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Infinity

For the automobile brand, see Infiniti.

Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. Popular usage is often not in accordance with the term's more technical meanings.

In theology, for instance in the work of Duns Scotus, the infinity of God carries the sense not so much of quantity (leading to the question, quantity of what?) but of unconstrainedness. In philosophy, infinity can be attributed to space and time, as for instance in Kant's first antinomy. In popular culture, we find Buzz Lightyear's rallying cry, "To infinity -- and beyond!" which might also be called the rallying cry of set theorists considering large cardinals, which are quantitative infinities, defining the number of things in a collection, so large that they cannot be proven to exist in the ordinary mathematics of Zermelo-Fraenkel plus Choice (ZFC), and which might be so large they embody a contradiction.

In mathematics, some articles relevant to the subject can be found at limit (mathematics), aleph number, class (set theory), Dedekind infinite, large cardinal, Russell's paradox, hyperreal numbers, projective geometry, extended real number and absolute infinite. In philosophy and theology, one can investigate the Ultimate, the Absolute, God, and Zeno's paradoxes.

For a discussion about infinity and the physical universe, see Universe.

Contents

History

Ancient view of infinity

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; the number of parts that can be taken always surpasses any assigned number." [Physics 207b8]

This is often called "potential" infinity; however there are two ideas mixed up with this. 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. The other is that we may quantify over finite numbers without restriction. For example "For any integer n, there exists an integer m > n such that P(m)". The second view is found in a clearer form by medieval writers such as William of Ockham:

"Sed omne continuum est actualiter existens. Igitur quaelibet pars sua est vere existens in rerum natura. Sed partes continui sunt infinitae quia non tot quin plures, igitur partes infinitae sunt actualiter existentes." (But every continuum is actually existent. Therefore any of its parts is really existent in nature. 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.)

The parts are actually there, in some sense. 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". Aquinas also argued against the idea that infinity could be in any sense complete, or a totality [reference].

Views from the Renaissance to modern times

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). 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:

1, 2, 3, 4, ...
2, 4, 6, 8, ...

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; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; 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. 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. Our idea of infinity is merely negative or privative.

"Whatever positive ideas we have in our minds of any space, duration, or number, let them be never so great, they are still finite; but when we suppose an inexhaustible remainder, from which we remove all bounds, and wherein we allow the mind an endless progression of thought, without ever completing the idea, there we have our idea of infinity ... 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. For let a man frame in his mind an idea of any space or number, as great as he will, it is plain the mind rests and terminates in that idea; which is contrary to the idea of infinity, which consists in a supposed endless progression." (Essay, II. xvii. 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. 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. Potentiality lies in the definitions of this operation, as well-defined and interconsistent mathematical axioms. A potential infinity is allowed by letting an infinitely-large quantity be cancelled out by an infinitely-small quantity.

Modern philosophical views

Modern discussion of the infinite is now regarded as part of set theory and mathematics, and generally avoided by philosophers. 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". (see also Logic of antinomies)

"Does the relation m = 2n correlate the class of all numbers with one of its subclasses? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but which are never related as class and subclass. Neither is this infinite process itself in some sense or other such a pair of classes ... In the superstition that m = 2n correlates a class with its subclass, we merely have yet another case of ambiguous grammar." (Philosophical Remarks § 141, cf Philosophical Grammar p. 465)

Unlike the traditional empiricists, he thought that the infinite was in some way given to sense experience.

"... I can see in space the possibility of any finite experience ... we recognise [the] essential infinity of space in its smallest part." "[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."

Mathematical infinity

Infinity in real analysis

In real analysis, the symbol $\infty$, called "infinity", denotes an unbounded limit. $x \rightarrow \infty$ means that x grows beyond any assigned value, and $x \rightarrow -\infty$ means x is eventually less than any assigned value. Points labeled $\infty$ and $-\infty$ can be added to the real numbers as a topological space, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat $\infty$ and $-\infty$ as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions. Infinity is represented by "∞".

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

• $\int_{0}^{1} \, f(t) dt \ = \infty$ means that f(t) does not bound a finite area from 0 to 1
• $\int_{0}^{\infty} \, f(t) dt \ = \infty$ means that the area under f(t) increases without bound as its upper bound increases
• $\int_{0}^{\infty} \, f(t) dt \ = 1$ means that the area under f(t) approaches 1, though its upper bound increases limitlessly.

Infinity symbol

It is unclear what the exact origins of the infinity symbol are, but the most commonly cited explanation says that it is derived from the shape of a Möbius strip twisted to look like $\infty$ since if one were to stand on a the surface of a Möbius strip, one could walk along it forever. In addition, the lemniscate curve looks like the infinity symbol, and its name is derived from the Latin lemniscus, meaning "ribbon," which is what a Möbius strip can be made of. The symbol itself is also sometimes referred to as the lemniscate.

This explanation may not be correct however since the symbol had been in use to represent infinity even before August Ferdinand Möbius had discovered the Möbius strip. John Wallis is often credited for introducing this symbol through his book Arithmetica Infinitorum, which was published more than a century before Möbius was born. Conjectures of why Wallis chose this symbol say that he derived it from the Etruscan numeral for 1000, which looked somewhat like CIƆ and is sometimes used to mean "many," or that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet.

The Dustbin of History by Paul JJ Payack recounts how the infinity symbol came into widespread use in mathematics.

Infinity in set theory

A different type 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 ($\aleph_0$), 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. 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. One example of this is Hilbert's paradox of the Grand Hotel.

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.

Use of infinity in common speech

In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forever to get tickets." In video games, "infinite lives" and "infinite ammo" usually mean a truly never-ending supply of lives and ammunition. Another accurate usage is an infinite loop in computer programming, a conditional loop construction whose condition always evaluates to true. As long as there is no external interaction (such as switching the computer off, or the heat death of the universe), the loop will continue to run for all time. In practice however, most programming loops considered as infinite will halt by exceeding the (finite) number range of one of its variables. See halting problem.

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. 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 (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.

This point of view does not mean that infinity cannot be used in physics. For convenience sake, calculations, equations, theories and approximations, often use infinite series, unbounded functions, etc., and may involve infinite quantities. 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.

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? Does space "go on forever"? This is an important open question of cosmology. Note that 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 walking/sailing/driving straight long enough, you'll return to the exact spot you started from. 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.

Infinity in science fiction

The Hitchhiker's Guide to the Galaxy contains the following definition of infinity:

"Bigger than the biggest thing ever and then some, much bigger than that, in fact really amazingly immense, a totally stunning size, real 'Wow, thats big!' time. Infinity is just so big that by comparison, bigness itself looks really titchy. Gigantic multiplied by colossal multiplied by staggeringly huge is the sort of concept we are trying to get across here."

External links

• A Crash Course in the Mathematics of Infinite Sets, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1-59. The stand-alone appendix to Infinite Reflections, below. A concise introduction to Cantor's mathematics of infinite sets.
• 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.

03-10-2013 05:06:04
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