is bigger than you think". Numberphile. Brady Haran.
- Hotel Infinity
- 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.
- 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)