
:''For the automobile brand, see Infiniti. For the radio company, see Infinity Broadcasting.'' Infinity 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 : "In-finite", ''is not ended''. In theology, for example in the work of List of Christian 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 (leading to the question, an unlimited quantity of what?). In philosophy, infinity can be attributed to space and time, as for instance in Immanuel 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 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 popular culture, we have Buzz Lightyear's rallying cry, "To infinity — and beyond!", which may also be viewed as the rallying cry of set theory considering large cardinals. For a discussion about infinity and the physical universe, see Universe. ==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 bisection 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 (mathematics) 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". Thomas 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 Galilei (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. John 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 Thomas 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 Ludwig Wittgenstein, who made an impassioned attack upon axiomatic set theory, and upon the idea of the actual infinite, during his "middle period". (see also [http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&type=html&an=0724.03003&format=complete 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 , called "infinity", denotes an unbounded limit. means that x grows beyond any assigned value, and means x is eventually less than any assigned value. Points labeled and 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 and 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 * means that ''f''(''t'') does not bound a finite area from 0 to 1 * means that the area under ''f''(''t'') increases without bound as its upper bound increases limitlessly * 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 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. ===Infinity in set theory=== A different type of "infinity" are the ordinal and cardinal number infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph number (), 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 set. 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 subset" parts; this notion of infinity is called Dedekind infinite. Cantor defined two kinds of infinite numbers, the ordinal numbers and the aleph number. 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 Map (mathematics) 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 hyperbole 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 observable could have an infinite value, for instance by taking an infinite value in an extended real number line 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. ==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." 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." ==See also== * Infinitesimal * Axiom of infinity ==External links== *''[http://www.earlham.edu/~peters/writing/infapp.htm 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. *''[http://www.earlham.edu/~peters/writing/infinity.htm 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. *[http://pespmc1.vub.ac.be/INFINITY.html ''Infinity'', Principia Cybernetica] *[http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html Hotel Infinity] * [http://samvak.tripod.com/infinite.html The concepts of finiteness and infinity in philosophy] ==Note== Large cardinals are quantitative infinities defining the number of things in a collection, which are so large that they cannot be proven to exist in the ordinary mathematics of ZFC (ZFC), to the extent that they embody a contradiction. Science Philosophy Philosophy of mathematics Theology jbo:ci'i simple:Infinity
There's an article about how to compare infinities at [http://www.wasteflake.com/tiki-read_article.php?articleId=1 Counting Games]. Should it be an external or "see also" link? ---- What about a comment about how to compare infinities? Is the total of all odd numbers greater or lesser than the total of all numbers? I'm not qualified to write on this, but I do know that some infinities are greater than others. User:F. Lee Horn This is discussed under :Cardinal number. The Infinity article does mention this, but perhaps it needs to be made clearer. --user:Zundark, 2002 Jan 7 ---- I'm a little unsure about the last paragraph. Do modern theologans and philosophers (it's not really a question of mathematics) have any interest in relating infinity to God? --User:Robert Merkel I think infinity is typically taken to be one of the properties of God. My problem with the last paragraph is that Goedel did not use infinity at all, he defined God as "absolute perfection" and came up with some axioms which establish the existance of an entity which is absolutely perfect. I don't see how that relates to infinity at all. --AxelBoldt ---- I like the history section, but it looks like it belongs in a different article. I don't think the Arabs used "1001" to mean infinity. Nor did the French with "million", nor Buddha with "10^421", nor the Romans with ''decies centena milia''. The only sentence that might be relevant is the one about infinity being called "zero denominator". Other than that sentence, how about moving the rest to number names? --user:LC I agree. Also, the claim that "infinity has greatly increasd in size over the years" is pretty hilarious. user:AxelBoldt ---- Isn't it generally assumed by astonomers and physicists that our universe, forget about any other ones, is not infinite? -User:Tubby :Yep - finite. However at the same time it is unbounded (meaning that there is no inherent max size). --mav :Universe by definition is finite, containing within itself the root for one, ''uni''; infinity is by definition some process which can continue indefinitely, for example, imagine a faster than light ship using a drive which (because of the peculiarities of faster than light travel accelerates as a function of the square of the product of speed and distance traveled). A journey on such a ship in a finite universe would nevertheless never reach the "edge" and "fall off". User:Fredbauder 10:33 Oct 15, 2002 (UTC) ---- In my opinion the definition at the beginning (the one before the TOC) is severely wrong. Even if in common use the term infinity is also the one describe there, this is not the precise tecnical (especially in Math) definition. The definition given is that of unlimited or unbounded not of infinty. Infinity means with no end, a set is infinity if when you count the number of its member you can not arrive at the point you have cont all the member. This definition is consistence with the rest of the article. (Phereps I have to rewrite it in a better way). An equivalent (but more difficult to understand) definition is that a set is infinite if there exist an its proper parts that is as big as the wole set (where as big as is to be understood in a sense proper to this branch of math) Also the traslation of the etimologhy is wrong: Infinitum in latin is not ''without limit'' but it is ''without end'' or ''not eneded''. As a prove of that consider that from the word finitum and fines derived the Italian Fine and French Fin that mean end in English Neverless the word infinity and infinite are common used in the meanig stated there and also to describe a very big set but finite. Maybe it will be worthly to add a section on this and on the difference on these term. A tecnical mathematical note to use the terms limited, unlimited (or the equivalent bounded and unbounded) you have to fix the way you do the measure of distance (you have to be in a Metric space). You have not to have this to speak of infinity/infinte User:AnyFile 18:16, 8 Sep 2004 (UTC) == Is the dispute over? == I believe the points raised on the talk page are now covered. Of course using "infinity" in describing a finite is wrong, but it is a popular mistake that needs to be included, clearly labeled as such. User:Kyz 10:49, 11 Sep 2004 (UTC) == Symbol for Infinity == Since I think the people reading this will know - it was my impression that the symbol for Infinity was the Möbius Band however maybe it's accurate to say that the Möbius Band is a specific case of the lemniscate? Thanks, R. :As far as I can tell, this isn't true at all. The Möbius strip doesn't look very much like the symbol for infinity, and the apocryphal explanation that it symbolises infinity because you can go on forever on it is equally true of the circle. They've got nothing to do with each other, and I've always thought that the infinity symbol was chosen because it was a fairly regular symbol that wasn't taken up already, freeing the precious ω for redefinition (much as ∈ used to be written ε). No idea whether that's true either. :User:Prumpf 16:02, 16 Oct 2004 (UTC) ::Apocryphal or not, these explanations are widely circulated (by math professors even !!!) and should at least be mentioned in the article so that viewers can realize that they may not be true rather than continue in ignorant bliss. I'll go ahead and add a bit of text about it... --User:Umofomia 12:29, 14 Mar 2005 (UTC) == What is this article about? == This article seems to be a mushy mix of philosophy, intellectual history, and mathematics, and has a lot of outright errors. Can someone explain if this should be made more mathematical, by clarifying what the purpose of it is? user: Gene Ward Smith Historically, "a mushy mix of philosophy, [religion,] and mathematics" is pretty much what people thinking about infinity used to do. Still, this article is still in need of a lot of attention. I'd suggest starting by ripping out all the parts referring to current mathematics (after Cantor's Absolute Infinite, I think). In ''current'' mathematics, infinity pops up in exactly two places: *in the axiom of infinity (and its various cognates in other axiom systems for set theory): for the mathematical formalism, this is just another axiom like all the others, but for those who (still) take a more platonism view, it might have deeper meaning. *as a name or symbol to be used however you choose. This isn't much of an exaggeration. Depending on what I'm doing, I might very well end up in a situation where &infinity; is the same thing as the real number 1 (or any other real number), or the set of natural numbers, or what-have-you. When this is made explicit, I'd probably avoid the symbol, since it might be a bit confusing, but it wouldn't be wrong. Note that for strict formalists, the axiom of infinity fits in here as well. We just define some sets to be finite, and if one of them is not, well, that's an infinite set. So many things are called finite, infinite, or infinity in mathematics that a complete list of the various usages would probably be completely useless (as well as virtually impossible). Just off the top of my head: *maximal element of an ordered set *extra point(s) in one-point/any other compactification *non-affine points in projective space (a special case of the preceding for the real numbers, different in general) *limit notation, even when no compactification is involved *objects in any category that can't be cancelled in direct sums *elements of monoids that can't be cancelled *positive hyperreal numbers whose standard part can't be defined *cardinals and ordinals, of course *various other things like finite CW-complexes. Depending on your definition, this might not be finite as a set. *for von Neumann algebras, the various -finite terms used to have conflicting definitions (though that's all cleared up now, I hope). Still, they can be hyperfinite, finite, infinite, purely infinite, and of course none of this will hold true for the sets. Furthermore, the trivial von Neumann algebra is usually considered finite and purely infinite. I'd be willing to turn these into a separate article if anyone thinks it'd help. I don't, but they're certainly just confusing in an article about infinity. As an analogy, it's a bit like talking about rational numbers in the rationality article. They happen to use the same term, but that's it, as far as their relationship goes. Today, of course, mathematical objects and properties are commonly named after their inventors, which might be the only reason Noetherian rings aren't called finite. So, to sum things up, let's throw out all the mathematics, redirect people who're just looking for an article about the mathematical term to a separate article (I think that'd be most of them), then get back to writing an article about the philosophical issues. :I don't think we should "throw out all the mathematics", the mathematics is inherently intertwined with the philosophical and physical issues. User:Paul August 17:47, Oct 21, 2004 (UTC) User:Prumpf 16:38, 16 Oct 2004 (UTC) ==hey!== Hi newbies, if you didn't realise, Wikipedia says that removing swathes of material, especially without permission or discussion beforehand, is wrong. Infinity is supposed to be a general article, so don't delete explanations even if they're covered in independent, separate articles. The coverage before was excellent. If there are mistakes, say so here [or there] and fix them rather than being stupid. User:Lysdexia 03:25, 21 Oct 2004 (UTC) :Sorry, but Gene Ward Smith's edits seemed to be improving the article, as far as I can tell. Certainly it doesn't warrant being called abuse or "being stupid". I suggest we restore some of the obvious improvements, at least. :My edits removed material that was just plain wrong. We simply should not start the article with a claim that "Infinity is a theoretical value which is larger than any other value". This vaguely stated claim might be OK to begin a discussion of limits or the two-point compactificiation of the real line, but as an introductory sentence it is unacceptable, simply flat-out wrong. "To count to infinity is to count without end" is incredibly naive in a post-Cantor world; we can certainly start out positing this to get the ball rolling, but it is not acceptable as it stands. "Infinite is the quantity which of being greater than anything" is self-contradictory and illiterate. User:Lysdexia said that removing large swaths of material is wrong, and then removed large swaths of material by someone with a PhD in mathematics who also has a background in philosophy--in other words, someone who knows what the hell is talking about. I'm restoring my edit and then I'll try to incorporate edits by people who do not, as Lysdexia did, resort to vandalism. User:Gene Ward Smith 02:55, 1 Nov 2004 (UTC) :I still think the version of the article prior to your reversion would make a better starting point for editing the article. Still, ideally it shouldn't matter too much. :If you want to keep the largely misleading mathematics section, please comment on this talk page. My proposal of dropping it seems to be unopposed so far. User:Prumpf 16:57, 21 Oct 2004 (UTC) ::I'm not sure why you call the maths section "misleading". Is it an exhaustive list of all mathematical concepts related to infinity? No. These should all be linked to as seperate articles, with a little introduction for each link as to how infinity relates to that topic. However, the main core of mathematical infinity (its unbounded, unquantified nature) should be kept. IMHO, infinite sets should definitely migrate to their own article, part of the set theory category. Currently, "Infinite set" is a redirect to infinity. User:Kyz 18:01, 21 Oct 2004 (UTC) :::I'm not sure what you mean by "unquantified", but when I want to refer to something with an unbounded nature, I tend to use the term "unbounded", not infinite. In mathematics, except in various highly specialised contexts (where it's just another mathematical term to be defined as the writing mathematician pleases), "infinite" refers to infinite sets; infinity can have a couple of meanings, but those have nothing in common, as far as I can tell. Of course, unbounded is a rough translation of infinite, but the latter is essentially just a "free" term which can be defined as fits the context. As for the "misleading" comment, the very first sentence uses a term that isn't defined ("unbounded quantity") and makes the wrong claim that infinity is such a thing (and thus, such a thing only), and that it is meant to be compared to real numbers. That's one use of the symbol, in extending the real numbers to an ordered half-ring, but it's hardly the only one. User:Prumpf 22:25, 21 Oct 2004 (UTC) :This article has been on my list of articles needing attention for a long time. I've dithered, because I wasn't quite sure what to do about it. I think most of the lead section is problematic. e.g. :*Infinity is a theoretical value that is larger than any other value. :*To count to infinity is to count forever, without end. :I like for the most part what User:Gene Ward Smith added to the article, especially with the new lead section. I'm less pleased about the deletions, those need to be discussed more I think. As a general rule if you are going to make large deletions they should at least be moved to the talk page for discussion.User:Paul August 17:23, Oct 21, 2004 (UTC) :: I'm not so taken with the new lead section. For a start, it doesn't actually define anything. It begins with "oh, I suppose it could mean anything, really". Instead of giving a basic mathematical definition, it gives a huge list of related articles. Are we meant to read all of those articles to get a basic idea of what infinity is? All we need are two very important words: '''unlimited'' and '''unbounded''. The mindless link dump can come later in the article (if at all). Sure, I agree, it takes several articles for differing mathematical concepts (Infinite Set, Complex Infinity, point at infinity), but they all use the "infinite/infinity" in their name to mean the same thing, linguistically. That's what this article must describe, not pawn off. User:Kyz 18:17, 21 Oct 2004 (UTC) :::The article should began by telling you "infinity" means a lot of different things, that it said the opposite was one of the things wrong with the stuff I removed. There is no basic mathematical definition to give, and therefore it should not begin by giving such a definition. There is no "basic idea of what infinity is". And no, all we need is not merely "unlimited" and "unbounded". You should take a look at the "mindless link dump"; you'd find there is much more to this subject that you think there is. Are links now a bad thing? Do we not want to inform people? User:Gene Ward Smith 03:56, 1 Nov 2004 (UTC) ::: For a start, this is infinity, not infinity (mathematics). I agree the most common usages of the infinity symbol ought to be put in the first paragraph, but we must by no means make it sound as though that's the only meaning of it (or the term "infinite", which somehow seems to have more meanings). "unlimited" and "unbounded" sound like the same thing to me, and they're both quite literal translations of the latin. Maybe we should just point out what it translates as? ::: Note that an unbounded point violates accepted mathematical notation. Boundedness is a property of sets, and any singleton is trivially bounded. I'm not sure how an element of a monoid with a + a = a (those are sometimes called infinite) is either unbounded or unlimited. User:Prumpf 22:25, 21 Oct 2004 (UTC) ::: "Infinity" as a compactification makes the reals compact, which is to say, bounded. I suppose in a bad mood I might claim "infinity" meant "bounded" :) User:Gene Ward Smith 03:56, 1 Nov 2004 (UTC) == Physical infinity -- impossible or not? == In the context: :... It is therefore assumed by physicists that no measurable quantity could have an infinite value, ... 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. ... I don't think so. Let's consider Electrical resistance. There are conductors with zero resistance (Superconductors). Now consider Electrical conductance. They are related by : So the conductance of an object with zero resistance (superconductors!) is infinity. --User:Kenny TM~ 11:59, Nov 23, 2004 (UTC) :Are you sure the resistence is zero? If you cool down a superconductor you see that the resistence go down (very sharply or more slowly, this depends on the type of superconductor). It go down a lot, you can see a lot of current passing. Comparing to to room temperature resistence this resistence is incredible smaller and in current speacking world we can say that this is an infinite difference. Often physicist say a quantity to be infinite to meaning very very much more than anything else in the surroundig. But can you really (in strict math and phys sense) say tha resistence is zero and conductance infinite? For a physicist to say tha means that you have measure. How could you have measure it! We have not an instrument to measure really zero resistence. When you have a real electric circuit (I am meanig a board with electrical contact, wires, resistence, like the board of a computer) and you schematic it, we usally say that wire have zero resistence and 2 not connected point have infinite resistence between them. This is not completly true. I am not saying tha is not true. It is true in the sense we usually meaning it (and that is the sense physicist use when speacking of infinite). It is true that between 2 not connected have between them a very high impedence compared to any other resistence you have put in the board, but if you measure it (with the appropiate instrument) you will find that the reistence is finite, very high but finite. You may found out value of houndreds of gigaohm or more (but sometime less, sometimes only 1 GΩ). Usually all the tester you use display OVL but this not mean infinite, means more than the tester can measure. I have worked with circuit where we deliberately put resistor of 10-100 GΩ and if I did not considere this I would have some problem in finding where some current were going. In the same faschion a wire is not a zero resistence conductor, and if you are dealing with supercoductor you have to take this well in consideration. :You have found out a real thing. If infinite does not exist in physics so does not exist zero. (I teacher of mine Giuliano Preparata said infinite and zero does not exist). (Of course exist in the case of natural number if you have an apple and it it you have zero apple)User:AnyFile 15:26, 24 Nov 2004 (UTC) == ∞ == I notice the character for infinity (∞) is never actually included in this article, other than in graphics. I was wondering why this was... Perhaps it was simply not known? Well, anyway... does anyone else think the the graphics should be replaced by the character? user:Oscar Evans December 14th 2004. == Infinity in real analysis == I do not like very much the susection ''Infinity in real analysis''. The difference between infinity as a real number (opps ... extended-real) and as a limit is not clear. In my opinion it should be point out that when infinity is a limit you could not treat it as a number. This to avoid that someone could thing that he/she could do ∞ - ∞ = 0 User:AnyFile 21:30, 30 Jan 2005 (UTC) == 1/1=1, 1/0=Infinity, (0/0=1 and Infinity) Please verify? == 1/1=1, 1/0=Infinity, (0/0=1 and Infinity) Please verify? Why are the past, the present and the future is the same time as January 2005, some cluster of stars(many galaxies) we can see by light speed only and after that human mind memory recall in only 0.0001 second, it means infinity light speed can touch by human mind in 0.0001 second too. I don't know what the rest of the stuff you're saying means, but I'm pretty confident 1/0 is ''not'' infinity, since the inverse of infinity is the infinitesimal, while the inverse of 1/0 is simply zero. User:Citizen Premier 15:54, 10 Jun 2005 (UTC) In mathematics, for real numbers or complex numbers, division by zero is ''undefined''. So 1/1 = 1, but 0/1 does not equal infinity, and 0/0 does not equal 1 or infinity, they simply don't equal ''anything''. The IEEE floating-point standard does allow for division by zero for floating-point computations for computers. It defines ''a''/0 to be "positive infinity" when ''a'' is positive, "negative infinity" when ''a'' is negative, and NaN ("not a number") when ''a'' = 0. User:Paul August User_talk:Paul August 18:18, Jun 10, 2005 (UTC) == Transfinite == Would you consider interesting to put a note why infinity numbers are called (starting form Cantor) transfinite numbers? User:AnyFile 10:45, 2 May 2005 (UTC) == ∞ infinity is not a number == Because infinity is not exactly equal to any finite number, and acts differently than any (other) number, many people find it easier to understand explainations that state right up front that "infinity is not a number". * http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM#roriginf * http://www.kuro5hin.org/story/2003/6/3/95744/71866 "While zero is a concept and a number, infinity is not a number. Infinity is the name for a concept. Infinity cannot be considered as a number since since it does not follow numbers' properties. "Infinity" is not a number. ... When mathematicians say "x approaches infinity", or write "x→∞", they mean "x grows arbitrarily large. And when they say that the limit of something is 0, they mean that it can go as close to 0 as you want it to, but it may or may not be actually equal to 0." OK if I use that definition on Infinity and Infinity plus 1 ? If no one comes up with a better suggestion the next time I come by, I'll just stick that definition in the article and watch the reaction. --User:DavidCary 01:58, 12 May 2005 (UTC)
See other meanings of words starting from letter:
Words begining with Infinity:
Infinity
Infinity
Infinity,_Inc.
Infinity-Man
Infinity_(album)
Infinity_(album)
Infinity_Abyss
Infinity_Broadcasting
Infinity_Broadcasting_Corporation
Infinity_Broadcasting_stations
Infinity_cove
Infinity_Crusade
Infinity_Engine
Infinity_Gauntlet
Infinity_Gem
Infinity_Gems
Infinity_hotel
Infinity_Inc.
Infinity_Inc._members
Infinity_Minus_One
Infinity_of_Heaven
Infinity_of_heaven
Infinity_plus_1
Infinity_plus_one
Infinity_symbol
Infinity_trick
Infinity_tricks
Infinity_tricks
Infinity_War
Infinity_Ward
Infinity_Watch
Infinity_Watch_members
Infinity_Within
Infinity
:''For the automobile brand, see Infiniti. For the radio company, see Infinity Broadcasting.'' Infinity 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 : "In-finite", ''is not ended''. In theology, for example in the work of List of Christian 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 (leading to the question, an unlimited quantity of what?). In philosophy, infinity can be attributed to space and time, as for instance in Immanuel 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 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 popular culture, we have Buzz Lightyear's rallying cry, "To infinity — and beyond!", which may also be viewed as the rallying cry of set theory considering large cardinals. For a discussion about infinity and the physical universe, see Universe. ==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 bisection 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 (mathematics) 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". Thomas 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 Galilei (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. John 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 Thomas 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 Ludwig Wittgenstein, who made an impassioned attack upon axiomatic set theory, and upon the idea of the actual infinite, during his "middle period". (see also [http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&type=html&an=0724.03003&format=complete 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 , called "infinity", denotes an unbounded limit. means that x grows beyond any assigned value, and means x is eventually less than any assigned value. Points labeled and 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 and 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 * means that ''f''(''t'') does not bound a finite area from 0 to 1 * means that the area under ''f''(''t'') increases without bound as its upper bound increases limitlessly * 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 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. ===Infinity in set theory=== A different type of "infinity" are the ordinal and cardinal number infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph number (), 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 set. 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 subset" parts; this notion of infinity is called Dedekind infinite. Cantor defined two kinds of infinite numbers, the ordinal numbers and the aleph number. 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 Map (mathematics) 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 hyperbole 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 observable could have an infinite value, for instance by taking an infinite value in an extended real number line 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. ==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." 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." ==See also== * Infinitesimal * Axiom of infinity ==External links== *''[http://www.earlham.edu/~peters/writing/infapp.htm 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. *''[http://www.earlham.edu/~peters/writing/infinity.htm 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. *[http://pespmc1.vub.ac.be/INFINITY.html ''Infinity'', Principia Cybernetica] *[http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html Hotel Infinity] * [http://samvak.tripod.com/infinite.html The concepts of finiteness and infinity in philosophy] ==Note== Large cardinals are quantitative infinities defining the number of things in a collection, which are so large that they cannot be proven to exist in the ordinary mathematics of ZFC (ZFC), to the extent that they embody a contradiction. Science Philosophy Philosophy of mathematics Theology jbo:ci'i simple:Infinity
Infinity
There's an article about how to compare infinities at [http://www.wasteflake.com/tiki-read_article.php?articleId=1 Counting Games]. Should it be an external or "see also" link? ---- What about a comment about how to compare infinities? Is the total of all odd numbers greater or lesser than the total of all numbers? I'm not qualified to write on this, but I do know that some infinities are greater than others. User:F. Lee Horn This is discussed under :Cardinal number. The Infinity article does mention this, but perhaps it needs to be made clearer. --user:Zundark, 2002 Jan 7 ---- I'm a little unsure about the last paragraph. Do modern theologans and philosophers (it's not really a question of mathematics) have any interest in relating infinity to God? --User:Robert Merkel I think infinity is typically taken to be one of the properties of God. My problem with the last paragraph is that Goedel did not use infinity at all, he defined God as "absolute perfection" and came up with some axioms which establish the existance of an entity which is absolutely perfect. I don't see how that relates to infinity at all. --AxelBoldt ---- I like the history section, but it looks like it belongs in a different article. I don't think the Arabs used "1001" to mean infinity. Nor did the French with "million", nor Buddha with "10^421", nor the Romans with ''decies centena milia''. The only sentence that might be relevant is the one about infinity being called "zero denominator". Other than that sentence, how about moving the rest to number names? --user:LC I agree. Also, the claim that "infinity has greatly increasd in size over the years" is pretty hilarious. user:AxelBoldt ---- Isn't it generally assumed by astonomers and physicists that our universe, forget about any other ones, is not infinite? -User:Tubby :Yep - finite. However at the same time it is unbounded (meaning that there is no inherent max size). --mav :Universe by definition is finite, containing within itself the root for one, ''uni''; infinity is by definition some process which can continue indefinitely, for example, imagine a faster than light ship using a drive which (because of the peculiarities of faster than light travel accelerates as a function of the square of the product of speed and distance traveled). A journey on such a ship in a finite universe would nevertheless never reach the "edge" and "fall off". User:Fredbauder 10:33 Oct 15, 2002 (UTC) ---- In my opinion the definition at the beginning (the one before the TOC) is severely wrong. Even if in common use the term infinity is also the one describe there, this is not the precise tecnical (especially in Math) definition. The definition given is that of unlimited or unbounded not of infinty. Infinity means with no end, a set is infinity if when you count the number of its member you can not arrive at the point you have cont all the member. This definition is consistence with the rest of the article. (Phereps I have to rewrite it in a better way). An equivalent (but more difficult to understand) definition is that a set is infinite if there exist an its proper parts that is as big as the wole set (where as big as is to be understood in a sense proper to this branch of math) Also the traslation of the etimologhy is wrong: Infinitum in latin is not ''without limit'' but it is ''without end'' or ''not eneded''. As a prove of that consider that from the word finitum and fines derived the Italian Fine and French Fin that mean end in English Neverless the word infinity and infinite are common used in the meanig stated there and also to describe a very big set but finite. Maybe it will be worthly to add a section on this and on the difference on these term. A tecnical mathematical note to use the terms limited, unlimited (or the equivalent bounded and unbounded) you have to fix the way you do the measure of distance (you have to be in a Metric space). You have not to have this to speak of infinity/infinte User:AnyFile 18:16, 8 Sep 2004 (UTC) == Is the dispute over? == I believe the points raised on the talk page are now covered. Of course using "infinity" in describing a finite is wrong, but it is a popular mistake that needs to be included, clearly labeled as such. User:Kyz 10:49, 11 Sep 2004 (UTC) == Symbol for Infinity == Since I think the people reading this will know - it was my impression that the symbol for Infinity was the Möbius Band however maybe it's accurate to say that the Möbius Band is a specific case of the lemniscate? Thanks, R. :As far as I can tell, this isn't true at all. The Möbius strip doesn't look very much like the symbol for infinity, and the apocryphal explanation that it symbolises infinity because you can go on forever on it is equally true of the circle. They've got nothing to do with each other, and I've always thought that the infinity symbol was chosen because it was a fairly regular symbol that wasn't taken up already, freeing the precious ω for redefinition (much as ∈ used to be written ε). No idea whether that's true either. :User:Prumpf 16:02, 16 Oct 2004 (UTC) ::Apocryphal or not, these explanations are widely circulated (by math professors even !!!) and should at least be mentioned in the article so that viewers can realize that they may not be true rather than continue in ignorant bliss. I'll go ahead and add a bit of text about it... --User:Umofomia 12:29, 14 Mar 2005 (UTC) == What is this article about? == This article seems to be a mushy mix of philosophy, intellectual history, and mathematics, and has a lot of outright errors. Can someone explain if this should be made more mathematical, by clarifying what the purpose of it is? user: Gene Ward Smith Historically, "a mushy mix of philosophy, [religion,] and mathematics" is pretty much what people thinking about infinity used to do. Still, this article is still in need of a lot of attention. I'd suggest starting by ripping out all the parts referring to current mathematics (after Cantor's Absolute Infinite, I think). In ''current'' mathematics, infinity pops up in exactly two places: *in the axiom of infinity (and its various cognates in other axiom systems for set theory): for the mathematical formalism, this is just another axiom like all the others, but for those who (still) take a more platonism view, it might have deeper meaning. *as a name or symbol to be used however you choose. This isn't much of an exaggeration. Depending on what I'm doing, I might very well end up in a situation where &infinity; is the same thing as the real number 1 (or any other real number), or the set of natural numbers, or what-have-you. When this is made explicit, I'd probably avoid the symbol, since it might be a bit confusing, but it wouldn't be wrong. Note that for strict formalists, the axiom of infinity fits in here as well. We just define some sets to be finite, and if one of them is not, well, that's an infinite set. So many things are called finite, infinite, or infinity in mathematics that a complete list of the various usages would probably be completely useless (as well as virtually impossible). Just off the top of my head: *maximal element of an ordered set *extra point(s) in one-point/any other compactification *non-affine points in projective space (a special case of the preceding for the real numbers, different in general) *limit notation, even when no compactification is involved *objects in any category that can't be cancelled in direct sums *elements of monoids that can't be cancelled *positive hyperreal numbers whose standard part can't be defined *cardinals and ordinals, of course *various other things like finite CW-complexes. Depending on your definition, this might not be finite as a set. *for von Neumann algebras, the various -finite terms used to have conflicting definitions (though that's all cleared up now, I hope). Still, they can be hyperfinite, finite, infinite, purely infinite, and of course none of this will hold true for the sets. Furthermore, the trivial von Neumann algebra is usually considered finite and purely infinite. I'd be willing to turn these into a separate article if anyone thinks it'd help. I don't, but they're certainly just confusing in an article about infinity. As an analogy, it's a bit like talking about rational numbers in the rationality article. They happen to use the same term, but that's it, as far as their relationship goes. Today, of course, mathematical objects and properties are commonly named after their inventors, which might be the only reason Noetherian rings aren't called finite. So, to sum things up, let's throw out all the mathematics, redirect people who're just looking for an article about the mathematical term to a separate article (I think that'd be most of them), then get back to writing an article about the philosophical issues. :I don't think we should "throw out all the mathematics", the mathematics is inherently intertwined with the philosophical and physical issues. User:Paul August 17:47, Oct 21, 2004 (UTC) User:Prumpf 16:38, 16 Oct 2004 (UTC) ==hey!== Hi newbies, if you didn't realise, Wikipedia says that removing swathes of material, especially without permission or discussion beforehand, is wrong. Infinity is supposed to be a general article, so don't delete explanations even if they're covered in independent, separate articles. The coverage before was excellent. If there are mistakes, say so here [or there] and fix them rather than being stupid. User:Lysdexia 03:25, 21 Oct 2004 (UTC) :Sorry, but Gene Ward Smith's edits seemed to be improving the article, as far as I can tell. Certainly it doesn't warrant being called abuse or "being stupid". I suggest we restore some of the obvious improvements, at least. :My edits removed material that was just plain wrong. We simply should not start the article with a claim that "Infinity is a theoretical value which is larger than any other value". This vaguely stated claim might be OK to begin a discussion of limits or the two-point compactificiation of the real line, but as an introductory sentence it is unacceptable, simply flat-out wrong. "To count to infinity is to count without end" is incredibly naive in a post-Cantor world; we can certainly start out positing this to get the ball rolling, but it is not acceptable as it stands. "Infinite is the quantity which of being greater than anything" is self-contradictory and illiterate. User:Lysdexia said that removing large swaths of material is wrong, and then removed large swaths of material by someone with a PhD in mathematics who also has a background in philosophy--in other words, someone who knows what the hell is talking about. I'm restoring my edit and then I'll try to incorporate edits by people who do not, as Lysdexia did, resort to vandalism. User:Gene Ward Smith 02:55, 1 Nov 2004 (UTC) :I still think the version of the article prior to your reversion would make a better starting point for editing the article. Still, ideally it shouldn't matter too much. :If you want to keep the largely misleading mathematics section, please comment on this talk page. My proposal of dropping it seems to be unopposed so far. User:Prumpf 16:57, 21 Oct 2004 (UTC) ::I'm not sure why you call the maths section "misleading". Is it an exhaustive list of all mathematical concepts related to infinity? No. These should all be linked to as seperate articles, with a little introduction for each link as to how infinity relates to that topic. However, the main core of mathematical infinity (its unbounded, unquantified nature) should be kept. IMHO, infinite sets should definitely migrate to their own article, part of the set theory category. Currently, "Infinite set" is a redirect to infinity. User:Kyz 18:01, 21 Oct 2004 (UTC) :::I'm not sure what you mean by "unquantified", but when I want to refer to something with an unbounded nature, I tend to use the term "unbounded", not infinite. In mathematics, except in various highly specialised contexts (where it's just another mathematical term to be defined as the writing mathematician pleases), "infinite" refers to infinite sets; infinity can have a couple of meanings, but those have nothing in common, as far as I can tell. Of course, unbounded is a rough translation of infinite, but the latter is essentially just a "free" term which can be defined as fits the context. As for the "misleading" comment, the very first sentence uses a term that isn't defined ("unbounded quantity") and makes the wrong claim that infinity is such a thing (and thus, such a thing only), and that it is meant to be compared to real numbers. That's one use of the symbol, in extending the real numbers to an ordered half-ring, but it's hardly the only one. User:Prumpf 22:25, 21 Oct 2004 (UTC) :This article has been on my list of articles needing attention for a long time. I've dithered, because I wasn't quite sure what to do about it. I think most of the lead section is problematic. e.g. :*Infinity is a theoretical value that is larger than any other value. :*To count to infinity is to count forever, without end. :I like for the most part what User:Gene Ward Smith added to the article, especially with the new lead section. I'm less pleased about the deletions, those need to be discussed more I think. As a general rule if you are going to make large deletions they should at least be moved to the talk page for discussion.User:Paul August 17:23, Oct 21, 2004 (UTC) :: I'm not so taken with the new lead section. For a start, it doesn't actually define anything. It begins with "oh, I suppose it could mean anything, really". Instead of giving a basic mathematical definition, it gives a huge list of related articles. Are we meant to read all of those articles to get a basic idea of what infinity is? All we need are two very important words: '''unlimited'' and '''unbounded''. The mindless link dump can come later in the article (if at all). Sure, I agree, it takes several articles for differing mathematical concepts (Infinite Set, Complex Infinity, point at infinity), but they all use the "infinite/infinity" in their name to mean the same thing, linguistically. That's what this article must describe, not pawn off. User:Kyz 18:17, 21 Oct 2004 (UTC) :::The article should began by telling you "infinity" means a lot of different things, that it said the opposite was one of the things wrong with the stuff I removed. There is no basic mathematical definition to give, and therefore it should not begin by giving such a definition. There is no "basic idea of what infinity is". And no, all we need is not merely "unlimited" and "unbounded". You should take a look at the "mindless link dump"; you'd find there is much more to this subject that you think there is. Are links now a bad thing? Do we not want to inform people? User:Gene Ward Smith 03:56, 1 Nov 2004 (UTC) ::: For a start, this is infinity, not infinity (mathematics). I agree the most common usages of the infinity symbol ought to be put in the first paragraph, but we must by no means make it sound as though that's the only meaning of it (or the term "infinite", which somehow seems to have more meanings). "unlimited" and "unbounded" sound like the same thing to me, and they're both quite literal translations of the latin. Maybe we should just point out what it translates as? ::: Note that an unbounded point violates accepted mathematical notation. Boundedness is a property of sets, and any singleton is trivially bounded. I'm not sure how an element of a monoid with a + a = a (those are sometimes called infinite) is either unbounded or unlimited. User:Prumpf 22:25, 21 Oct 2004 (UTC) ::: "Infinity" as a compactification makes the reals compact, which is to say, bounded. I suppose in a bad mood I might claim "infinity" meant "bounded" :) User:Gene Ward Smith 03:56, 1 Nov 2004 (UTC) == Physical infinity -- impossible or not? == In the context: :... It is therefore assumed by physicists that no measurable quantity could have an infinite value, ... 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. ... I don't think so. Let's consider Electrical resistance. There are conductors with zero resistance (Superconductors). Now consider Electrical conductance. They are related by : So the conductance of an object with zero resistance (superconductors!) is infinity. --User:Kenny TM~ 11:59, Nov 23, 2004 (UTC) :Are you sure the resistence is zero? If you cool down a superconductor you see that the resistence go down (very sharply or more slowly, this depends on the type of superconductor). It go down a lot, you can see a lot of current passing. Comparing to to room temperature resistence this resistence is incredible smaller and in current speacking world we can say that this is an infinite difference. Often physicist say a quantity to be infinite to meaning very very much more than anything else in the surroundig. But can you really (in strict math and phys sense) say tha resistence is zero and conductance infinite? For a physicist to say tha means that you have measure. How could you have measure it! We have not an instrument to measure really zero resistence. When you have a real electric circuit (I am meanig a board with electrical contact, wires, resistence, like the board of a computer) and you schematic it, we usally say that wire have zero resistence and 2 not connected point have infinite resistence between them. This is not completly true. I am not saying tha is not true. It is true in the sense we usually meaning it (and that is the sense physicist use when speacking of infinite). It is true that between 2 not connected have between them a very high impedence compared to any other resistence you have put in the board, but if you measure it (with the appropiate instrument) you will find that the reistence is finite, very high but finite. You may found out value of houndreds of gigaohm or more (but sometime less, sometimes only 1 GΩ). Usually all the tester you use display OVL but this not mean infinite, means more than the tester can measure. I have worked with circuit where we deliberately put resistor of 10-100 GΩ and if I did not considere this I would have some problem in finding where some current were going. In the same faschion a wire is not a zero resistence conductor, and if you are dealing with supercoductor you have to take this well in consideration. :You have found out a real thing. If infinite does not exist in physics so does not exist zero. (I teacher of mine Giuliano Preparata said infinite and zero does not exist). (Of course exist in the case of natural number if you have an apple and it it you have zero apple)User:AnyFile 15:26, 24 Nov 2004 (UTC) == ∞ == I notice the character for infinity (∞) is never actually included in this article, other than in graphics. I was wondering why this was... Perhaps it was simply not known? Well, anyway... does anyone else think the the graphics should be replaced by the character? user:Oscar Evans December 14th 2004. == Infinity in real analysis == I do not like very much the susection ''Infinity in real analysis''. The difference between infinity as a real number (opps ... extended-real) and as a limit is not clear. In my opinion it should be point out that when infinity is a limit you could not treat it as a number. This to avoid that someone could thing that he/she could do ∞ - ∞ = 0 User:AnyFile 21:30, 30 Jan 2005 (UTC) == 1/1=1, 1/0=Infinity, (0/0=1 and Infinity) Please verify? == 1/1=1, 1/0=Infinity, (0/0=1 and Infinity) Please verify? Why are the past, the present and the future is the same time as January 2005, some cluster of stars(many galaxies) we can see by light speed only and after that human mind memory recall in only 0.0001 second, it means infinity light speed can touch by human mind in 0.0001 second too. I don't know what the rest of the stuff you're saying means, but I'm pretty confident 1/0 is ''not'' infinity, since the inverse of infinity is the infinitesimal, while the inverse of 1/0 is simply zero. User:Citizen Premier 15:54, 10 Jun 2005 (UTC) In mathematics, for real numbers or complex numbers, division by zero is ''undefined''. So 1/1 = 1, but 0/1 does not equal infinity, and 0/0 does not equal 1 or infinity, they simply don't equal ''anything''. The IEEE floating-point standard does allow for division by zero for floating-point computations for computers. It defines ''a''/0 to be "positive infinity" when ''a'' is positive, "negative infinity" when ''a'' is negative, and NaN ("not a number") when ''a'' = 0. User:Paul August User_talk:Paul August 18:18, Jun 10, 2005 (UTC) == Transfinite == Would you consider interesting to put a note why infinity numbers are called (starting form Cantor) transfinite numbers? User:AnyFile 10:45, 2 May 2005 (UTC) == ∞ infinity is not a number == Because infinity is not exactly equal to any finite number, and acts differently than any (other) number, many people find it easier to understand explainations that state right up front that "infinity is not a number". * http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM#roriginf * http://www.kuro5hin.org/story/2003/6/3/95744/71866 "While zero is a concept and a number, infinity is not a number. Infinity is the name for a concept. Infinity cannot be considered as a number since since it does not follow numbers' properties. "Infinity" is not a number. ... When mathematicians say "x approaches infinity", or write "x→∞", they mean "x grows arbitrarily large. And when they say that the limit of something is 0, they mean that it can go as close to 0 as you want it to, but it may or may not be actually equal to 0." OK if I use that definition on Infinity and Infinity plus 1 ? If no one comes up with a better suggestion the next time I come by, I'll just stick that definition in the article and watch the reaction. --User:DavidCary 01:58, 12 May 2005 (UTC)
See other meanings of words starting from letter:
I
IA | IB | IC | ID | IE | IF | IG | IH | IJ | IK | IL | IM | IN | IO | IP | IR | IS | IT | IU | IW | IX | IY | IZ |Words begining with Infinity:
Infinity
Infinity
Infinity,_Inc.
Infinity-Man
Infinity_(album)
Infinity_(album)
Infinity_Abyss
Infinity_Broadcasting
Infinity_Broadcasting_Corporation
Infinity_Broadcasting_stations
Infinity_cove
Infinity_Crusade
Infinity_Engine
Infinity_Gauntlet
Infinity_Gem
Infinity_Gems
Infinity_hotel
Infinity_Inc.
Infinity_Inc._members
Infinity_Minus_One
Infinity_of_Heaven
Infinity_of_heaven
Infinity_plus_1
Infinity_plus_one
Infinity_symbol
Infinity_trick
Infinity_tricks
Infinity_tricks
Infinity_War
Infinity_Ward
Infinity_Watch
Infinity_Watch_members
Infinity_Within
These materials are based on Wikipedia and licensed under the GNU FDL
YouTube.com videos better site than Turbo Tax 2007
