Infinity is something which is boundless, limitless, or endless. It is denoted by , called the infinity symbol.
From the time of the ancient Greeks, the philosophical nature of infinity has been the subject of debate. In the 17th century, with the introduction of the infinity symbol and infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them.
In physics and cosmology, it is an open question whether the universe is spatially infinite or not.
History
Ancient cultures had various ideas about the nature of infinity. The ancient Indians and the 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 in Greece may be that of Anaximander (c. 610 – c. 546 BC) a pre-Socratic Greek philosopher. He used the word apeiron, which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
Aristotle (350 BC) distinguished potential infinity from actual infinity, which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic Greeks had a "horror of the infinite"Maor, p. 3 which would, for example, explain why Euclid (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers." It has also been maintained, that, in proving the infinitude of the prime numbers, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate, sometimes translated:
Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", thus avoiding the implication that Euclid was comfortable with the notion of infinity. Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.
Zeno: Achilles and the tortoise
Zeno of Elea ( 495 – 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 farther. Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet farther. Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet farther. And this continues infinitely. Basically,
Step #(n+1): Achilles advances to where the tortoise was at the end of Step #n while the tortoise goes yet farther.
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. As a member of the Eleatics 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 millennia 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 ,
Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with and . Achilles does overtake the tortoise; it takes him
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: Enumerable: lowest, intermediate, and highest Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable Infinite: nearly infinite, truly infinite, infinitely infinite
17th century In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. 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 (1656), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in "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. Extract of p. 62
Symbol
The infinity symbol (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at and in LaTeX as \infty.
It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
Calculus Gottfried 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.
Real analysis In real analysis, the symbol , called "infinity", is used to denote an unbounded limit. It is not a real number itself. The notation means that increases without bound, and means that decreases without bound. For example, if for every , thenThese uses of infinity for integrals and series can be found in any standard calculus text, such as, means that does not bound a finite area from to means that the area under is infinite. means that the total area under is finite, and is equal to
Infinity can also be used to describe infinite series, as follows: means that the sum of the infinite series converges to some real value means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
In addition to defining a limit, infinity can be also used 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.
Complex analysis
In complex analysis the symbol , called "infinity", denotes an unsigned infinite limit. The expression means that the magnitude of 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 (which leads to the one exception 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 . 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 (see Möbius transformation § Overview).
Nonstandard analysis
The original formulation of infinitesimal calculus by Isaac Newton and Gottfried Leibniz used infinitesimal quantities. In the second half of the 20th 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 in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in .
Set theory
A different form of "infinity" is the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, 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 19th century from works 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 (derived from Euclid) that the whole cannot be the same size as the part. (However, see Galileo's paradox where Galileo concludes that positive integers cannot be compared to the subset of positive square integers since both are infinite sets.) An infinite set can simply be defined as one having the same size as at least one of its proper subsets (a strictly smaller subset); this notion of infinity is called Dedekind infinite. The diagram to the right 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". Extract of page 118