Infinity

Georg Cantor makes an appearance as a character, and the hero finds a physical correlate for Cantor's Continuum Problem. His feast day in the Roman Catholic and Eastern Orthodox Churches is 3 June. Rudy Rucker's novel White Light describes a mathematician who leaves his body and travels to a kind of afterworld that includes a mountain whose Absolute Infinite height matches that of the class of all ordinals. His legend says that he lived to the age of 120. Looking up into the night sky is looking into infinity -- distance is incomprehensible and therefore meaningless.". Eventually, Glendalough, with its seven churches, became one of the chief pilgrimage destinations in Ireland. Another quote from The Hitchhiker's Guide to the Galaxy states: "Infinity itself looks flat and uninteresting. This monastery was to become the parent of several others.

The Hitchhiker's Guide to the Galaxy contains the following definition of infinity:. He was at Disert-Coemgen for a time and later established a church for his own community at Glendalough. Brouwer, David Hilbert, Bertrand Russell, Kurt Gödel and Georg Cantor. It maintains that he was descended from a royal line, given the name Coemgen, which means "fair-begotten", and educated by Petrock during that saint's sojourn in Ireland. The footnote on p.335 of his book suggests the consideration of the following names: Abraham Robinson, Plato, Thomas Aquinas, L.E.J. His legend is particularly untrustworthy, as very little contemporaneous material exists. Rudy Rucker, in his book Infinity and the Mind -- the science and philosophy of the mind (1982), has worked out a model list of representatives of each of the eight possible standpoints. "Kevin" is the English-language spelling of the Irish name Caoimhín (Coemgen in Old Irish, latinized as Coemgenus).

And in between there are the various possibilities. 498 - 618) is a Christian saint who was the Abbot of Glendalough in County Wicklow, Ireland. There are scientists who hold that all three really exist and there are scientists who hold that none of the three exist. Saint Kevin (c. Besides the mathematical infinity and the physical infinity, there could also be a philosophical infinity. If the universe is indeed ever expanding as science suggests then you could never get back to your starting point even on an infinite time scale.

The universe, at least in principle, might have a similar topology; if you fly your space ship straight ahead long enough, perhaps you would eventually revisit your starting point. By walking/sailing/driving straight long enough, you'll return to the exact spot you started from. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. Note that the question of being infinite is logically separate from the question of having boundaries.

An intriguing question is whether actual infinity exists in our physical universe: Are there infinitely many stars? Does the universe have infinite volume? Does space "go on forever"? This is an important open question of cosmology. 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. Physicists however require that the end result be physically meaningful. For convenience sake, calculations, equations, theories and approximations often use infinite series, unbounded functions, etc., and may involve infinite quantities.

This point of view does not mean that infinity cannot be used in physics. 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. There exists the concept of infinite entities (such as an infinite plane wave) but there are no means to generate such things. It is for example presumed impossible for any body to have infinite mass or infinite energy.

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. counting). In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. The number Infinity plus 1 is also used sometimes in common speech.

These terms describe things that are only potential infinities; 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. See halting problem. In practice however, some programming loops considered as infinite will halt by exceeding the (finite) number range of one of its variables. In theory, as long as there is no external interaction, the loop will continue to run for all time.

An infinite loop in computer programming is a conditional loop construction whose condition always evaluates to true. In video games, infinite lives and infinite ammo refer to a never-ending supply of lives and ammunition. For example, "The movie was infinitely boring, but we had to wait forever to get tickets.". In common parlance, infinity is often used in a hyperbolic sense.

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. One example of this is Hilbert's paradox of the Grand Hotel. Our intuition gained from finite sets breaks down when dealing with infinite sets. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.

Cantor's views prevailed and modern mathematics accepts actual infinity. If a set is too large to be put in one to one correspondence with the positive integers, it is called uncountable. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. 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.

Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences. 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. Cantor defined two kinds of infinite numbers, the ordinal numbers and the cardinal numbers. 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.

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. 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. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (), the cardinality of the set of natural numbers. A different type of "infinity" are the ordinal and cardinal infinities of set theory.

This is because zero times infinity is undefined. Notice that . Infinity is not a real number but may be considered part of the extended real number line, in which arithmetic operations involving infinity may be performed.
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One important example of such functions is the group of Möbius transformations. The domain of a complex-valued function may be extended to include the point at infinity as well. In this context is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of at the poles. When this is done, the resulting space is still a one-dimensional complex manifold and called the extended complex plane or the Riemann sphere.

A point labeled can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. means that the magnitude | x | of x grows beyond any assigned value. As in real analysis, in complex 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.

Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions. We can also treat and as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Adding algebraic properties to this gives us the extended real numbers. Points labeled and can be added to the real numbers as a topological space, producing the two-point compactification of the real numbers.

means that x grows beyond any assigned value, and means x is eventually less than any assigned value. In real analysis, the symbol , called "infinity", denotes an unbounded limit. The infinity symbol is represented in Unicode by the character ∞ (∞). Another conjecture is that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet.

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". John Wallis is usually credited with introducing ∞ as a symbol for infinity in 1655 in his De sectionibus conicus. However, this explanation is improbable, since the symbol had been in use to represent infinity for over two hundred years before August Ferdinand Möbius and Johann Benedict Listing discovered the Möbius strip in 1858. Again, one can imagine walking along its surface forever.

A popular explanation is that the infinity symbol is derived from the shape of a Möbius strip. One can imagine walking forever along a simple loop formed from a ribbon. One possibility is suggested by the name it is sometimes called — the lemniscate, from the Latin lemniscus, meaning "ribbon". The precise origins of the infinity symbol ∞ are unclear.

Unlike the traditional empiricists, he thought that the infinite was in some way given to sense experience. 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".². Modern discussion of the infinite is now regarded as part of set theory and mathematics, and generally avoided by philosophers. A potential infinity is allowed by letting an infinitely-large quantity be cancelled out by an infinitely-small quantity.

Potentiality lies in the definitions of this operation, as well-defined and interconsistent mathematical axioms. Such seeming paradoxes are resolved by taking any finite figure and stretching its content infinitely in one direction; the magnitude of its content is unchanged as its divisions drop off geometrically but the magnitude of its bounds increases to infinity by necessity. Not reported, this motivation of Hobbes came too late as curves having infinite length yet bounding finite areas were known much before. 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.

Our idea of infinity is merely negative or privative. They believed all our ideas were derived from sense data or "impressions", and since all sensory impressions are inherently finite, so too are our thoughts and ideas. Locke, in common with most of the empiricist philosophers, also believed that we can have no proper idea of the infinite. 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.

He thought this was one of the difficulties which arise when we try, "with our finite minds", to comprehend the infinite. 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. For example, we can match up the "set" of even numbers {2, 4, 6, 8 ...} with the natural numbers {1, 2, 3, 4 ...} as follows:. Galileo (during his long house arrest in Siena after his condemnation by the Inquisition) 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).

Aquinas also argued against the idea that infinity could be in any sense complete, or a totality. 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". The parts are actually there, in some sense. The second view is found in a clearer form by medieval writers such as William of Ockham:.

For example, ∀n∈Z(∃m∈Z[m>n∧P(m)]), which reads, "for any integer n, there exists an integer m > n such that P(m)". The other is that we may quantify over infinite sets without restriction. One is that it is always possible to find a number of things that surpasses any given number, even if there are not actually such things. This is often called potential infinity; however there are two ideas mixed up with this.

In Europe, the traditional view derives from Aristotle:. [1] [2] The concept of different orders of infinity would remain unknown in Europe until the late 19th century. It recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite.

The Indian Jaina mathematical text Surya Prajinapti (ca. The earliest known documented knowledge of infinity is presented in the Veda- Yajur Veda which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". . For a discussion about infinity and the physical universe, see Universe.

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.¹. By some, infinity is considered to be not a number but a concept of increase beyond bounds. In mathematics, infinity is relevant to, or the subject matter of, articles such as mathematical 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 both theology and philosophy, infinity is explored in articles such as the Ultimate, the Absolute, God, and Zeno's paradoxes.

In philosophy, infinity can be attributed to space and time, as for instance in Kant's first antinomy. In 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. The word infinity comes from the Latin infinitas, "unboundedness". Popular or colloquial usage of the term often does not accord with its more technical meanings.

Infinity refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. and . and . If then and .

If then and . and . and . means that the area under f(t) approaches 1.

means that the area under f(t) is not finite. means that f(t) does not bound a finite area from 0 to 1.