Philosophers have speculated about the nature of the infinite, for example Zeno of Elea , who proposed many paradoxes involving infinity, and Eudoxus of Cnidus , who used the idea of infinitely small quantities in his method of exhaustion. Modern mathematics uses the general concept of infinity in the solution of many practical and theoretical problems, such as in calculus and set theory , and the idea also is used in physics and the other sciences. In mathematics, "infinity" is often treated as a number i.
In the theory he developed, there are infinite sets of different sizes called cardinalities. 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 comes from Anaximander , a pre-Socratic Greek philosopher who lived in Miletus.
He used the word apeiron which means infinite or limitless. Aristotle called him the inventor of the dialectic. 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.
However, recent readings of the Archimedes Palimpsest have found that Archimedes had an understanding about actual infinite quantities. According to Nonlinear Dynamic Systems and Controls , Archimedes has been found to be "the first to rigorously address the science of infinity with infinitely large sets using precise mathematical proofs.
The Jain mathematical text Surya Prajnapti c. Each of these was further subdivided into three orders: In this work, two basic types of infinite numbers are distinguished. European mathematicians started using infinite numbers and expressions in a systematic fashion in the 17th century. In Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas. Hermann Weyl opened a mathematico-philosophic address given in with: It was introduced in by John Wallis ,   and, since its introduction, has also been used outside mathematics in modern mysticism  and literary symbology.
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.
Infinity is also used to describe infinite series:. Infinity can be used not only to define a limit but as a value in the extended real number system.
Adding algebraic properties to this gives us the extended real numbers. 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.
The domain of a complex-valued function may be extended to include the point at infinity as well. 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. This approach to non-standard calculus is fully developed in Keisler A different form of "infinity" are the ordinal and cardinal infinities of set theory.
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: Cantor defined two kinds of infinite 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.
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.
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. 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. Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the s and s. 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. It is therefore assumed by physicists that no measurable quantity could have an infinite value, [ citation needed ] 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 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 [ disputed — discuss ] [ citation needed ].
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.
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. The first published proposal that the universe is infinite came from Thomas Digges in 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. 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.
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.More...