Talk:Infinity/Archive1

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Crack[edit]

I'll take a crack at expanding this over the weekend. - User $\scriptstyle (6\sum _{n=1}^{\infty }n^{-2})^{1/2}$ 22:20, 8 April 2009 (EDT)

I renew the promise, this is the very next thing I will work on. - π 02:07, 29 June 2009 (UTC)

What about adding "misconceptions of infinity"? The stuff you reverted could go well under that header and it'd be more RW-like than just trying to get an accurate definition, which we usually go to wikipedia for anyway. I mean, infinity is a complicated thing in maths so oversimplifications and people throwing the term around without grasping what it means could be interesting to look at. ^narchist 02:13, 29 June 2009 (UTC)

That is not a bad idea. My favourite rule of thumb, at least in calculus, is that if you can't rewrite the sentence without using the word infinity you are using it wrong. - π 02:27, 29 June 2009 (UTC)

The divide by 0 thing is true though. I remember talking about it with someone in AP Calc back in high school. The graph lines in a hyperbola go on forever, or until the calculator can't handle anymore. That x=0 limit is indeed negative and positive ~~infaggotry~~ infinity. Kettle o' fish 02:17, 29 June 2009 (UTC)

There are ways of finding integrals of undefined objects, have a look at Gabriel's Horn. - π 02:24, 29 June 2009 (UTC)

The one I know is "mathematically undefined" it's not as simple as "infinity" because you first need to define which infinity (some of which are bigger than others, although still infinite). When you talk about infinity you're either talking about a process - which is the commonly accepted, albiet slightly out of date, way of dealing with it - or possibly cardinality which is numbers like Aleph-zero.

^narchist 02:20, 29 June 2009 (UTC)

I think we should have an short, snarky but above all accurate article on infinity and its many associated concepts. Cardinality and limits are probably the most important two. Also we could have some fun with the Hilbert hotel example. - π 02:24, 29 June 2009 (UTC)

But how can you describe the Hilbert hotel better than it is in Science of Discworld? That's a great coverage of the subject.

^narchist 02:33, 29 June 2009 (UTC)

Universe[edit]

Isn't there a slight discrpency on the actual universe and the observable universe. I'm not as hot on cutting edge cosmology. ^narchist 18:03, 17 August 2009 (UTC)

I would guess that the "observable universe" is finite by definition. My impression was that the rest of it was finite as well, but I'm also no astrophysicist.--Bob ^{Not Jim} 18:09, 17 August 2009 (UTC)

(from interested inorance) Isn't it finite in 3 dimensions (you'll supposedly end up back where you started if you go far enough in a straight line), infinite in 4 dimensions (spacetime) and absolutely mad in any other number of dimensions? ^{I am eating}

_& honey^chat 18:19, 17 August 2009 (UTC)

The universe is a big, flat, finite things, which is a bit lumpy in places. That help? - π 13:26, 7 September 2009 (UTC)

Depends what you mean by finite. Unless my geometry's failing me, I don't think flatness and finiteness are really compatible. Doesn't "finite" mean that the (spacetime) manifold is closed (or at least compact), and flat mean that it's non-compact? And yeah, that may sound pretentious, but a lot of this stuff doesn't translate too easily into natural language.--Star trooper man 19:50, 6 October 2009 (UTC)

I think it is bounded but open. - π 10:20, 30 October 2009 (UTC)

Those two are incompatible, at least in my understanding of the words from geometry.--Star trooper man 12:11, 30 October 2009 (UTC)

It is actually a rather thorny question. Inflation theory states that the universe expanded rapidly, perhaps faster than the speed of light, so the universe we see is limited by the distance light has traveled. Hence it appears finite but boundless as the further light travels, the seemingly greater the universe appears to be expanding. Interesting things like this - sort of thing, dubbed "dark flow" suggests that there are further reaches beyond which our observation allows. Ace ^McWicked 11:15, 30 October 2009 (UTC)

Sometimes I wish I'd studied GR back when I was an undergrad...--Star trooper man 12:11, 30 October 2009 (UTC)

Incompleteness[edit]

I like the article, but I don't think the statement "Unfortunately, owing to Godel's incompleteness theorem, this question is unanswerable." is really true. Incompleteness shows that statements like this exist. It's a separate result of Cohen (which does not require incompleteness) that the continuum hypothesis is actually one of these statements (in the sense that it's independent of ZFC). --Pyfgcr 17:19, 30 October 2009 (UTC)

I think you're right, and I'll have a look at that at some stage, unless a real mathematician comes along first...--Star trooper man 17:31, 30 October 2009 (UTC)

Delete (or at least reform)[edit]

This article needs to be deleted, or at least completely re-written from a different perspective. It is inconsistent with the statement of purpose at RationalWiki. Moreover, it is unfocused and simply wrong. It contains several mathematical errors (such as the claim that the Continuum Hypothesis follows from Godel's Incompleteness Theorem). Rationalwiki is not an encyclopedia. Wikipedia and other sites do a fairly good job of explaining mathematical and scientific ideas. See here for Wikipedia's Infinity article. It's not perfect, but it hits most of the main points. Note how many auxiliary articles are needed to explain all of the mathematical concepts surrounding "infinity." Is that really a project we want to take on?

What this article should be is a listing of all the fallacious appeals to infinity by crackpots and the misuse of the mathematical idea of infinity for ideological reasons. Something along these lines.-Antifly 20:28, 30 October 2009 (UTC)

Here, here! I suppose

. But it's interesting & has taken me to things that I never knew existed. Since reading it I've been all over t'web looking at infinity. ^{I am eating}

_& honey^chat 20:42, 30 October 2009 (UTC)

I agree in the sense that we needn't go in depth with the concept of infinity, but more toward revealing misuse by those who don't understand. We should, however, at least leave some background information about the actual concept.

By the way, welcome back! Aboriginal Noise ^{Theist, barely hanging by a nail} 20:47, 30 October 2009 (UTC)

RW might not be an encyclopedia, but pop-science and pop-maths articles to act as a background to the more on mission stuff is essential. Especially as Wikipedia has a very dry and sometimes very inacessible style. I say that there is no reason at all to delete this. If there are errors, correct them.

^narchist 20:51, 30 October 2009 (UTC)

Knew there was a reason for keeping it. ^{I am eating}

_& honey^chat 20:54, 30 October 2009 (UTC)

True, true. Perhaps expanding the "Misapplications" section could add that much needed snark? Aboriginal Noise ^{Theist, barely hanging by a nail} 20:55, 30 October 2009 (UTC)

Let's take that little section at the end and move it to the top (and pump it up a bit), since it's really the "RW Article", but leave all the technical stuff below it? ħuman

20:56, 30 October 2009 (UTC)

I have made a start at re-writing this "article." Feel free to jump right in! I don't agree with Human's suggestion. The article is called "Infinity" for better or for worse, and should begin with an explanation of infinity. We could re-name it "Abuse of Infinity" an re-order the sections, or create a new article "Abuse of Infinity" that links here. Also, the misuse section is in dire need of improvement. Surely we can do better than the examples currently given...-Antifly 05:17, 31 October 2009 (UTC)

Also, I know very little about the use of infinity in physics, philosophy and theology. Perhaps our resident experts (or at least journeymen) can fill those sections out.-Antifly 05:19, 31 October 2009 (UTC)

I still think the "abuse" section should be prominent, before all the math. The lead should do a decent "math tag" free description, then the abuses, then the delightful geek stuff. ħuman

06:07, 31 October 2009 (UTC)

OK, I just looked at it again, and this article is far worse than it was 12 hours ago. Then, it was actually pretty good, and possibly even understandable and interesting to the layman. Now it's just gibberish, concepts introduced out of order and poorly explained, etc. I am sorely tempted to revert back to a readable version. Oh, and I cut a bunch of empty headers, I would have pasted them here but I lost them. Empty headers are for sandboxes. Antifly, you have wrecked was was on the way to being a decent, readable article. It is now once again full of undefined, unclear, and unnecessary math symbols, and flows like a glassful of broken razor blades. I'll leave it alone for now, but I am sorely tempted to revert back to before this change, which is when it became undecipherable. Perhaps we should discuss phrasing and major changes here? Maybe? PS, I disagree with your arguments about the title versus the content. It's called "infinity", but it is still an RW article, not a definition of the term in encyclopedic language. ħuman

06:16, 31 October 2009 (UTC)

I agree with Human. I've not worked through all the revisions and my comments refer to the earlier "mathematical" versions of the article. We can arbitrarily divide the world into two types of people. Those who understand mathematics and mathematical symbols and those who don't. In terms of our mission I suspect that those who understand maths are the least likely to be taken in by woo and pseudoscience. Those who don't understand mathematical symbols won't read anything that contains them. Consequently:

Those who understand maths don't need it.
Those who don't understand maths won't read it.

What we need then is a symbol-free version which talks about how infinity is misused in pseudoscience. If infinity is not used in woo and pseudoscience then we don't need the article in any form.--Bob ^{Not Jim} 07:51, 31 October 2009 (UTC)[http://tap3x.net/EMBTI/j8gonsowski.html

I slightly agree, but maths is used in woo extensively, the question is of how notable woo has to be for us to mention it. This is pretty bad, but I'm not sure it counts as notable. Perhaps we need some notability guidlines and/or a blanket policy for dealing with maths articles? What do you guys think.--Star trooper man 07:58, 31 October 2009 (UTC)

Sorry for wrecking the article. I may have jumped the gun, so I reverted back to the old version. But to say that the article was better before really gets to the heart of why mathematics is abused by cranks. Sure the article was easier to read before. That's mostly because it was wrong. In order to make it correct you have to add math (you know undefined, unclear, unnecessary symbols). That was exactly my point in beginning this section. Look at the Wikipedia article, look at how many auxiliary pages are needed to explain all of the jargon. That's why we shouldn't be writing encyclopedia articles.- Antifly^{Now with 50% less retirement!} 15:10, 31 October 2009 (UTC)

It's a good point tbh. Whilst I don't think the article before (or in it's current state) is terrible, it's not great either. And yes, to whomever removed it, the stuff I put in about Godel was misleading, so sorry. I think most of my other changes were positive though.--Star trooper man 16:07, 31 October 2009 (UTC)

I'm starting to wonder if we need two different articles.--Bob ^{Not Jim} 18:16, 31 October 2009 (UTC)

I've scribbled some notes on a piece of paper I may try to use to fix this up a little without removing anything. Hopefully motivation will strike shortly... ħuman

21:47, 31 October 2009 (UTC)

Problems With the Article[edit]

Here is a list of problems with this article:

I largely agree, I've added a few remarks.

1. The first sentence is completely inaccurate. Infinity is not a number and certainly isn't "bigger than that." There are many different infinite objects in different areas of mathematics. The only objects I can think of that could reasonably be called "infinity" are the elements $+\infty ,-\infty$ in the extended real line. But to suggest that infinity only arises in this context, or in the context of numbers in general is wrong. If we want to explain "infinity" in mathematics we need (at least) a discussion of: cardinals, ordinals, and convergence/divergence.

I agree.

2. No mention is made of the idea of infinity outside mathematics.

True, though I'm not so convinced that's a problem.

3. The fact that it is "still rejected by some people" needs a cite and further explanation (I don't really know who is rejecting what).

That is a problem indeed.

4. I'm not a physicist, and could be wrong here. Have we really shown that time and space are finite?

I'm not convinced that we have, though I'm no expert in GR.

5. Second paragraph, first two sentences: What?

Doesn't seem too bad.

Seeing as there is no such thing as infinity, starting the paragraph by purporting to explain what it is, is silly. Especially considering that no definition is ever given. Then we claim to define infinity as the opposite of finite, which it isn't. The opposite of finite is infinite.

6. Why does the article, from the 2nd paragraph onward, only focus on infinite cardinals?

Good point.

7. Second paragraph, last three sentences: This is blatantly wrong. The idea of an uncountable set has been conflated with the idea of a dense order which has, rather bizarrely, been conflated with the ability to list things.

Can you explain what's wrong with it? I thought that was how we defined an uncountable set - if an injection from a set to the natural numbers is possible, the set is countable. Otherwise, not.

What you just said is true. What the article says is not true. An uncountable set is a set that is not countable. A set is countable if there is an injection from the set to the natural numbers. (As you rightly point out).

What the article says is that an uncountable set is a set that is impossible to list, "there is no "next" real number, so they can't be listed." Unfortunately, the concept of "next" is outside set theory (which is where we are operating). In order to talk about "next" you need to be talking about a total order (or a partial order with a suitably modified definition of "next"). It is true that if we take the standard ordering of the real numbers, a real number x has no successor. But that is also true of the standard ordering of the rational numbers, a countable set (simply note that if x and y are rational, there is a rational number z between them). This property is called density and is a property of total orders, or topological spaces, but not of sets per se. Also, the inability to "list" a set is ambiguous. If by "list" you mean construct an injection to the natural numbers, then inability to list is equivalent to being uncountable, but only because you have replaced the word "countable" with "list" (it simply muddies the explanation). Once you explain what is meant by "list" we can resume the discussion, but you may want to take a closer look at transfinite ordinals first.

8. The "Now it gets tricky" section is pretty good for a six sentence introduction to convergence. But it is unclear how it is supposed to relate to the material before or after it, or to infinity in general.

It doesn't, scrap.

9. The "Counting" section could be presented better, but again is pretty good. It is again unclear why we have skipped from cardinality to convergence and back to cardinality.

Hmmm...

10. The "Now Counting gets tricky" section should be deleted. It consists of jargon, and a fairly poor illustration of why there are as many non-negative rational numbers as there are natural numbers.

Probably, or at least needs to be clearer.

11. "The paradox rests ...": There is no paradox. Infinite mathematical objects can have counter-intuitive properties.

I disagree that it isn't a paradox. A paradox is when a language is stretched outside of the domain to which it is usually applicable. Ideas of size for finite objects are clearly outside of their proper domain when stretched to infinite sets.

I suppose you are technically right. But I don't like using the word "paradox" while explaining math to a lay audience, as the alternate definition of paradox implies the existence of a logical contradiction. This leads people like Andy-pants to reject the Axiom of Choice because of the Banach-Tarski Paradox, without trying to understand what either actually means. Counter-intuitive is a better "word" for what we are talking about.

12. The "Cardinality, the reals, and the continuum hypothesis" is entirely jargon.

Yep, needs rewriting.

13. The final sentence in that section is still wrong. The idea of independence has been conflated with consistency. All that can be said is that the continuum hypothesis is independent of the ZFC axioms. That means that there is no proof of the continuum hypothesis assuming only the ZFC axioms. It says nothing about whether or not ZFC is consistent, nor about whether or not ZFC+(continuum hypothesis) is consistent.

Agree.--Star trooper man 05:52, 1 November 2009 (UTC)

I maintain that this article needs a complete rewrite, but I am happy to discuss fixes to these problems here.-- Antifly^{Now with 50% less retirement!} 23:45, 31 October 2009 (UTC)

Apparently I didn't do my divinely inspired version yet (which may, of course, end up reverted without comment), but I have a germ of an idea of how to proceed here. Unless someone else beats me to it, of course ;) ħuman

00:12, 1 November 2009 (UTC)

Fodder![edit]

This is probably the type of thing we are looking for. I will take a look when I have a chance. You probably need to go to a university (or even better be a student) to access the file.-- Antifly^{Now with 50% less retirement!} 00:46, 2 November 2009 (UTC)

Oh you can't read it. Do you want me to send you a copy? - π 02:17, 2 November 2009 (UTC)

I can read it, thanks. I was just warning non-student types that they may not be able to access it...-- Antifly^{Now with 50% less retirement!} 02:19, 2 November 2009 (UTC)

I agree. This crap and the "misuses" needs to be the first header, IMO. Then we do fun maths below that. ħuman

02:02, 2 November 2009 (UTC)

You actual need the maths (particularly the cardinality stuff) to understand what is wrong with the article. - π 02:07, 2 November 2009 (UTC)

Yes, but it's ok for the explanation to be below the stupidity. Articles need not be read in a linear fashion? ħuman

02:11, 2 November 2009 (UTC)

True, but it is a lot harder to read, as you can not go back to the spot you were reading after a detour. - π 02:14, 2 November 2009 (UTC)

How hard is it to scroll back up? Or implement inter-article links? Really? How hard is it to write "for more mathy explorations, see below" and "back to the snark", with links? ħuman

02:33, 2 November 2009 (UTC)

You are assuming everyone stops reading and scrolls down at the same place. - π 02:56, 2 November 2009 (UTC)

No, I am positing that we can use "anchor" tags that send people back and forth. ħuman

05:20, 2 November 2009 (UTC)

Density and non-discreteness[edit]

As mentioned above, countability is being conflated with "density". However, unlike what is pointed out above, density is the wrong word. Density applies to subsets; the appropriate term is, I believe, "non-discrete". ^{— Unsigned, by: Imarcuson / talk / contribs} Now signed by Imarcuson (talk) 06:39, 14 May 2010 (UTC)

Talk:Infinity/Archive1

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