Talk:Infinity

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A vital article.

1 Is the dispute over?
2 Symbol for Infinity
3 What is this article about?
4 hey!
5 Physical infinity -- impossible or not?
6 ∞
7 Infinity in real analysis
8 1/1=1, 1/0=Infinity, (0/0=1 and Infinity) Please verify?
9 Transfinite
10 ∞ infinity is not a number
11 Wanna know what I hate???
12 Arithmetic properties of infinity
13 Infinity society
14 modular arithmatic with infinity
15 explanation of revert
16 Comment on 'Explanation of revert'
17 Infinite integrals
18 Jaina conceptions of infinity
19 Nissan Infiniti
20 Integrals that evaluate to infinity
21 So are some infinities bigger than others or not?
22 Recurring numbers
23 Lead
24 Minus Infinity is less than Infinity - True or False
25 undefined operation?
26 The Mathematics of Quantification - Infinity Mathematically Defined
27 Accountable and Unaccountable
28 One infinite greater than another
29 Negative infinity plus infinity? / The middle of infinity? [Borges]
30 [citation needed] tag for Cosmology
31 ∞/2 = ∞.
32 a strange varient
33 Pi and Fractals
34 Hindu Wisdom?
35 Infinity (mathematics)
36 At least two different infinites in mathematics
37 Source?

There's an article about how to compare infinities at 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. F. Lee Horn

This is discussed under Cardinal number. The Infinity article does mention this, but perhaps it needs to be made clearer. --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? --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? --LC

I agree. Also, the claim that "infinity has greatly increasd in size over the years" is pretty hilarious. AxelBoldt

Isn't it generally assumed by astonomers and physicists that our universe, forget about any other ones, is not infinite? -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". 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 AnyFile 18:16, 8 Sep 2004 (UTC)

[edit] 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. Kyz 10:49, 11 Sep 2004 (UTC)

[edit] 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,

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 &in; used to be written ε). No idea whether that's true either.

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... --Umofomia 12:29, 14 Mar 2005 (UTC)

My recollection is that the infinity symbol first appeared in Roman times (B.C.E.) and it represented 1,000,000 sesterce. I'm sorry I can't cite the source it was some dusty old mathematics history book. I have a photo of the artifact stashed somewhere files in storage. I'll see if I can dig it out sometime in the coming months.--YORD-the-unknown 23:46, 7 July 2006 (UTC)

[edit] 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 formalist, this is just another axiom like all the others, but for those who (still) take a more platonistic 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. Paul August 17:47, Oct 21, 2004 (UTC)

Prumpf 16:38, 16 Oct 2004 (UTC)

[edit] 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. 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. 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. 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. 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. 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.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. 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? 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. 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" :) Gene Ward Smith 03:56, 1 Nov 2004 (UTC)

[edit] 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

$S = \frac{1}{R}$

So the conductance of an object with zero resistance (superconductors!) is infinity. --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)AnyFile 15:26, 24 Nov 2004 (UTC)

As a number wouldn't infinity = 9 repeat

[edit] ∞

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? Oscar Evans December 14th 2004.

[edit] 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 AnyFile 21:30, 30 Jan 2005 (UTC)

I have put in a section regarding the properties of infinity on the real set. It explicitly says that you can not do infinity minus infinity (or plus minus infinity) in order to get zero as it is an undefined property. I have put in some properties regarding undefined operations and some regarding operations on the real set. ~Econ Schol (Wednesday, Spetember 14, 2005; 2237 EST)

[edit] 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. 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. Paul August ☎ 18:18, Jun 10, 2005 (UTC)

Instead of asking why is ${1 \over 0}$ undefined ask what is the value of x when $1 = x \cdot 0$ . The value can not be pinned down to any unique and existent value in the real number set. ~Econ Schol

(modified from a post in the Q&A area) I think that ${0 \over 0}$ and 0 * infinity are indeterminate, not undefined? $\lim_{x \to 0} x \cdot \frac{a}{x} = {0a \over 0} = 0 \cdot \infty$ , but also $\lim_{x \to 0} \frac{ax}{x} = \lim_{x \to 0} a = a$ , so $0 \cdot \infty$ can be whatever a is allowed to be? See indeterminate form. --AySz88^-^ 02:44, 12 December 2005 (UTC)

You can not divide anything by zero

[edit] Transfinite

Would you consider interesting to put a note why infinity numbers are called (starting form Cantor) transfinite numbers? AnyFile 10:45, 2 May 2005 (UTC)

I think it might be. Let me give my understanding of the reasons, and see if anyone can confirm or deny or (most importantly) give a reliable source:

"Infinite" means "without limit", but the infinite objects Cantor was considering were in some sense limited, in that we could consider them all at once as a set, and consider abstract operations performed on their totalities. This is in distinction to what Cantor called "inconsistent multiplicities" (what we now call "proper classes") such as the collection of all sets.
Cantor was a deeply religious man, and didn't want any confusion between his transfinite numbers and the Infinite; that is, God.

I think both of these points (if I'm right about them) could be useful additions to the article. --Trovatore 01:47, 25 July 2005 (UTC)

[edit] ∞ 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".

"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.

--DavidCary 01:58, 12 May 2005 (UTC)

No, I don't think that's a good idea. The word "number" is pretty vague; it's not clear what content is expressed by the claim that something is (or is not) a number. It's true that contemporary set theorists usually drop the word "number" from the locutions "cardinal (number)" or "ordinal (number)", but it's not clear whether this is from queasiness about calling such things "numbers"; could just be for economy. BTW your quote above seems to be restricted to the notion of "potential infinity", which is only one of the concepts developed in the article. --Trovatore 20:53, 24 July 2005 (UTC)

Agree with Trovatore. Infinity is not one concept, but many. Point on Riemann sphere, cardinal, ordinal, extended real, etc. --MarSch 10:45, 30 August 2005 (UTC)

[edit] Wanna know what I hate???

I hate it when people try to claim there is such a thing as "negative infinity". That's the most absurd thing I've ever been taught in school. Like the comment above, infinity is not a number and therefore cannot be negative. You can go approach infinity in a negative direction, but you can't go to "negative infinity". Phew... got that off my chest. --Lord Voldemort ^{(Dark Mark)} 21:26, 22 August 2005 (UTC)

In the extended real numbers, there most definitely is a negative infinity, whether you hate it or not. --Trovatore 02:41, 23 August 2005 (UTC)

No, since "infinity" is not a value, there can be no negative infinity. Only numbers can be negative. Infinity can be a distance, and there cannot be such a thing as a negative mile, negative meter, etc. Likewise, there cannot be a negative infinity. Just because someone says it is so, doesn't make it so. --Lord Voldemort ^{(Dark Mark)} 13:56, 23 August 2005 (UTC)

What's wrong with a negative mile? If I go a negative mile east I end up a mile west. Paul August ☎ 17:12, August 23, 2005 (UTC)

No, you are walking a mile in the direction of West, not going a negative mile East. Likewise, you can travel for infinity in the negative direction, but you cannot go towards "negative infinity". --Lord Voldemort ^{(Dark Mark)} 17:16, 23 August 2005 (UTC)

Are you familiar with a number line, in particular Extended real number line? Going in one direction goes through positive numbers, the other way through negative numbers. If positive infinity is all of the way out through the positive numbers, negative infinity is all the way out through the negative numbers. Bubba73 17:31, August 23, 2005 (UTC)

Yes, I understand number lines very well. However, I think it is erroneous to label infinity "positive" or "negative". Infinity is definitely not a number and therefore does not fall on a number line. Can you plot infinity? --Lord Voldemort ^{(Dark Mark)} 17:34, 23 August 2005 (UTC)

Negative infinity and positive infinity are not real numbers, but they are part of the extended real numbers (Extended real number line). Positive infinity is infinitely far in the positive direction and negative infinity is infinitely far in the negative direction. For instance, you frequently see an integral from negative infinity to positive infinity. This means that you get the area under the curve from infinitely far in the negative direction to infinitely far in the positive direction. Bubba73 17:46, August 23, 2005 (UTC)

Okay, so you are saying that the area goes infinitely far in one direction or another. That is what I agree with. Labeling infinity as positive or negative seems to be making "infinity" an integer, which I believe is false. I never said you could not go infinitely far in the negative direction, I only said that the could be no such thing as "negative infinity". Is it just a labeling problem? I believe I am smart enough to understand most mathematical theories, I just don't get this one. --Lord Voldemort ^{(Dark Mark)} 17:55, 23 August 2005 (UTC)

Quoting you Labeling infinity as positive or negative seems to be making "infinity" an integer, - no one ever said that positive infinity or negative infinity were integers. No one ever said that positive infinity or negative infinity were real numbers. They are extended real numbers - the real numbers plus positive infinity and negative infinity. They are both infinitely far away, but in opposite directions on the number line, and they are not the same. Is it just a labeling problem? Those are the labels we use. They could have been named positive prixibeen and negative prixibeen. I believe I am smart enough to understand most mathematical theories... I think you're having problems with negative as well as infinity. You said that you can't have a negative mile. Can you have negative money? Bubba73 18:11, August 23, 2005 (UTC)

Moved over here for easier discussion... First, please do not try and belittle me. I quite understand both "negative" and "infinity". You don't need to try and be condescending. Secondly, no, you cannot have a "negative mile". A mile is a measure of distance. And you could have negative money since it is a real object. But on second thought, perhaps you cannot have "negative money". You may owe people money, but you cannot posses "negative money". I don't recall ever seeing that term in an economic book or journal. But I digress... I don't care what you label infinity, if prixibeen means the same thing as infinity, you would not be able to have "negative prixibeen". Basically all I am saying is that you may be able to go prixibeen in the negative direction, but how can you label something as "negative" if it is not a number? Can you have "negative democracy", "negative art", "negative file cabinets"? (and I am talking about mathematical "negative", not the generic adjective) I just think it is absurd to call something negative when it cannot be negative. --Lord Voldemort ^{(Dark Mark)} 18:36, 23 August 2005 (UTC)

I give up. You asked us to help, we've explained it to you, but you're not listening. Bubba73 18:42, August 23, 2005 (UTC)

I asked for help, and you try and belittle me. Oh gee golly, thanks a bunch! I just feel like this whole thing is a crock, and wanted to know if someone could explain it. Hopefully you are not a math teacher, because if you are I would be amazed if your students could learn anything from you. I was asking a serious question. Some people are honestly trying to help. I thank them for their work, although I fear it may be in vain. I may never get the whole "negative infinity" thing, and thanks to responses like yours, I might not want to ask you guys for help again. Thank you to all who tried to help. I guess I was made to live in the real world, and not the world of theoretical mathematics. --Lord Voldemort ^{(Dark Mark)} 18:48, 23 August 2005 (UTC)

You can plot infinity in projective geometry. Fredrik | talk 17:49, 23 August 2005 (UTC)

From my quick perusal of the article, it seems like they define infinity as the point at which parallel lines will meet. However, isn't that just theory, and no such point actually exists. I think of it like I think oflimits. You will come really, really close to the value, but you will never quite get there. No one can get to "the line of infinity" as the article says, it is just a made-up myth. Perhaps I need to go read the article again. --Lord Voldemort ^{(Dark Mark)} 18:00, 23 August 2005 (UTC)

Well I must say that Point at infinity is a very interesting article (however false and made up it is). But I understand people want to push mathematics into the hypothetical realm, and that's okay. --Lord Voldemort ^{(Dark Mark)} 18:06, 23 August 2005 (UTC)

Voldermort, the same way you feel about "negative infinity", that it is "made up" and "hypothetical" others have felt about zero, negative numbers, rational numbers, irrational numbers, imaginary numbers, complex numbers, quaternions, … As Leopold Kronecker famously said, "God made the natural numbers, all the rest is the work of man." Paul August ☎ 18:38, August 23, 2005 (UTC)

See stereographic projection for a more concrete example. Fredrik | talk 18:15, 23 August 2005 (UTC)

Back to your original post, I think you misunderstand what actually infinity is in mathematics. I think it's easier to look at infinity as a concept rather than some object. Then, as you write, "approaching infinity in negative direction" is acceptable for you. This is like saying "there is some sequence of numbers, and for any chosen negative number, since a certain point in the sequence, all next numbers in the sequence are less than the chosen number". Then the sequence is approaching infinity in negative direction, or, as mathematical definition goes, it has negative infinity as its limit. But this definition is very similar to definition of finite limit, where a sequence is approaching to a certain number. So, in a sense (as far as limits are involved), just like limit of some sequences are real numbers (finite), you may understand negative infinity as a special number (that is, understand it as an object rather than as a concept), which is a limit of some other sequences. This is nicer, because instead of having two very similar definitions of what is limit, you get one definition. The same goes for positive infinity. The properties of these are elaborated in extended real number line. So the negative and positive infinity are not exactly numbers, but have some similar traits. So saying "infinity is not a number, so it cannot be negative" is only partially true. It's possibly true for infinity as a concept, but it's not quite true for infinity as an object in the above sense. As a side note, in complex numbers, there is only one infinity, because "positive" and "negative" don't make sense for complex numbers (so maybe you would be happier with that? :)). Samohyl Jan 21:47, 23 August 2005 (UTC)

To add the confusion ;) -- I believe "negative infinity" is mathematically incorrect. In contexts (such as the extended real numbers) where "infinity" is positive, then it's negation is "minus infinity". Arthur Rubin 01:36, 24 August 2005 (UTC)

I don't understand. I consider the terms "negative infinity" and "minus infinity" to be semanticaly equivalent (although honestly, I tend to rather use "minus infinity"); you can disagree with that, but either way that doesn't make the object (minus infinity as extended real number) ill-defined (mathematically incorrect). I think the real problem here is that "infinity" in mathematics is used in much more senses than just this, and thus the exact semantics depends on context. Samohyl Jan 09:13, 24 August 2005 (UTC)

"Negative" (and "positive") are predicates (the mathematical analogue of an adjective). "Minus" is a function; if "infinity" is "positive", then "minus infinity" must be "negative", for any context in which "positive" and "negative" have a more-or-less standard meaning. Arthur Rubin 20:22, 24 August 2005 (UTC)

Voldemort, I had a very similar argument when being interviewed for an undergraduate university place. I was asked the value of x / sin x at 0. I replied that the limit as you approach 0 is 1, but I insisted that actually at 0 it is undefined, since there is a divide by 0. I have no idea what bearing this has on the existence or not of a minus infinity. --stochata 16:17, 26 August 2005 (UTC)

A long-ish comment by Nowhither:

There is an important step in learning to do mathematics, which many of us, who have passed it, forget. That step is learning about abstraction. We talk about things like "minus infinity" as if they have some independent existence. Well, they don't. And we know they don't. But we forget that the thought processes we have when we consider "minus infinity", and things like it, are not the same as what others are thinking when we discuss it when them.

So, Lord Voldemort, here is what is really going on.

Sometimes, we have need of an object that is considered to "come before" every real number. A common situation is when we want to distinguish between the behavior of sequences like -1, -2, -3, -4, -5, -6 ... and sequences like 0, 1, 0, 1, 0, 1 .... Both of these diverge, in the real numbers, but they do it in different ways. The first one is heading "to the left" forever. The second just jumps around.

The way we solve problems like these is to "extend" the real numbers. How? Start with the real numbers and add a new symbol to the mix. How about k. This k is not going to participate in the arithmetic of the real numbers, as it is usually defined, but it will be part of the order relation. We already have an order relation defined on the usual real numbers: the standard symbol for it is "<". Thus, it is true that 5 < 6, but it is not true that 4 < 3. So, we extend this order relation as follows:

The statement "k < x" is true for every standard real number x.
The statement "x < k" is false for every standard real number x.
The statement "k < k" is false.

Now, we have an order relation defined on our new set, which includes all the standard real numbers, along with k.

Of course, we don't usually call it "k". We usually call it "minus infinity". If you don't like that name, well, you are entitled to your opinion. However, the name we choose for it has no bearing on the correctness of the mathematics involving it.

It turns out that this thing we call "minus infinity" (along with a commonly defined companion value at the other end, "plus infinity", or just "infinity") acts very much like a real number, in some ways, but not in others. Experienced mathematicians know the ways in which it does and the ways in which it does not, and we are ready to avoid the pitfalls before we hit them. Thus, we can get away with lazy ways of speaking; we know what we really mean.

However, as I said earlier, we forget that others do not.

— Nowhither 01:29, 28 August 2005 (UTC)

Infinity in space, time has alway been an issue in philosophy (Aristotle, Zenon, etc), physics, theology, and of course mathematics (starting with Bolzano, Cantor).

According to Aristotle, "infinitum actu non datur". There is no actual infinity, infinity exists only in potential.
According to Descartes, space is indefinitely (indefini) rather than infinitely (infini) large. We can not imagine a limit to space, without imagining that there is something beyond this limit.
In contrast, Newton always thought that the actual infinity extists. Consider a triangle and let one of the base angles increase. The point of intersection of two sides will go to infinity as they become parallel. "and noone can say that this point of intersection of parallel lines exists only in imagination. there is always a third point in the triangle, even if it is beyond any limit"

I can go on to infinity with this remark. However, I would like to point out the view of Wittgenstein. In mathematics, they operate with symbols according to some pre-set rules. There is nothing infinite in this operation, in the symbols, in the grammar, or in the rules. It is just a game, where they try to eliminate contradictions. A claim that there is no infinity has nothing to to with the symbol $\pm\infty$ .

— Igny 18:31, 30 August 2005 (UTC)

[edit] Arithmetic properties of infinity

I've done some minor cleanup on this section (original author didn't mention extended reals, for example). But wording and organization are still strange, and frankly I question the need for the section at all, given the prominent links to extended real number line elsewhere in the article. What do others think? --Trovatore 15:17, 15 September 2005 (UTC)

Hi Trov, yes I saw this also, and was dithering around trying to decide whether to just remove it, move it here for discussion, just do a quick and dirty edit to make it correct or fix it up completely. In the end I intended to do exactly what you did, with exactly your questions. I think this sections does probably belong here. But if it is going to be here it should be fixed up, and probably be as complete as the corresponding section in the extended reals article. I since I also think that article should be formatted more like this one, I was going to fix that one first, but I've been distracted dealing with the Ed Poor case and its fallout :( (Amazingly after a 14 months and 14,000 edits that's the first time on Wikipedia that I've had the need to use that particular emoticon!) Paul August ☎ 19:14, 15 September 2005 (UTC)

The section on extended arithmetic is a valuable addition that should be kept. It illustrates one of the main principles of the mathematical method. Like extending from natural numbers to integers to rationals to reals to complex. Or from real functions to generalised (infinitely differentiable) functions. −Woodstone 21:13, 15 September 2005 (UTC)

I'm sorry I did not do a good job on it. I just copied it down right from my notes from class and put it here. My professor put the multiplication of infinity with the real numbers in a seperate heading than arithametic properties (I do not know why). He has some more stuff under the same headings but as different points/bullets like zero times infinity and zero times negative infinity as two different bullets that I combined into one. ~Econ Schol

[edit] Infinity society

I've removed the link to the "Infinity Society", http://www.infinitysociety.org/index.html . The site is definitely worth a look, though. I like this passage:

Jean-Pierre Ady Fenyô, is an internationally-recognized philosopher currently residing in Budapest whose main work is on the penultimate positive social benefits of contemplating the infinite.

He doesn't tell us what the last benefits are. Anyway he does seem a decent sort of person, at least from what you can tell from what he says; I'm just not convinced his site is a correct link for an encyclopedia article on infinity. --Trovatore 02:32, 23 November 2005 (UTC)

I've put a note on Wikipedia talk:WikiProject Philosophy to see if anyone's heard of this Fenyo character, and in the mean time I've removed the reference to him on Infinity. --Trovatore 03:38, 23 November 2005 (UTC)

I've put Jean-Pierre Ady Fenyo on AfD. If anyone knows anything about this, please contribute, especially if you can back up the claim that he's notable and the article should be kept. --Trovatore 20:49, 23 November 2005 (UTC)

I found somthing about him here: http://www.reference.com/browse/wiki/Jean-Pierre_Ady_Fenyo. The site claims to be "Powered by Wikipedia", but I'm not sure what they mean by that.--67.188.80.67 23:39, 7 February 2006 (UTC)

It means that they're a mirror site of WP, and on articles like this they're what you might call a "mirror into the past", because WP's article was deleted more than two months ago. See Wikipedia:Articles for deletion/Jean-Pierre Ady Fenyo. --Trovatore 01:39, 8 February 2006 (UTC)

[edit] modular arithmatic with infinity

Assuming that when we write $x mod y$ we mean the remainder (there are other meaning of the symbol mod) then $\infty\bmod x$ is undefined.

The statement " ${\pm\infty} \bmod {\pm\infty}$ in undefined" is not true.

First we never define $x\;\bmod y$ if $y$ is negative so we should only look at ${\pm\infty} \bmod {\infty}$ .

I claim we can consistantly define

${\pm\infty} \bmod {\infty} = {\pm\infty}$

Mungbean 12:05, 25 November 2005 (UTC)

When there is only one infinity in a statement, e.g. p(∞), we can define this to be limit(x->∞) p(x). If the limit exists (possibly an infinity), p(∞) is well-defined. When there is more than one infinity in a statement, e.g. p(∞,∞), then the limit must be defined regardless of how the infinities are approached: limit(x->∞,y->∞)p(x,y) (F1). This is not necessarily the same as limit(x->∞)limit(y->∞)p(x,y) (F2) or vice versa (F3), and is certainly not the same as limit(x->∞)p(x,x) (F4) -- though if F1 is defined then F1=F2=F3=F4; and if it is not true that F2=F3=F4, then F1 is undefined (this is not an iff).

So for example ∞+∞ is unambiguously ∞, but ∞-∞ and ∞/∞ are undefined. In the mod case, consider for example limit(x->∞) (x mod x+1) = limit(x->∞)(x) = ∞ vs. limit(x->∞) (x mod x) = 0.

So ∞ mod ∞ is undefined. --Macrakis 19:28, 29 November 2005 (UTC)

Why would anyone expect it to be defined? The extended reals are generalizations of the reals, not of the naturals, and we generally don't define "mod" on the reals. One could do so, I suppose, but in practice it's rarely done. This sort of playing around with the definitions is a good thing for the curious to do on their own, but I don't see that it belongs in the encyclopedia article. I'm going to remove it. If anyone wants to put it back, please provide a source where it's been published. --Trovatore 19:36, 29 November 2005 (UTC)

a) Limits at infinity (though not for finite numbers) are just as meaningful over Z extended with ∞ as over R extended with ∞. b) It is common to define the natural extension of integer division and remainder over R: (a>0, b>0) a div b = floor(a/b); a rem b = a-(a div b)*b (definitions vary for neg a,b). But it's no big deal to me whether it's included. In any case, I think we agree that ∞ mod ∞ is undefined, not ∞. --Macrakis 21:00, 29 November 2005 (UTC)

Point taken, I agree ${\pm\infty} \bmod {\infty}$ is undefined. Mungbean 12:02, 30 November 2005 (UTC)

Should we also add a section for other common functions. e.g.

$e^{\infty}=\infty$

$e^{-\infty}=0$

etc Mungbean 12:02, 30 November 2005 (UTC)

Functions aren't defined in structures unless you so specify, and the extended reals aren't the reals. The source I have at hand (Folland, Real Analysis, Wiley-Interscience 1984, ISBN 0 471-80958-6) defines only addition, subtraction, and multiplication. Does someone have a source that defines more? --Trovatore 16:43, 30 November 2005 (UTC)

[edit] explanation of revert

I have reverted this claim:

In mathematics, infinity is not a number but a concept of increase beyond bounds.

The notion of "number" in mathematics is not sufficiently well-defined to say whether infinity is or is not a number. In the extended real numbers there are explicit values ∞ and −∞. See also cardinal number and ordinal number, which include infinite examples of each (though none of them is called simply "infinity"). --Trovatore 19:14, 29 November 2005 (UTC)

[edit] Comment on 'Explanation of revert'

The statement that 'infinity is not a number' is a direct quote from: Mathematics: from the Birth of Numbers, by Jan Gullberg, 1997, ISBN 0-393-04002-X, pub. by W.W. Norton & Co., Inc..

It also is supported by Comment 10, above.Duncan.france 19:56, 29 November 2005 (UTC)

No doubt there are sources that will say that. It's a commonplace among high school math teachers who are trying to keep their students from treating infinity according to the usual algebraic rules they're used to, and getting into trouble that way. But it's not a meaningful claim in the categories most contemporary mathematicians use, and it's not helpful in the context of the article. --Trovatore 20:02, 29 November 2005 (UTC)

The most recent version looks fine, even if "a concept of increase beyond bounds" falls oddly on my ear--I want to read "increase" as a verb, so it brings back memories of scrolls of confuse monsters. But that's OK. --Trovatore 22:48, 29 November 2005 (UTC)

[edit] Infinite integrals

The statements (which I have modified):

$\int_{0}^{\infty} \, f(t) dt \ = \infty$ means that the area under f(t) increases without bound as its upper bound increases limitlessly
$\int_{0}^{\infty} \, f(t) dt \ = 1$ means that the area under f(t) approaches 1, though its upper bound increases limitlessly.

are problematic. What is this business about upper bounds increasing limitlessly?:

For f(t)=(t<1: 1/t^2; t>=1: 0), the integral is already infinite for [0,1].
For f(t)=(t<1:1 ; t>=1: 0), the integral is 1 for [0,1] and remains 1.

The new wording isn't ideal, but at least it isn't as wrong.... --Macrakis 01:55, 1 December 2005 (UTC)

[edit] Jaina conceptions of infinity

The ancient Indian conceptions of infinity are very interesting. Apparently they are documented in R.C. Gupta, "The first unenumerable number in Jaina mathematics", Ganita Bharati 14 (1-4) (1992), 11-24. Could someone please look up this article and report back on what it says? Apparently Jaina mathematics included more than one magnitude of infinity, which is fascinating. It seems like a leap, though, to compare that to Cantor's work on transfinite numbers. --Macrakis 17:21, 5 December 2005 (UTC)

[edit] Nissan Infiniti

There is no need to have a separate disambig from Nissan Infiniti; that's one of the articles covered at Infinity (disambiguation), which is linked to from the "For other uses ..." line at the top of the page. --Trovatore 18:34, 11 February 2006 (UTC)

[edit] Integrals that evaluate to infinity

In reference to

$\int_{0}^{\infty} \, f(t) dt \ = \infty$ means that the area under f(t) is infinite

where a couple of editors have wanted to say the area is "not bounded". That doesn't make sense; it's a single quantity. Now, you could say, if you wanted to, that the set of all integrals $\int_a^b f(t)\,dt$ , where 0<a<b<∞, is not bounded. That would be one way for the integral to have an infinite value (though not the only way, as the integral could already be infinite for some particular a and b). If it were specified that f is continuous, then it would be the only way. --Trovatore 16:39, 17 February 2006 (UTC)

Actually the whole paragraph has problems. I had to change the third example to "equals 1" from "approaches 1". What does it mean for a single quantity to "approach" 1? The examples do look kind of tautological now, but at least they say something accurate. Rephrasing the original intent into an accurate statement would result in awkward language, I think. Maybe the paragraph should just be deleted.

One possibility would be to say something like

$\int_{0}^{\infty} \, f(t) dt \ = \infty$ means that the region under f(t) has subsets of arbitrarily large finite area

which strictly speaking I disagree with (it's equivalent but it's not what it means; what it means is that the area is really infinite) but would probably be acceptable. --Trovatore 18:08, 17 February 2006 (UTC)

The difficulty comes from the occurrence of infinity in the left hand expression. The integral is not a priori a number, but a limit. The proper way to express it would be: for every y there is a value x>0 such that $\int_0^x f(t)\,dt > y$ . Or stated otherwise: the set of $\int_0^x f(t)\,dt$ for all x>0, is not bounded above.−Woodstone 19:04, 17 February 2006 (UTC)

The region of the plane bounded by the x-axis, the y-axis, and the graph of f, has an area (more precisely, a two-dimensional Lebesgue measure), and that area is ∞. Your formulation is equivalent but stipulates an unnecessarily specific, and therefore awkward, collection of approximations to the region (bounding it on the right by vertical lines). --Trovatore 19:12, 17 February 2006 (UTC)

You seem to assume that the function f is continuous, how else can you have an area bounded by its graph. In integration of a function like this, only a lebesgue measure is defined on the integration domain (here indicated by t). There is no need for a 2-dimensional measure. The definition of the integral as a limit does not need the function to be continuous and is therefore more general (and correct). −Woodstone 21:34, 17 February 2006 (UTC)

If you look at the set in the plane given by {(x,y)|x>0 & y > 0 & y<f(x)}, then that set has a 2-d Lebesgue measure provided that f is anything reasonable; it certainly doesn't need to be continuous. (It could even be, say, the characteristic function of the rationals, which is not Reimann integrable. Proving the existence of functions for which it's not defined requires the axiom of choice.) No, the definition using the limit is not more general, not at all. --Trovatore 23:27, 17 February 2006 (UTC)

The limit formulation does not require Riemann integration, it can just as well be Lebesgue integration. Still no 2-D measure is needed. (But yes, the set as defined above would be well defined for "reasonable" functions, assuming a 2-D measure.) Still, defining infinite in terms of infinite does not seem right. −Woodstone 13:24, 18 February 2006 (UTC)

The limit formulation is a fifth wheel when considering the Lebesgue integral, but sure, you can do it. Not sure why you'd want to. Still, I don't object to some accurate characterization of these integrals in terms of limits or (perhaps better) suprema; just please don't say things like "the area approaches 1" or "the area is not bounded" when you're talking about the area of a fixed subset of the plane. --Trovatore 16:27, 18 February 2006 (UTC)

[edit] So are some infinities bigger than others or not?

There are an infinite number of numbers between 0 and 1. There are an infinite number of numbers between 0 and 2. Is the second infinity twice as large as the first? Or are they equal? The Disco King

There is a one-to-one mapping between those two sets given by f(x) = 2x. If there is a one-to-one mapping between two sets, we say they have the same number of elements. —Keenan Pepper 04:55, 22 February 2006 (UTC)

But, there are infinitely many integers and also infinitely many reals, but there is no 1-1 mapping between them; so there are really more reals than integers. There are many different "infinite" cardinalities of sets. −Woodstone 14:24, 22 February 2006 (UTC)

[edit] Recurring numbers

e.g. 0.9 recurring. There are infinite 9s. Does this number necessarily equal 1?

Yes. You can see this by subtracting 1.000... - 0.999.... What is the first non-zero digit of the result? The answer is that there is no non-zero digit in the result. You can formalize this using limits, etc. I'm assuming of course that we're talking about the ordinary real number system (not non-standard analysis with infinitesimals etc.). --Macrakis 21:48, 19 March 2006 (UTC)

That 0.99999... is equal to 1, came out from the way we choose in order to give a decimal rappresentation of real number (and of course from the definition of real number). AnyFile 10:43, 23 April 2006 (UTC)

[edit] Lead

This article needs a better lead - it weasels its way around, without saying what infinity actually is - which it should say in the first paragraph, if not the first sentence. zafiroblue05 | Talk 04:57, 2 April 2006 (UTC)

The problem is, there's no one thing that "infinity actually is"; it's a word having a variety of distinct, though related, meanings. That's exactly what the lead section is trying to get across. --Trovatore 18:21, 2 April 2006 (UTC)

I understand that, but it doesn't mean it can't be improved. What if the lead section was as follows?

The word infinity comes from the Latin infinitas or "unboundedness". The word refers to several distinct concepts 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.

(just rearranging, basically.) zafiroblue05 | Talk 00:23, 3 April 2006 (UTC)

I don't have any strong objection to that. --Trovatore 01:37, 3 April 2006 (UTC)

I have seen that this problme has already discussed previously. However I too consider that this article need a better lead. It was state previously (in this discussion) that this article Infinity is not Infinity (mathematics), however the article Infinity (mathematics) do not exists and it is currently just a redirect to Infinity. In mathematics (or at least in some sectors of mathematics) unboundness and infinity are two different concept (for instance a segment has infinte points, but is not unbounded. Morover to speak about unboundness a distance, or a metric system, should has been defined, while the same is not required to speak about infinite). I agree with a previous statement that speaking of infinite as a proprierty describing a set that can not be counted is naive in a post Cantor era. However since many people do really mean numerical infinity (the cardinality of N) when speaking of infinite and it is easier to start speaking of mathematical infinity in this way anote to this may be useful 10:55, 23 April 2006 (UTC)

[edit] Minus Infinity is less than Infinity - True or False

I need clarification after reading the previous discussions (perhaps I missed the definitive answer on this)...

Is the statement: $(-\infty) < \infty$ ...true or false?

-- PCE 20:25, 5 May 2006 (UTC)

The article states $-\infty < x < \infty$ for any real number x. As a direct consequence the statement $-\infty < \infty$ is true as well. −Woodstone 20:47, 5 May 2006 (UTC)

Okay I see it now... Just below the point where I was looking. Thanks. -- PCE 00:08, 6 May 2006 (UTC)

Is $\infty -\infty = 0$ ?...Kenosis 00:33, 6 May 2006 (UTC)

If you mean: $\infty -\infty = 0$ then I think it is undefined instead of zero. If you mean $\infty - (-\infty) = 0$ then I think it is infinity instead of zero. Before investing your life savings, however, on what I think I would wait for someone else's more qualified opinion. -- PCE 03:51, 6 May 2006 (UTC)

I think it depends on how you're moving to those infinities; for example, if you take x - x as x reaches infinity, it would be zero. But if you take (x+1) - x as x reaches infinity, that would still be "infinity - infinity", but now it equals one. If you take (2x) - x as x reaches infinity, that would be infinity.

That expression you wrote is currently under the Undefined operations section of the article. --AySz88 ^-^ 03:57, 6 May 2006 (UTC)

[edit] undefined operation?

The article says the following is an undefined "operation": $1^{\pm\infty}$

I interpret it to mean the number "1" raised to an infinitely large (or small) power. But my understanding is that multiplying one by itself always equals one, no matter how many times multiplied. So I would think it equals "1". What have I missed? Dagoldman 08:33, 17 May 2006 (UTC)

Well, zero times any finite number is zero, but that doesn't mean zero times infinity is zero. Analogously, one raised to any finite power is one, but that doesn't mean one raised to the power infinity is one. If you have an expression involving limits and you get $1^\infty$ by simple substitution, then it doesn't necessarily equal one, and you have to take the log and use l'Hôpital's rule. In fact, that's a good way to see why it's undefined. It's logarithm is $log(1^\infty) = log(1)\cdot\infty = 0\cdot\infty$ , another undefined expression. —Keenan Pepper 01:01, 18 May 2006 (UTC)

Yes, the same "intuitive reasoning" can also be applied to "zero times infinity". It seems to be carrying out a known simple operation (adding zero and zero) an infinite number of times. So it seems that the outcome is always "0". So what's the proof that "zero times infinity" is not zero? Or what's the logical flaw in this "intuitive reasoning"? I might buy your second explanation as a proof that "one raised to the power infinity" is not one. As a first step, I would have to be convinced that $log(1^\infty) = log(1)\cdot\infty$ is a valid equation. But is this operation valid if the exponent is infinity? I would not assume that it's valid, since "infinity" is not a real number. Does anyone have a reference or proof that you can move "infinity" out of the exponent in this manner? By the way, I'm treating "1" as EXACTLY 1, not a limit. Ditto for "0". Of course, if these quantities were limits, then "zero times infinity" and "one raised to the power infinity" would be undefined. But if you want to treat these quantities as limits, I think they need to be differently expressed, which is certainly easily done. Dagoldman 06:42, 18 May 2006 (UTC)

Has anyone defined zero or infinity mathematically? 88.109.19.139 19:32, 28 June 2006 (UTC)

Yes; they're mathematical concepts. Zero is the cardinality of the empty set, and a set is infinite if there exists a bijection from the whole set to a proper subset of itself. —Keenan Pepper 21:54, 28 June 2006 (UTC)

[edit] The Mathematics of Quantification - Infinity Mathematically Defined

The excerpt from: "The CIDR Network Descriptor expands the size of the IPtX

                  Address Space beyond the IPv6 IP Addressing Specification"

    ‘draft-terrell-cidr-net-descrpt-expands-iptx-add-spc-04.txt'

" Nevertheless, while the foregoing proof are confirmed by the

 conclusions from the proof of “Fermat’s Last Theorem”: However,
 a more thorough understanding of the results from the successive
 additions of '1', and the Basic Theory of Mathematics, which define
 Sequential Counting is required to understand the error in the
 current definition of Infinity; where 1/0 = INFINITY. And how is
 it possible to define an Infinitely Large Address Pool to the
 IPtX Specification.

   The proof;

   First, accepting that 1/0 = INFINITY, is not true, since Division
   is defined [3] as Subtraction, then the results from the
   continuous operation defined by eq-a, is pointless, because the
   result from this equation defines the REMAINER, and if the
   difference between the Remainder and the Quotient equals Zero,
   then the solution, by eq-b, of 1/0, is equal to Zero.

        eq-a.     1 - 0 = 1 - 0 = ... = (1 - 0)vN = 1

        eq-b.     1 / 0 = 1 - 1 = 0

   And if it can be said that the relationship between the
   Unary Set and the Binary defines a { 1 : 11 } Ratio, where
   Cardinatility representing this Ratio, given by eq-c, is True.

        eq-c.     ' 1 = 2 ' - Unary Set = {1}, Binary Set = {11}
                  "Given that:  11 = 1 + 1 = 2"

   Then the Definition of a Prime Number is Given By:‘

        eq-d.     '1 = 2’ - 'Prime Numbers'

             A 'Prime Number' or 'Prime Integer', is a positive
             integer, 'p Greater Than or Equal to 1', that has
             no positive integer divisors other than itself, 'p',
             and '1'.

   In which case, from the definition of a Prime Number, it can be
   concluded that if every Number except '1', is defined as the
   Progressive Additions of '1', then "1" defines the Absolute
   Number, which is equal to the Cardinal Number that defines the
   Universal Multiplicative Identity Element, representing every
   Item defining itself [2].

   Clearly, the Modern Numbering System evolve defining the Zero
   distinction as the Multiplicative Coefficient defining the
   Count of the Members in the Set of ' 1's ' containing 10
   Elements. Nevertheless, Nature's Mathematical System defines
   the Binary pair {0, 1}, as the symbols to be used for numerology,
   which ultimately refined into the more mathematical symbolic
   numerical representation used today.

   In other words, since '1' is the absolute Numeral, and the Binary
   and Unary Sets must, if they are equal and enumerate to infinity,
   map equally, in a One-to-One Correspondence with the Real Number
   Set, which is counted using successive additions of '1'.

   Then the Ratio defining the relationship between the Binary and
   the Unary Sets, and represented in the mathematical expression
   defining eq-c, since any One-to-One defines a sequential count
   defined by the Unary Set as the successive additions using  1’s,
   also defines 2 Infinities.

        eq-c. ' 1 = 2 '

     Unary Set = {1} = Infinity  =  Binary Set = {11} = Infinity

   Furthermore, since Infinity is not enumerable, then Infinity
   must define a Prime Number, because only the mathematical
   operations involving ‘ITSELF’ and ‘1’ are defined.

   Hence, "Given that:  11 = 1 + 1 = 2", and since eq-c is true,
   then by eq-e, we have;

        eq-e.  'Infinity’ = (Infinity + Infinity) = 'Infinity’

  And from eq-d, since multiplication is equal to the quantified
  sum of addition, we have by eq-f;

        eq-f. 'Infinity’ = (Infinity + Infinity) = Infinities,
                    (Infinity + Infinity) = 'Infinity’

  Therefore, since eq-d, “ 1 = 2 - Prime Numbers ”, is true, then;

                      Infinity  =   Infinity

  And this, since division is the quantified difference of the
  repeated subtractions performed on a constant, further implies;

     Infinity/Infinity = ‘ 1 ‘ and (Infinity - Infinity) = ' 0 '

  Nevertheless, since the foregoing equations defines the
  mathematical operations involving Infinity, and since these
  operations also defines the enumeration of the Elements defining
  the respective members contained in the Binary and Unary Sets.
  Then the Elements these Set respectively contain, each
  enumerate an individual count using successive additions of '1'
  in a One-to-One Correspondence, to reach an Infinite Count of
  the Members each Set contains.

  In other words, by definition [3], since the

                            Unary Set = {1},

  and the

                            Binary Set = {0, 1}

  Then Infinity is the definition of 2 Sets, which defines;

  The Binary Set defines; 1 either followed by an Infinite Zero
  String, or a Decimal Point followed by an Infinite Zero String
  terminated with a 1.

  1) A ‘1’ followed by an Infinite String of Zeros

  2) 1 + 0vINFINITY...00000000000000000000000000000000000000.0

  3) A ‘1’ proceeded by an Infinite String of Zeros

  4) 0.00000000000000000000000000000000000000...0vINFINITY + 1

  The Unary Set defines an Infinite String of ‘ 1’s ‘, either before
  or after the Decimal Point.

  5) A ‘1’ followed by an Infinite String of ‘1’s’

  6) 1 + 1vINFINITY...1111111111111111111111111111111111111.0

  7) A ‘1’ proceeded by an Infinite String of ‘1’s’

  8) 0.1111111111111111111111111111111111111...1vINFINITY + 1

     Note: Both examples define every element contained in the Set
           of Real Numbers. And more importantly, if the Infinite
           Zero String is countable, then succeeding or following
           the Zero String is also countable... Also, this proof
           can be construed as equating the Count of the '1's' in
           the Unary Set with the Zeros, and the '1' (occurring at
           a Count one unit beyond the Infinite Zero String) in
           the Binary Set."

Author:

  Eugene Terrell

  Principle Director
  Research & Development

  Engineering Theoretical Technologies
  Research & Development Publications
  (ETT-R&D Publications)67.124.90.191 04:13, 18 May 2006 (UTC)

[edit] Accountable and Unaccountable

I have no idea what i am talking about, but i heard "accountable and unaccountable infinity" mentioned. it was something like accountable is where you have say 1, 3, 5, 7, 9... and you pair those numbers up with 1, 2, 3, 4, 5... etc. and so every number can be assigned another number and so on. Unaccountable was where if you invent a whole lot of random numbers, say:

238973847692384

238477456902843076587323

7823465874638597263498574

982643586348576398456

9384658346587648

382746821947378468921

etc.

and this list is infinite. and then you go diagonally from the top right, so you'd get 232666... and this number can never appear in the first list.

a friend tried to explain this to me and i didn't get it so i went on google and then wikipedia to see if there was anything about it. does anyone know what i am talking about? --58.107.95.163

The terms are countable and uncountable. See also Cantor's diagonal argument. --Zundark 11:59, 18 July 2006 (UTC)

[edit] One infinite greater than another

Ok there is an infinte way to make a chair, but the is also an inifine ite way not to make a chair, there is more ways not to make a chair than there is ways to make a chair, so one infinity is greater than another, how can this be, (maybe reading the atrical more closely would help) There is an Infite number of things that arn't a chair 12:26, 26 July 2006 (UTC) sorry if i have done makeing this question bad, i am new to wiki langualge There is an Infite number of things that arn't a chair 12:26, 26 July 2006 (UTC)

Was that a question or a statement? Could you explain why you think "there is [sic] more ways not to make a chair than there is [sic] ways to make a chair"? —Keenan Pepper 23:23, 26 July 2006 (UTC)

well technically is supose it was a statement, but i intended it as a question, i wish to recieve guidence from another on this veiw and its truth or falsitiude. I belive there is infinite nuber of ways to make a chair as there in an infinite number of materials (alloys of various percentages,) and d colours of paint etc. but the is also an infinite number of ways not to make a chair (even if there are only two things that arn't a chair) these non-chair have an infinite nuber of materials they could bemade form etc. There is an Infite number of things that arn't a chair 11:07, 27 July 2006 (UTC)

Surely they are both infinite, but I don't understand why you say one infinity is bigger than the other. Personally I think the question is meaningless because you haven't defined anything in a mathematically rigorous way. —Keenan Pepper 18:47, 27 July 2006 (UTC)

perhaps you are splitting hairs (i intend no offence)?, my statement/question is intended to illistrate the fact that one infity might be greater than another (though i don't know the truth of this). i will do my best to explain mathimatically (even if i do have to let the number of ways to make a thing be infinte), a chair is one thing thus in my formulae it will be shown as 1, Let the number of ways to make a thing (including a chair) be is infinite, let the number of things that can be made that are not a chair be "a",

$1<a\qquad1\infty<a\infty$ --OXINABOX 09:31, 28 July 2006 (UTC)

I have replied on your user talk page, because this article talk page is only supposed to be about changes to the article, and some people get mad if you don't follow that rule. —Keenan Pepper 19:04, 28 July 2006 (UTC)

[edit] Negative infinity plus infinity? / The middle of infinity? [Borges]

In a footnote to his short story 'The Library of Babel' Jorge Luis Borges describes:

A volume of ordinary format, printed in nine or ten point type, containing an infinite number of infinitely thin leaves. (In the early seventeenth century, Cavalieri said that all solid bodies are the superimposition of an infinite number of planes.) The handling of this silky vade mecum would not be convenient: each apparent page would unfold into other analogous ones; the inconceivable middle page would have no reverse.

FIRST QUESTION: The pages are infinitely thin and therefore, if I am not mistaken, have a mass that can be computed at 0; however, they are infinite in number. What is the total mass of the book? In other words, if you multiply something of infinite smallness an infinite number of times, what is the resulting mass? Would an infinite number of gravitational singularities be infinite in size, or the same size as a single singularity?

SECOND QUESTION: Can anyone tell me why the middle page of this book has no reverse? I believe there is an answer, involving the mobius strip.

Thanks.

(I ask these questions because their solution might eventually feed into the article. Borges writes obsessively of infinity. It'd be nice to see an Infinity in Literature section to this article.)

Hmm, wasn't Borges an early magic realist? I think comparing his writings to the notions of mathematicians might kind of miss the point. If he were a hard SF writer we'd expect him to get it right.

Anyway, the answer to the first question is "it depends"; there are lots of different contexts in which you might formulate the question, and they give different answers. I have no idea why Borges would think the middle page should have no reverse. --Trovatore 19:33, 31 July 2006 (UTC)

Isn't this article about infinity in general? And what do you mean about a hard Sci-Fi writer being expected to get it "right". I can't be sure, but I suspect half of what you are saying is crap. "Correctness" should not be the single criterion for the inclusion of material in an article. The treatment of infinity in literature -- a part of the cultural impact of the idea of infinity -- has an unquestionable right to be included in this article; as much right as theological ideas in an article on cosmogony.

The book with an infinite number of pages appears to be an analogy for the real number line. The weight of each individual page must be zero, or else the book would be infinitely heavy. The product of a finite number of 0s is 0 but the product of an infinite number of 0s is undefined. The pages on the left side of the book correspond to negative numbers and those on the right to positive numbers. The page number 0, if it existed, would be in the middle of the book with no counterpart.Awis 00:22, 10 August 2006 (UTC)

[edit] ^{[citation needed]} tag for Cosmology

Everything that I have currently read on Cosmology and the Universe seems to suggest a finite universe not an infinite one. Anyone care to provide facts to backup a case for a infinite one? Since the Bang Bang, I don't believe we deal with the universe in terms of infinity any more. (Simonapro 14:10, 28 August 2006 (UTC))

There is no consensus on any of the three options as yet. See shape of the Universe and ultimate fate of the universe. EdC 14:53, 28 August 2006 (UTC)

Is an infinite universe as an alternative for big bang cosmology. (88.101.172.224 20:24, 28 August 2006 (UTC))

Not as such; big bang cosmology is compatible with the universe being flat or open. EdC 03:09, 29 August 2006 (UTC)

[edit] ∞/2 = ∞.

∞/2 = ∞

Is this logical or illogical? (Simonapro 09:32, 29 August 2006 (UTC))

An example of cardinality and bijection from set theory might help. Consider N, the set of all naturals {0,1,2,3,...}, and E, the set of even naturals {0,2,4,6,...}. We can define a bijection between the two sets, f(e) = ^e/₂, for every e in E. Clearly, every member e in E is mapped to a unique member ^e/₂ in N. It's also clear that both E and N are infinite sets. So that's one way of thinking about ^∞/₂ = ∞. — Loadmaster 00:16, 31 October 2006 (UTC)

[edit] a strange varient

i'v been charged by the group who came up with this to spread there idea on infinity. they claim that inifinity is the numbers 0-9. they numbers that appear everywhere.

thats it.

eh? 86.16.223.203 14:51, 25 October 2006 (UTC)

[edit] Pi and Fractals

Perhaps something can be mentioned about the infinite scope of mathematical entities such as Pi and some types of Fractals? HighInBC 02:04, 13 September 2006 (UTC)

Fractal would be nice but I think Pi, like all irrational numbers, can be represented with infinite number of non-repeating decimals. So I'm not sure what you exactly mean by infinite scope of mathematical entities such as Pi.Exteray ^T ^.C 02:23, 13 September 2006 (UTC)

By infinite scope, I meant that that it has an infinite number of non-repeating decimals, and thus involve infinity. But I don't know much about Pi, other than what you just told me. But I do know that the Mandelbrot set has infinite detial. HighInBC 02:28, 13 September 2006 (UTC)

So, does 1/3 have "infinite scope" too? —Keenan Pepper 03:26, 14 September 2006 (UTC)

I dunno HighInBC 03:38, 14 September 2006 (UTC)

Well, there you go. —Keenan Pepper 01:52, 15 September 2006 (UTC)

[edit] Hindu Wisdom?

Is this external link appropriate?

[edit] Infinity (mathematics)

I would like to suggest splitting off the mathematical concept of infinity to its own article and leaving a summary here, with {{main}}. There's a lot more to be said:

Cardinal infinities like the &aleph;s and &beth;s
Ordinal infinities like the ωs
The extended real line
Projective space, including the one-point compactification of the real line
Space-filling curves (showing that 1-dimensional and 2-dimensional 'infinities' aren't actually different)

and so forth. Some of these are already represented, but not in much depth. Further, there are few citations for what little content is there. I have three concerns that this solution would address:

A 'hard math' article should not be presented in the context of the social and literary works that are actually rather unrelated
Math content in a general page like this needs to be explained in more detail than is convenient, because of a wider audience, making it harder for readers who want technical details and not 'fluff'
A general page like this one should not be overwhelmed by mathematical content, because readers concerned with other meanings of infinity should not have to scroll through lots of content, let alone slog through it.

What do you think? CRGreathouse (t | c) 23:17, 6 October 2006 (UTC)

I think the mathematical meaning of infinity is the "main" one, so I don't agree with this project. The topics you mention above all have their own individual articles (well, not sure about the space-filling curves, but my guess is it's somewhere in WP). If any of these articles don't have wikilinks, then of course they should. --Trovatore 23:31, 6 October 2006 (UTC)

I counted 26 headings and subheadings in the article, of which 10 were mathematical. The mathematical meaning isn't first, or even in the first page -- unless you have a 3000px high display. At first I thought about ways to "fix" this as a math article, and then I realized that it wasn't a math article. It doesn't read like one now, and certainly the content isn't there.

Admittedly, some of these concepts should be discussed mostly elsewhere. Doesn't it strike you as odd, though, that in the article on infinity there's little talk about projective space and no mention (that I could see) about space-filling curves?

If the mathematical meaning of infinity is to be the main part of this article, a lot has to be changed. Rather than change the article so drastically, when it seems like a fine (non-mathematical) article, doesn't make much sense to me. I don't want to end up deleting lots of content. CRGreathouse (t | c) 04:00, 7 October 2006 (UTC)

It's difficult to discuss infinity in a solely mathematical context; since there are so many different mathematical infinities, a certain amount of historical and philosophical context is essential. Remember, many of the mathematicians who helped pin down infinities were also philosophers. I think splitting the article would of necessity duplicate a lot of content. Better to make sure that all relevant concepts are covered in brief with wikilinks to the actual articles. -- EdC 11:25, 7 October 2006 (UTC)

I would say that it is actually easy and proper to split mathematical and philosophical discussions of infinity.

The philosophical discussion is somewhat mathematical in nature, but never really gets into the nuts and bolts of mathematical infinity. The construction of the real numbers from the natural numbers is really the only mathematical example you need in order to discuss most of the philosophy (potential vs. actual infinity, etc.).

By contrast, mathematical objects such as compactified spaces are philosophically uninteresting, but are very important within mathematics.

I would strongly suggest either 1) split off an article about mathematical infinity; talk only about the philosophical ramifications in the non-mathematical article. Or 2) Split the mathematical section of this article into two sections, one on the philosophy of mathematical infinity, and one on the mathematics of mathematical infinity.

Maybe I'll try doing the second option, as that seems at least less controversial. There should at least be a clear space in this article for mathematical concepts, because most of them are missing from the discussion right now. Perhaps if we can get enough mathematics into this article, there will be more support for a split. Worldworld 18:57, 14 October 2006 (UTC)

Sorry, missed your response. I'd rather see what mathematics there is to be added before countenancing a split. If it starts to get unweildy, then sure, a split might start to look like a good idea. EdC 22:00, 31 October 2006 (UTC)

[edit] At least two different infinites in mathematics

I'd love to read about the fact that there are at least two kinds of infinites:

one is that of natural numbers and rational numbers
and the other is that of real numbers.

It can be mathematically proven that the numerosity (?) of the above two is different, that is, the second constitutes a "bigger infinite" than the first. Adam78 11:15, 25 October 2006 (UTC)

Look at Countable set. I added the link to the article as well. CMummert 13:53, 25 October 2006 (UTC)

There are infinitely many different infinite cardinal numbers. The two you are referring to are aleph-null and the cardinality of the continuum. --Zundark 14:01, 25 October 2006 (UTC)

Thank you! It's an interesting bit of info so I hope it won't be lost in the article. Adam78 20:49, 25 October 2006 (UTC)

Just to add my two cents worth: The two mathematical infinities could also be designated:

Discrete infinity, such as $\sum_{n=1}^\infty{f(n)}$ . The ∞ here is the countable kind of infinity, i.e., the sum of a countably infinite number of values of f(n).
Continuous infinity, such as $\int_{0}^{\infty} f(x) dx$ . The ∞ here is the uncountable kind, i.e., the sum of an uncountably infinite number of f(x)dx values. — Loadmaster 00:02, 31 October 2006 (UTC)

Hm. I don't think that works. Consider: the Cantor set is a disconnected dust but of cardinality C, while the rationals are dense on the reals but of cardinality $\aleph_0$ . EdC 22:03, 31 October 2006 (UTC)

Cantor dust is a good counter-argument. It is also, if I'm not mistaken, non-integratable. But I don't think a countably dense set (the rationals) is a good counter-example to integration of a countinuous function. But I could be wrong. (Are there continuous functions, other than trivial ones like f(x)=1, that contain only rational values?) — Loadmaster 23:58, 6 November 2006 (UTC)

In the reals, no; but in the rationals $f: \mathbb{Q} \rarr \mathbb{Q}: x \mapsto x$ is continuous, assuming the standard topology on $\mathbb{Q}$ . Similarly, the Lebesgue integral doesn't exist on $\mathbb{Q}$ , but (I think) one can use the Daniell integral to construct an integral on $\mathbb{Q}$ . EdC 12:05, 7 November 2006 (UTC)

These seem to be special cases not usually encountered by calculus students. So I think my point is valid, that we can in general distinguish between countable infinite using summation and uncountable infinity using intgration. Even if it's not 100% technically correct, it can still be used to introduce the difference between the two types to beginner-level students. — Loadmaster 23:33, 7 November 2006 (UTC)

[edit] Source?

Pardon, but what is the reference for the section, "Infinities as part of the extended real number line?" specifically the equations listed --JohnLattier 07:32, 15 November 2006 (UTC)

One source for many of the equations is Affinely Extended Real Numbers from Wolfram MathWorld. -- Schapel 15:21, 15 November 2006 (UTC)

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