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Infinity
Infinity is a theoretical value that is larger than any other value. To count to infinity is to count forever, without end. Infinity comes from the Latin "infinitus", meaning without end and usually denoted by the symbol "∞" (lemniscate). Infinite is the quality of being greater than anything. It is also used to denote the quality of being unbounded or having no limit. The Infinite is usually defined as that which has no bounds in space or time.
Infinity is often used in mathematics and has a precise mathematical definition, as given below. It is also used in common speech, often inaccurately as an exaggerated synonym for "very great", rather than "unending".
See also infinitesimal, the theoretical value that is nearer to the value zero (without actually being zero) than any other value.
Contents |
1.1 Ancient view of infinity |
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 Phi(m)". The second view is found in a clearer form in 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 a set of infinite numbers into one-to-one correspondence with one of its proper subsets (any part of the set, that is not equivalent to 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 appearance or "impressions", and since all sense impression is inherently finite, so too for 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.
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."
Use of infinity in mathematics
In mathematics, infinity is an unbounded quantity that is greater than every real number. [1]
A distinction is made between different "sizes" of infinity because it can be shown that some infinite sets have greater cardinality than others. 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.
The modern mathematical conception of the infinite developed in the late nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of sets. Their 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.
Thus Cantor showed that infinite sets can even have different sizes, distinguished between countably infinite and uncountable sets, and developed a theory of cardinal numbers around this. His view prevailed and modern mathematics accepts actual infinity. Certain extended number systems, such as the surreal 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.
It is worth mention that the infinite cardinal numbers (relating to set theory) and the infinity commonly encountered in algebra and calculus are two completely different concepts. In algebra and calculus, 2^{∞} is not technically a number, but taking a limit yields 2^{∞} = ∞. Real numbers are not used to measure the sizes of sets, so ∞ can be used for any quantity that grows indefinitely at a limit. The corresponding statement in set theory is that 2^{ℵ0} > ℵ_{0} because the former term is uncountable, while the latter is countable. Exponents in set theory are not the same as regular exponents in high school mathematics, and ∞ is not the same as aleph null.
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.
Mathematics without infinity
Leopold Kronecker rejected the notion of infinity and started a branch of mathematics, called finitism (see also mathematical constructivism).
"bounded" versus "unbounded"
In mathematics, the term bounded is use to designate a set whose elements in a container of finite size. More formally, the set is said to be bounded if there exists at least one point c (center) and a positive real number r (radius) such that the set , where is the set that contain all the points than are less or equal to distance r from c, in both directions, contains all the elements of X; .
This is the mathematical way of saying "how long is a piece of string? It's twice the distance from the middle to the end."
Sets are unbounded if, for any c and any r, does not include all elements of X.
For the above definitions to make sense, we have to have define what we mean by distance, We must define a metric to be in metric space. If we are not, the terms "bounded" and "unbounded" are meaningless. However, the term '"infinity" even without a metric.
These definitions of bounded and unbounded are the same, regardless of whether point c is part of X, or whether or not is exactly equal to X.
It can be easily shown that
- If a set is finite, it is bounded.
- If a set in unbounded, is infinite.
However,
- If a set is bounded, it is not necessarily finite. For example, a segment is bounded but has an infinite number of elements.
- If a set is infinite, it may not be unbounded.
If a set is bounded, we can define a diameter of the set:
Where d(x,y) is the distance between x and y in set X. We take the maximal d(x,y) returned after consideration all permutations of x and y.
The diameter of a set is always a positive real number or zero, if the set is bounded. It can be zero if and only if the set is empty or has only one member.
If the set Y is unbounded, we can write but it must be understood that this is only a convention for stating that Y is unbounded. It does not literally mean that the diameter is infinite.
Calculus and mathematical analysis
A very common use of infinity is in calculus and mathematical analysis, for example:
Infinity is not part of the set of real numbers; and cannot be used in places where a real number can normally be used. For example, is true, but is undefined.
There are only few cases when you can consider ∞ as a regular number. In these unusual cases, you are in the so-called extended real number field, denoted by
In limit analysis, we can make statements which include the theoretical case that we put infinity in the place of a real number, for example . This states that as x continues to grow in magintude (tends towards infinity), 1/x becomes closer and closer to zero (tends toward zero). The limit case, is undefined, however if x was the largest finite value known to us, 1/x would be the closest finite value to zero known to us.
Limits do not literally consider the case of x=∞ If the definition did include ∞, the properties of the definition change, and some properties that were valid before may no longer be valid. For example, when you extend the definition of integrals, you get improper integrals. Without fully understanding this and correctly assessing the consequences of using infinity in place of a real number, error and paradox may occur. For an example, see the explanation of why the mean of the Cauchy distribution is undefined.
It is important to note that not all limits, series and integrals are convergent.
In the usual ordered real number field, it is common to distinguish between +∞ and -∞.
Use of "infinity" outside of mathematics
Use of infinity in common speech
Outside the realm of strict mathematical interpretation, "infinity" often loses its quality of being unbounded and is deliberately used to mean very large finite quantities. For example, the statement "I tried to call him an infinite number of times" is wrong; only a finite number of calls could be made, however many that may be. Similarly, the statement "we had to wait forever and a day to get the tickets" or "built to last forever!" do not recognise that "forever" is an infinite time period. It convenient to misuse infinity as an exaggeration, because it is impossible to extend the exaggeration beyond infinity without appearing silly or childish.
One common real-world use where the term is used accurately is in video games, where "infinite lives" and "infinite ammo" usually means a truly never-ending supply of lives or 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), the loop will continue to run forever.
Use of infinity in Physics
From a physical point of view, infinity is not an acceptable result. It is assumed by physicists that no measurable quantity (observable quantity in the quantum physics language) could have an infinite value. For example, if an object is moving towards a point and calculations about it show that it will reach it after an infinite amount of time, in real terms this means that the object will never reach the point. Likewise, it is impossible for any body to have infinite mass or infinite energy, as if it exists in our finite universe, it must be composed of matter within the universe, and are therefore will be finite itself. 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 wrongly assuming it is possible for a machine to attain 100% efficiency or greater.
Time and space can be considered infinite, however the time available before the end of the Universe is considered finite. Nothing within the Universe can actually reach an infinitely distant point, unless given infinite time to do it. This point of view is in complete agreement with the etymology of the word "infinite" and with its dominant mathematical meaning of being unbounded.
This point of view does not mean that infinity cannot be used in Physics. Calculations, equations and theories often use infinite terms, non-continuous functions, etc., when the mathematical complexity of avoiding infinity would otherwise make them impossible to deal with. The mathematical work is done keeping in mind the physical meaning of infinity, and results are similarly dealt with. For example, substituting infinity with the largest possible finite value, and recognising that this is an approximation. In most situations the difference between using infinity and near-infinity is minimal and gives good enough approximations for the needs of physicists.
Avoiding infinite energy terms in modern field theory led to the introduction of renormalization methods.
The misunderstanding of the Infinite and of infinitesimal fractions of time and space led to the rise to Zeno's paradox.
For a discussion about the dimension and infiniteness of the Universe, see Universe.
See also
External link
Topics in mathematics related to quantity |