Talk:Infinity

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A Vital article, potential GA. Salix alba (talk) 12:56, 30 September 2006 (UTC)

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	The following comments have been left for this page: A Vital article, potential GA. Salix alba (talk) 12:56, 30 September 2006 (UTC) (edit)

1 Does the symbol for infinity actually make sense?
2 Incorrect statement removed
3 Comparing infinities
4 Physicial infinity
5 Cosmology sub-section
6 Extended real number line
7 Too much speculation about the symbol's origin
8 Citations for "Mathematical infinity" section
9 Infinity in the Arts
10 Not defined or infinity
11 Omni?
12 1/0=Infinity
13 "commonly represented as ∞"
14 Asaṃkhyeya
15 Infinity as a number

Talk:Infinity/archive1

[edit] Does the symbol for infinity actually make sense?

If you really think about it, the symbol for infinity suggests an infinite repeating loop, not a set of numbers that go on forever into increasing value. So, I guess maybe we should show a different main picture as a representation for infinity, and keep the circle loop thing an example of the mathmatical symbol for infinity. —Preceding unsigned comment added by Kaji132007 (talk • contribs) 06:14, 6 October 2007 (UTC)

[edit] Incorrect statement removed

I took out

"Likewise, perpetual motion machines theoretically generate infinite energy by attaining 100% efficiency or greater, and emulate every conceivable open system; the impossible problem follows of knowing that the output is actually infinite when the source or mechanism exceeds any known and understood system."

In fact, a perpetual motion machine could be detected in short order by measuring its finite energy output. The definition does not require infinite output, just output as great as the input.--Cherlin 23:31, 5 May 2007 (UTC)

[edit] Comparing infinities

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

There's an article about how to compare infinities at Counting Games. Should it be an external or "see also" link? --12.205.148.77

To answer your question: yes. There are more "numbers" than there are "odd numbers".--71.141.113.17 01:15, 27 July 2007 (UTC)

Sorry, but no. The cardinality of the odd, natural numbers is the same as that of the natural numbers. Thus any meaningful definition of "the same" will say that there are as many odd numbers as whole numbers. This is a simple result of Cantor's theory, which is the only way we know to define such relationships.

Unintuitive? Yes. But that doesn't change anything.--71.178.151.40 04:35, 9 September 2007 (UTC)

[edit] Physicial infinity

The article actually says this :

"It should be pointed out that this practice of refusing infinite values for measurable quantities does not come from a priori or ideological motivations, but rather from more methodological and pragmatic motivations."

Is there a source for this? Looks challengeable for the ideological part... Also, "methodological and pragmatic motivations" are somewhat "a priori" no? I further note that the example given (infinite gravitational mass) is an indication that no such body exists, but does not necessarily mean that no other infinite physical concept (such as space) can exist. --Childhood's End 15:36, 7 June 2007 (UTC)

I might have found something about this (seems to come from Glenn Learning Technologies Project (LTP), with some relationship to NASA) [1] :

"Books have been written on infinity. I shall not deal with infinity further here except to note (because I am a physicist) that physicists take great pains to avoid it. Whenever and wherever infinity appears in a calculation, or in the development of a theory, it must be eliminated. In quantum mechanics, this elimination is done via a formidable technique called, “renormalization.”

It is clear why physicists are so “antsy” about infinity: the equations of physics are built entirely out of numbers and their defined properties; infinity is simply NOT a number; it is an abstract and sometimes contradictory-seeming concept whose exact nature and definition continues to challenge mathematicians and philosophers even to this day."

I found that this was a nice description of what I had in mind... :) It also seems to support the article about the fact that the motivations are methodological and pragmatic, although I still think that it does not rule out a priori or ideological motivations. Any thoughts? --Childhood's End 13:57, 8 June 2007 (UTC)

A recently released computer game "Portal", allows the player to create a wormhole entrance and exit for the purpose of transeversing obsticles, however a common practice is to open a wormhole over another, creating an infinite chain of parralle realities which are openly visible and transversible by the player, a similar effect of infinitely extending vision involves placing two mirrors so that the reflective surfaces face one another, this could possibly add to the article some for explaining infinity in an easily accesible context Dagorlad 3 (talk) 00:35, 18 February 2008 (UTC)

[edit] Cosmology sub-section

The cosmology sub-section is interesting and wide-ranging despite that it is short, but perhaps someone could provide thoughts on a problem that I have with it:

"Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By walking/sailing/driving straight long enough, you'll return to the exact spot you started from."

This is a theory often encountered in cosmology debates about the Universe size/boundaries, but I still fail to grasp why this point has much merit. The fact that you can come back to your starting point on Earth and go round and round indefinitely is due to the fact that the Earth has physical boundaries and that you would give respect to them by following its defined surface. But as for the Universe, if it has no boundaries at all, you could not come back to your starting point by flying straight ahead. And even if the U would have a topology such as the Earth, you could still escape its boundaries by not respecting them while flying, unless we pretend there's a wall or something (what opens the door for at least a neighboring universe, and thus for an Infinite Universe if we call "Universe" the sum of all the universes...). Thus, it seems dubious to affirm that "the question of being infinite is logically separate from the question of having boundaries". Any thoughts? --Childhood's End 15:48, 7 June 2007 (UTC)

The Earth has a boundary -- its surface. The surface of the Earth does not have a boundary; there's no edge you can sail off of. The cosmological analogy is between the universe and the surface of the Earth. Melchoir 18:32, 7 June 2007 (UTC)

But then, this analogy seems to compare apples with oranges... Should not the analogy be made between the Earth and the Universe rather than with the Earth's surface and the Universe? I mean, of course, if the Universe, like the Earth, has a boundary - its surface - then its surface can be without boundaries. But what if the Universe does not have a boundary/surface like the Earth has? Again, unless I miss something, this analogy shows no logical separation of being infinite and the question of having no boundaries.

Further, for the analogy to hold even if we accept the dubious comparison, we must accept to comply with the Earth's gravity or topology and follow the surface so that it can be endless. Technically, there's an edge you can sail off a sphere's surface at every point of it. Unless there's some wall, the same could be said of any such boundary for the Universe and thus, again, we find that this shows no logical separation between the two ideas. Or perhaps I missed something? --Childhood's End 12:50, 8 June 2007 (UTC)

What you're missing is that this is an analogy to enable your intuition to deal with something with which it has no direct experience. To get closer to the (possible) real situation, you need to add one to each dimension. So instead of being the two-dimensional surface of a three-dimensional ball (like the surface of the Earth), the proposal is that our universe might be topologically equivalent to the three-dimensional surface of a four-dimensional ball. (More precisely, that could be the topology of a spacelike three-dimensional slice of four-dimensional spacetime.) --Trovatore 03:21, 15 June 2007 (UTC)

Yes, the problem is that the article is describing an inherently four-spatial-dimensional behavior that we, as creatures of three spatial dimensions, are just not built to understand. You shouldn't be blamed for this, but please don't let this hold the article up, because it is also quite true. It's an extremely common idea in modern cosmology, ever since Einstein's General Theory of Relativity introduced the idea (now experimentally verified) that 3D space (and 1D time) can be curved. Mathematically it makes sense, just like it makes sense to describe the 2D surface of a sphere as an unbounded, finite area. However, our intuition fails us in such cases, and we resort to analogous situations in lower dimensions. The mathematical description, however, is well defined in all dimensions, and this is one of the things that you just learn to live with when dealing with modern physics.--71.178.151.40 04:44, 9 September 2007 (UTC)

Well that's interesting, although learning to live with this when dealing with modern physics appears to me somewhat a biaised learning when I take renormalization into consideration. Or perhaps I dont make sense... --142.195.189.128 15:44, 21 September 2007 (UTC)

[edit] Extended real number line

The last few edits made to the "Infinity operated with a real number a" subsection contain equations with limits. This is probably incorrect, since that subsection is talking about the properties of the extended real number line, which includes the actual values +∞ and −∞. Thus ^a/₀ = ∞ (for all a ≠ 0), for example. — Loadmaster 05:13, 20 June 2007 (UTC)

A secondary point -- the phrases "infinity operated with itself" and "infinity operated with a real number a" are bad grammar. Presumably they were added by a non-native speaker? No offense meant to whoever wrote those phrases, but they absolutely can't remain. I won't fix them at the moment, first of all because I'm not sure exactly what to change them to, and perhaps more important, because I think the whole section should probably just be removed (the material should appear at extended real number line but is too detailed for this article). Anyone want to beat me to fixing it? --Trovatore 09:33, 20 June 2007 (UTC)

I support removing the section. --Zundark 09:44, 20 June 2007 (UTC)

Can you support that? that $a/0 = \infty$ for all a &ne 0 using the properties of the extended real line, where there's two infinities, and not a single infinity like in the Riemann sphere? A recent previous edit [2] made more sense in this section because it used the limits to explain how the operations worked. This may be a moot point, see below. Root⁴(one) 12:56, 20 June 2007 (UTC)

Which section? The extended real line needs to stay, or at least explicit some reference needs to be placed there, somehow. Maybe we should allow the extended real line explain these operations... yes, that would make sense. Hmmm, in fact our articles seem to contradict each other. To quote extended real number line#Arithmetic operations:

Note that 1 / 0 is not defined as either +∞ or −∞, because although it is true that whenever f(x) → 0 for a continuous function f(x), we must have that 1/f(x) is eventually in every neighborhood of the set {−∞, +∞}, it is not true that 1/f(x) must converge to one of these points. An example is f(x) = 1/(sin(1/x)).

Ok, this seems to be a fairly serious contradiction. My guess is that article is probably more correct than this one. Root⁴(one) 12:56, 20 June 2007 (UTC)

There is no contradiction. The article said nothing about 1 / 0. FilipeS 02:35, 22 June 2007 (UTC)

I can't see how you can say it said "nothing" about 1/0 ! It discussed 1/0 and an uncountable number of other expressions, but it never directly referred to 1/0. IIRC, we were talking about the expressions x/0 where x is real and $x \ne 0$ This set of expressions includes 1/0, but also all expressions are related to 1/0 by multiplication by some constant $c \ne 0$ .

No matter, I think the consensus here is that the algebraic manipulation of some symbolic concept of infinity should be only defined or handled on the pages which describe the addition of infinity or a multitude of infinities to some number system in question and not here. Root⁴(one) 14:10, 22 June 2007 (UTC)

Ugh..... reading [3], this is where Loadmaster got his facts. What's right? Is Affinely Extended different from Extended? Root⁴(one) 13:23, 20 June 2007 (UTC)

Affinely extended is the type discussed in the article (at least predominantly): This is the real line with a pair of points "at infinity". An alternative is the projectively extended real line (see [4]) which has a single point at infinity attached. Silly rabbit 13:54, 20 June 2007 (UTC)

(re-tab) After reading the Mathworld Affinely Extended Real Numbers link again, I see that it is $\left | \tfrac {x}{0} \right | = \infty$ that is defined, not $\tfrac {x}{0}$ . In any case, it seems the limits in that section are misplaced. Root⁴(one) 18:57, 20 June 2007 (UTC)

[edit] Too much speculation about the symbol's origin

If there is no documented evidence regarding the symbol's origin, then any explanation regarding it should be designated a folk explanation. For all that we know, he created it as a secret message to the freemasons and their alien masters. As with songs, I think that people tend to read too much into these simple characters. I'm a member of the "0 was already taken" party if you want to know the truth. Griff@66.229.255.90 21:19, 23 July 2007 (UTC)

[edit] Citations for "Mathematical infinity" section

The "Mathematical infinity" section has been tagged {{Unreferencedsection}} since June. It looks like the section discusses basic mathematics known for over a hundred years (except for the bit about nonstandard analysis), so what citations are necessary? — Loadmaster 16:47, 6 August 2007 (UTC)

I had wondered this very same thing myself. Perhaps a reference to a simple textbook on some of the mathematics involved would suffice? Surely these "required" references exist on other mathematics articles which deal with similar equations? — Metaprimer 05:55, 12 August 2007 (UTC)

[edit] Infinity in the Arts

I think we should start a (small) section "Infinity in the Arts", starting with a link to M.C.Escher. Obviously, we don't want this to grow without bound (pun intended), it should be just to show enough representative examples of artists who were known for their portrayals or use of infinity in their artwork. — Loadmaster 16:45, 7 August 2007 (UTC)

I started the "In the arts" section. Hopefully others will add some heft to it, especially links to other artists like Escher. — Loadmaster 17:10, 7 August 2007 (UTC)

I hope no-one will be offended that I removed the 'popular culture' section. The concept of infinity as used in popular culture probably deserves its own article, and can certainly be better exemplified than with a trading card and an album cover. --carelesshx _talk 04:09, 15 September 2007 (UTC)

[edit] Not defined or infinity

$\frac{1}{\frac{1}{9}+\frac{1}{9^2}+\frac{1}{9^3}+\frac{1}{9^4}\ldots}=\frac{1}{\sum_{n=1}^{\infty}\frac{1}{9^n}}=8$

$\frac{1}{\frac{1}{8}+\frac{1}{8^2}+\frac{1}{8^3}+\frac{1}{8^4}\ldots}=\frac{1}{\sum_{n=1}^{\infty}\frac{1}{8^n}}=7$

$\frac{1}{\frac{1}{7}+\frac{1}{7^2}+\frac{1}{7^3}+\frac{1}{7^4}\ldots}=\frac{1}{\sum_{n=1}^{\infty}\frac{1}{7^n}}=6$

$\frac{1}{\frac{1}{6}+\frac{1}{6^2}+\frac{1}{6^3}+\frac{1}{6^4}\ldots}=\frac{1}{\sum_{n=1}^{\infty}\frac{1}{6^n}}=5$

$\frac{1}{\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+\frac{1}{5^4}\ldots}=\frac{1}{\sum_{n=1}^{\infty}\frac{1}{5^n}}=4$

$\frac{1}{\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+\frac{1}{4^4}\ldots}=\frac{1}{\sum_{n=1}^{\infty}\frac{1}{4^n}}=3$

$\frac{1}{\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}\ldots}=\frac{1}{\sum_{n=1}^{\infty}\frac{1}{3^n}}=2$

$\frac{1}{\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}\ldots}=\frac{1}{\sum_{n=1}^{\infty}\frac{1}{2^n}}=1$

$\frac{1}{\frac{1}{1}+\frac{1}{1^2}+\frac{1}{1^3}+\frac{1}{1^4}\ldots}=\frac{1}{\sum_{n=1}^{\infty}\frac{1}{1^n}}=0$

The inverse of zero is ${\infty}$ User:Twentythreethousand 11:55, 18 September 2007

[edit] Omni?

I've called the ∞ lemniscate, the "omni" symbol, for over a decade and people usually seem to understand it through relation to the infinite nature of most words prefixed with "omni-" and to the dissimilarity of that grouping of sounds from any other English words. It just seems sensible to write ∞-scient, ∞-present, or ∞-potent. Anyhow, as it stands this is way too OR to put in an article, so I just wanted to know if others think this is a reasonable neologism or have seen it crop up anywhere before. Thecurran 01:54, 11 October 2007 (UTC)

I think you're maybe being influenced by Omni, a pretty good sci-fi mag from the eighties or so, that IIRC used to use the symbol prominently. --Trovatore (talk) 01:15, 9 December 2007 (UTC)

[edit] 1/0=Infinity

Infinity can never be defined, therefore shouldn't it be equal to 1/0. 1/0 is greater than any imaginable number, as well as infinity. 1/0 being greater than any other number can be shown if you start taking any real number and plugging it in for the zero. When this happen the answer to the algebraic equation gets greater as the denominator gets lower. This brings me to conclude that 0 when plugged into 1/x would be the largest number, infinity. Even in the graph 1/x, it can been seen that when x is equal to zero, it crosses the y-axis at a point that is undefined, though logically infinity. Therefore, through this equation is possible to postulate that infinity and negative infinity are one and the same. The reason for this is since the equation is a function, meaning it crosses the y-axis at one point, it cannot have two y-axis crossings, unless infinity and negative infinity are one and the same. —Preceding unsigned comment added by ARedens (talk • contribs) 05:43, 8 December 2007 (UTC)

Aredens, as far as I'm aware there is no such thing as "the largest number" in standard mathematical analysis. 1/0 is not equal to infinity. But the limit of 1/x as x approaches 0 is. In other words, infinity means "to grow without bound." Hoomank (talk) 23:32, 3 June 2008 (UTC)

Hoomank, a new sense to 1/0 = infinity was introduced by Martin Cooke in: "To Continue with Continuity" Metaphysica (2005) 6, pp. 91-109, http://www.metaphysica.de/texte/mp2005_2-Cooke.pdf. The cardinal number of points in a continuum is postulated to be equal to the length of a line of points divided by the length of a point = 1/0. Such a cardinal number (denoted by #) has the property that it is not a set-theoretical cardinal number (an aleph). So if the natural numbers form a potential infinity (as some supertasks might one day show) there could still be such an actual infinity as #. Or if Peirce was right, if the cardinality of the continuum was greater than any aleph, then there might be # points in a line (Peirce had the points being so many that they blurred into each other, so that they were and yet also were not there, as he thought the set-theoretic paradoxes indicated). Incidentally, although it is a reciprocal of the smallest number, # might not be the biggest number (even in this non-standard setting), since 2 to the power of # might be bigger (according to Cooke 2005, p. 103). Username12321 (talk) 17:12, 5 June 2008 (UTC)

[edit] "commonly represented as ∞"

On the recent exchange between FelipeS and Jwy: FelipeS is certainly correct that infinite ordinals and cardinals are not represented by the ∞ symbol. However, infinite ordinals and cardinals -- though certainly relevant enough to mention in this article -- are also not usually called "infinity". In fact when people speak of "infinity" in connection with ordinals, they usually mean something like iterating through all the ordinals, or something that's true for all ordinals -- compare infinity-Borel set. If I wanted to stretch a point I could find a philosophical undercurrent here, but the point might break if I stretched it that far, and I don't suppose any of us want broken points lying around where someone might step on them. --Trovatore (talk) 03:51, 15 January 2008 (UTC)

It depends on which people are speaking. People with some mathematical culture (and I don't just mean mathematicians, mind you) tend to treat the infinity of calculus (∞) as a mere uninteresting placeholder, barely worth a mention, and the infinite cardinals (aleph-0, c, etc.) as the real infinities. FilipeS (talk) 20:36, 15 January 2008 (UTC)

How interesting they are is not the point. They aren't called "infinity". They'll be called, for example, "infinite quantities", but not "infinity" unmodified. How would people know which ordinal or cardinal you meant, if you just said "infinity"? --Trovatore (talk) 20:46, 15 January 2008 (UTC)

And how do people know whether you mean positive infinity, negative infinity, or unsigned infinity, when you say "infinity"? When most people think of "infinity", they think of an infinite set (such as in the popular example of Hilbert's infinite hotel), not of limits. FilipeS (talk) 20:50, 15 January 2008 (UTC)

Infinite sets are called infinite sets, not infinity. When the term "infinity" is used in set theory (which is fairly rarely), it doesn't mean any particular ordinal or cardinal, nor does it mean the notion from calculus -- it most likely means Cantor's absolute infinite. And this is frequently represented by the ∞ symbol.

That's in mathematics done in English, of course; for all I know it might be different in Portuguese. --Trovatore (talk) 20:54, 15 January 2008 (UTC)

I think you are being unreasonably literal. People do not care about what is conventionally called "infinity" and what is not. They care about the concept of infinity. Of which there are several. Aleph-0 may not often be called "infinity", but it and its successors are certainly called infinite cardinals, sometimes even "infinite numbers".

The page should concern itself with the various notions of infinity in several disciplines, as it currently does. The infinite elements of R are but two ways to formalize infiniteness. FilipeS (talk) 21:02, 15 January 2008 (UTC)

[edit] Asaṃkhyeya

On the assumption that the two terms are synonymous and/or cognate, I linked "asaṃkhyāta" to Asaṃkhyeya. If someone knows better, by all means change it back. Lusanaherandraton (talk) 12:23, 4 May 2008 (UTC)

[edit] Infinity as a number

The opening section refers to infinity as a number but then goes on to say it is not like other numbers on the real line. I think this section is misleading. Except in some non-standard mathematical systems, infinity is not a number. The infinity symbol usually means "to grow without bound" as in lim x->0 1/x.

I would like the section that refers to infinity as a number removed.

Hoomank (talk) 23:40, 3 June 2008 (UTC)

Hmm. Strictly speaking it doesn't say infinity is a number, just that it's sometimes treated as one, which is certainly true. I would note that there is no uniform meaning of either "infinity" or "number", so the phrases infinity is a number and infinity is not a number are both fairly meaningless.

Just the same, I'm not convinced that the current wording, which says

In mathematics, "infinity" is often used in contexts where it is treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is a different type of "number" from the real numbers

is really all that useful or well-phrased; I wouldn't necessarily be opposed to a complete rewrite of the second paragraph of the lede. I don't think that just deleting the sentence is a good idea, as it would leave a one-sentence paragraph. --Trovatore (talk) 03:07, 4 June 2008 (UTC)

I'm not volunteering to formulate a better intro, but I think the lede ought to say that (also) in mathematics the terms "infinite" and "infinity" are used for a number of different concepts and have different uses and meanings in different contexts. --Lambiam 07:31, 4 June 2008 (UTC)