In mathematics, a number greater than any other number. The symbol ? was first used for infinity by the English mathematician John Wallis (16161703). Cantor and other mathematicians in the 19th-c and 20th-c showed the complexity of the concept of infinity, arising out of their work on set theory, investigating, for example, how the cardinal number of the set of all points in an infinite straight line compares with the cardinal number of the set of all points in an infinite plane. Cantor's introduction of an endless hierarchy of infinite sets of different sizes admitted paradoxes that were resolved by the axiomatic approach of German mathematician Ernst Zermelo (18711953) and others, but the whole enterprise was strongly rejected by Dutch mathematician Jan Brouwer (18811966) and other intuitionists as exceeding the capacity of the human mind. Later developments have shown how infinitely large and infinitely small numbers can be introduced into the calculus.
In Greek philosophy, for example in Anaximander, 'the Boundless' is the origin of all that is. He took the beginning or first principle to be an endless, unlimited primordial mass (apeiron)
In Judeo-Christian theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity.
In mathematics, infinity is relevant to, or the subject matter of, limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals, Russell's paradox, hyperreal numbers, projective geometry, extended real numbers and the absolute Infinite.
In popular culture, we have Buzz Lightyear's rallying cry, "To infinity — and beyond!", which may also be viewed as the rallying cry of set theorists considering large cardinals.
For a discussion about infinity and the physical universe, see Universe.
History
Early Indian views of infinity
Along with the early conceptions of infinite space proposed by the Taoist philosophers in ancient China, one of the earliest known documented knowledge of infinity was also presented in ancient India in the Yajur Veda (c. 1200–900 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. Infinite: nearly infinite, truly infinite, infinitely infinite.
The Jains were the first to discard the idea that all infinites were the same or equal. They recognized different types of infinities: infinite in one and two directions (one dimension), infinite in area (two dimensions), infinite everywhere (three dimensions), and infinite perpetually (infinite number of dimensions).
According to Singh (1987), Joseph (2000) and Agrawal (2000), the highest enumerable number N of the Jains corresponds to the modern concept of aleph-null (the cardinal number of the infinite set of integers 1, 2, ...), the smallest cardinal transfinite number. The Jains also defined a whole system of infinite cardinal numbers, of which the highest enumerable number N is the smallest.
In the Jaina work on the theory of sets, two basic types of infinite numbers are distinguished.
Huston Smith, born in China, a philosopher and religion scholar, has said that in Hinduism:
“The invisible excludes nothing, the invisible that excludes nothing is the infinite — the soul of India is the infinite.”
“Philosophers tell us that the Indians were the first ones to conceive of a true infinite from which nothing is excluded. The trouble is that boundaries also imprison — they restrict and confine.”
“India saw this clearly and turned her face to that which has no boundary or whatever.” “India anchored her soul in the infinite seeing the things of the world as masks of the infinite assumes — there can be no end to these masks, of course. If they express a true infinity.” And It is here that India’s mind boggling variety links up to her infinite soul.”
“India includes so much because her soul being infinite excludes nothing.” It goes without saying that the universe that India saw emerging from the infinite was stupendous.”
Early European views of infinity
In Europe, the traditional view derives from Aristotle:
"... It is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential, never actual; [Physics 207b8]This is often called potential infinity; The other is that we may quantify over infinite sets without restriction. But the parts of the continuum are infinite because there are not so many that there are not more, and therefore the infinite parts are actually existent. However, on this view, no infinite magnitude can have a number, for whatever number we can imagine, there is always a larger one: "There are not so many (in number) that there are no more."
Views from the Renaissance to modern times
Galileo was the first to notice that we can place an infinite set into one-to-one correspondence with one of its proper subsets (any part of the set, that is not the whole).
It appeared, by this reasoning, as though a set which is naturally smaller than the set of which it is a part (since it does not contain all the members of that set) is in some sense the same size. He thought this was one of the difficulties which arise when we try, "with our finite minds," to comprehend the infinite.
"So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; and finally the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities." [On two New Sciences, 1638]The idea that size can be measured by one-to-one correspondence is today known as Hume's principle, although Hume, like Galileo, believed the principle could not be applied to infinite sets.
Locke, in common with most of the empiricist philosophers, also believed that we can have no proper idea of the infinite. yet when we would frame in our minds the idea of an infinite space or duration, that idea is very obscure and confused, because it is made up of two parts very different, if not inconsistent. 7., author's emphasis)
Famously, the ultra-empiricist Hobbes tried to defend the idea of a potential infinity in the light of the discovery, by Evangelista Torricelli, of a figure (Gabriel's horn) whose surface area is infinite, but whose volume is finite. Not reported, this motivation of Hobbes came too late as curves having infinite length yet bounding finite areas were known much before.
Modern philosophical views
Modern discussion of the infinite is now regarded as part of set theory and mathematics. An exception was Wittgenstein, who made an impassioned attack upon axiomatic set theory, and upon the idea of the actual infinite, during his "middle period". Neither is this infinite process itself in some sense or other such a pair of classes... 465)
Unlike the traditional empiricists, he thought that the infinite was in some way given to sense experience. "[Time] is infinite in the same sense as the three-dimensional space of sight and movement is infinite, even if in fact I can only see as far as the walls of my room." what is infinite about endlessness is only the endlessness itself."
Infinity symbol
The precise origins of the infinity symbol ∞ are unclear.
A popular explanation is that the infinity symbol is derived from the shape of a Möbius strip.
John Wallis is usually credited with introducing ∞ as a symbol for infinity in 1655 in his De sectionibus conicis.
The infinity symbol is represented in Unicode by the character ∞ (U+221E).
Mathematical infinity
Infinity is the state of being greater than any finite (real or natural) number, however large.
Infinity in real analysis
In real analysis, the symbol , called "infinity," denotes an unbounded limit.
Infinity is often used not only to define a limit but as if it were a value in the extended real numbers in real analysis; if f(t) ≥ 0 then
means that f(t) does not bound a finite area from 0 to 1 means that the area under f(t) is infinite. means that the area under f(t) equals 1Infinity in complex analysis
As in real analysis, in complex analysis the symbol , called "infinity", denotes an unbounded limit.
Infinities as part of the extended real number line
Infinity is not a real number but the extended real number line adds two elements called infinity (), greater than all other extended real numbers, and minus infinity (), less than all other extended real numbers, in which arithmetic operations involving these new elements may be performed. In this system, infinity, and minus infinity have the following arithmetic properties:
Infinity with itself
Operations involving infinity and real numbers
If then If thenUndefined operations
Notice that is not equivalent to .
Infinities in nonstandard analysis
The original formulation of the calculus by Newton and Leibniz used infinitesimal quantities. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. 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 different infinite numbers.
Infinity in set theory
A different type 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. 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, the ordinal numbers and the 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.
Our intuition gained from finite sets breaks down when dealing with infinite sets.
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;
Mathematics without infinity
Leopold Kronecker rejected the notion of infinity and began a school of thought, in the philosophy of mathematics called finitism, which led to the philosophical and mathematical school of mathematical constructivism.
In computing
The IEEE floating-point standard specifies positive and negative infinity values;
Some programming languages (for example, J and UNITY) specify greatest and least elements, i.e.
Use of infinity in common speech
In common parlance, infinity is often used in a hyperbolic sense. An infinite loop in computer programming is a conditional loop construction whose condition always evaluates to true. In practice, however, some programming loops considered infinite will halt by exceeding the finite number range of their variables. it is impossible to play a video game for an infinite period of time or keep a computer running for an infinite period of time.
The number Infinity plus 1 is also used sometimes in common speech.
Physical infinity
In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. 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 (see also: hyperreal number), 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. There exists the concept of infinite entities (such as an infinite plane wave) but there are no means to generate such things. Likewise, perpetual motion machines theoretically generate infinite energy by attaining 100% efficiency or greater, and emulate every conceivable open system; the impossible problem follows of knowing that the output is actually infinite when the source or mechanism exceeds any known and understood system.
It should be pointed out that this 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. 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. 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 masss object, which is not what we can observe in reality.
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.
Infinity in cosmology
An intriguing question is whether actual infinity exists in our physical universe: Are there infinitely many stars? Does the universe have infinite volume? Note that the question of being infinite is logically separate from the question of having boundaries. If, however, the universe is ever expanding then you could never get back to your starting point even on an infinite time scale.
Three types of infinities
Besides the mathematical infinity and the physical infinity, there could also be a philosophical infinity.
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