samoth said:
See stuff by the Cantor guy that RedGuru mentioned earlier. Or you could look up the continuum hypothesis.
But, seriously, think about it mathematically. The set R contains all real numbers. Real numbers are isomorphic with the points on an infinitely long line, that is, they carry a one-to-one correspondence. So included in this set would be {1, 2, 2.01, 2.02, 3, 3.001, 4, 4.001, 5, 5.1x10^-32, 6, ..., i}. Upon inspection, it is obvious this set contains a larger number (more) elements than the set of integers Z != {0, 1, 2, 3, 4, 5, 6, ..., j}. While both are infinite sets, the set R obviously contains more elements than Z.
∑ { Aα | ∀ α ∈ ℝⁿ} > ∑ { Bβ | ∀ β ∈ ℤⁿ}
Do you see now? We're not
counting each element of each set to infinity; they both reach it. We're looking at the number of elements in each set. I know it's kinda tricky, but do you see what I mean more clearly now?
l is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from Latin : "Infinito", unending.
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. In philosophy, infinity can be attributed to space and time, as for instance in Kant's first antinomy. In both theology and philosophy, infinity is explored in articles such as the Ultimate, the Absolute, God, and Zeno's paradoxes.
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. By some, infinity is considered to be not a number but a concept of increase beyond bounds.
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
Ancient view of infinity
The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800 BC - 800 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. Jaina mathematicians were the first to conceive of different orders of infinity, including one they called unenumerable (innumerable).[1] [2] The concept of different orders of infinity would remain unknown in Europe until the late 19th century.
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; 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 infinite sets without restriction. 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 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.
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[3]
"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."
Infinity symbol
The precise origins of the infinity symbol \infty are unclear. One possibility is suggested by the name it is sometimes called — the lemniscate, from the Latin lemniscus, meaning "ribbon". One can imagine walking forever along a simple loop formed from a ribbon.
A popular explanation is that the infinity symbol is derived from the shape of a Möbius strip. Again, one can imagine walking along its surface forever. This possible explanation is probably incorrect, however, 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.
John Wallis is usually credited with introducing \infty as a symbol for infinity in 1655 in his De sectionibus conicus. 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". Another conjecture is that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet.
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 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) is not finite
* \int_{0}^{\infty} \, f(t) dt \ = 1 means that the area under f(t) approaches 1
Infinity in complex analysis
As in real analysis, in complex analysis the symbol \infty, called "infinity", denotes an unbounded limit. x \rightarrow \infty means that the magnitude | x | of x grows beyond any assigned value. A point labeled \infty 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 still a one-dimensional complex manifold and called the extended complex plane or the Riemann sphere. In this context is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of \infty 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.
Arithmetic properties of infinity
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.
Infinity with itself
1. \infty + \infty = \infty \cdot \infty = (-\infty) \cdot (-\infty) = \infty
2. (-\infty) + (-\infty) = \infty \cdot (-\infty) = (-\infty) \cdot \infty = (-\infty)
Operations involving infinity and real numbers
1. -\infty < x < \infty
2. x + \infty = \infty and x + (-\infty) = (-\infty)
3. x - \infty = -\infty
4. x - (-\infty) = \infty
5. {x \over \infty} = 0 and {x \over -\infty} = 0
6. If 0<x<\infty then x \cdot \infty = \infty and x \cdot (-\infty) = (-\infty).
7. If -\infty<x<0 then x \cdot \infty = -\infty and x \cdot (-\infty) = \infty.
Undefined Operations
1. 0 \cdot \infty and 0 \cdot (-\infty)
2. \infty + (-\infty) and (-\infty) + \infty
3. {\pm\infty \over \pm\infty}
4. {\pm\infty}^0
5. 1^{\pm\infty}
Notice that [{x \over \infty} = 0] \not\equiv [0 \cdot \infty = x]. This is because zero times infinity is undefined.
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 never-ending supply of lives and ammunition. An infinite loop in computer programming is 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. These terms describe things that are only theoretically infinite; 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. 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.
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.
Three types of infinities
Besides the mathematical infinity and the physical infinity, there could also be a philosophical infinity. There are scientists who hold that all three really exist and there are scientists who hold that none of the three exist. And in between there are the various possibilities. 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. The footnote on p.335 of his book suggests the consideration of the following names: Abraham Robinson, Plato, Thomas Aquinas, L.E.J. Brouwer, David Hilbert, Bertrand Russell, Kurt Gödel and Georg Cantor.
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, that's 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."
Another quote from The Hitchhiker's Guide to the Galaxy states: "Infinity itself looks flat and uninteresting. Looking up into the night sky is looking into infinity -- distance is incomprehensible and therefore meaningless."
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. Georg Cantor makes an appearance as a character, and the hero finds a physical correlate for Cantor's Continuum Problem.
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