Talk:Infinity

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)

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.

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)

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)

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)

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

<math>S = \frac{1}{R}<math>

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)

∞

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.

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)

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)

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)

∞ 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.