Talk:High School Mathematics Extensions/Set Theory and Infinite Processes

From Wikibooks, the open-content textbooks collection

Jump to: navigation, search

I'm going to be pedantic and whine about

"Infinity is unlike a normal number."

Well, no, it's not. In fact, it's not really a number at all. (Sometimes people treat it like a number -- as if \infty were just an unusual, special number, like π is an unusual number. But that leads to all kinds of confusion).

  • "The word "infinity" doesn't represent an actual number that is bigger than all the others; "infinite" just means "without end," and is a way of describing something that never comes to an end. There are infinitely many numbers, because there is no last number. And that's really all it means." http://mathforum.org/library/drmath/view/60400.html

If there's no objections, I'm going to change the article to say something like "Infinity is not number." and try to make it consistent with the "infinite process" idea.

--DavidCary 22:41, 29 Oct 2004 (UTC)

Contents

[edit] you have to face up to .03999... = .04000...

etc. in your proof that R is uncountable. One way is to toss out the things ending in all 9's. Then be careful in your construction of the new number.

~reader





[edit] examples of aleph-one, aleph-two, etc.

Could anybody please explain or (using the logic-style and/or ;) ) give an example of an aleph-two cardinality series? r3m0t 23:20, 1 Jan 2004 (UTC)

No not really. The set of all possible subsets (Called the power set) of a set is always bigger than that set. E.g. if A = {a,b,c} the P(A) = {{empty set}{a}{b}{c}{a,b}{a,c}{b,c}{a,b,c}}Even for infinite sets the cardinality of the power set is always higher than the set itself. I believe it's true that
P(\aleph^0) = \aleph^1

(I have to check on a book for the proof)

but I don't know if P(\aleph^1) = \aleph^2

Even so power sets aren't exactly easy to visualize. I think they are too obscure for this book. Theresa knott 13:31, 19 Jan 2004 (UTC)

Well, okay. By the way, isn't it usually written \aleph_0 and not \aleph^0? I have to go to lessons now (science in fact!) so bye... r3m0t 13:35, 19 Jan 2004 (UTC)
Yes - I dashed it off quick without thinking. Theresa knott 14:16, 19 Jan 2004 (UTC)

Here's some examples:

  • Aleph-zero: the set of positive integers -- countably infinite. (Any particular member of this set is a relatively harmless finite integer, like 2999).
  • Aleph-one: set of the real numbers between zero and one -- uncountable. Think of the binary expansion of a real number: 0.x1x2x3x4x5x6..., with a *countably infinite* number of bits (or decimal places, whatever). It takes a (countably) infinite amount of information just to describe a typical real number, a single member of the set. The set of all of them is larger.
  • Aleph-two: one example is the number of open curves of length 1. Imagine you're driving from the start to the end of one particular curve. Any particular point on the curve, you could be turning to the left or to the right. Since there's aleph-one particular points on the curve, it takes aleph-one bits of information just to describe one particular curve. The set of all of them is larger.

[edit] Direction

This chapter is already looking interesting. But I'm not sure what direction this chapter is taking. It seems to be more like a book then a textbook; more passive then interactive. I dont know enough about infinity, maybe that's why I dont see where the exercises come in. How do we involve the reader? Xiaodai 15:52, 13 Jan 2004 (UTC)

Don't worry, I'll be adding some exercises shortly.Theresa knott 16:12, 13 Jan 2004 (UTC)

Ooh! Let me join in in the big everybody-online-at-the-same-time party! Thing is, I got left behind when I started this... better catch up on some reading :) r3m0t 13:15, 19 Jan 2004 (UTC)

Me too, I'm coming to the end of what I can do off the top of my head or from memory. Perhaps you can start by reading what I've already written and making sure it all makes sense. Theresa knott 14:16, 19 Jan 2004 (UTC)

[edit] Page Needs to be split

I did a quick print preview and found that this module now runs to 10 pages. I still have more to write on infinite sets and transfinite numbers, so I propose that we split this page. What say you? Theresa knott

I say no. Other chapters are already collosal (primes, complex number...) and I think this one should follow. I certainly don't shy away from a long page without first reading a page at least. The reading style is easy and light here for that reason :) r3m0t 17:10, 19 Jan 2004 (UTC)

Whoops, sorry for that "fix", I see what happened now. Never mind :) r3m0t 20:17, 19 Jan 2004 (UTC)

That's OK It's part of what makes mathematics difficult. You have to be so precise with the language! one stray 1 in a subscript completely alters the meaning :-( Still with three people regualy checking this page those sort of errors should be picked up. Slightly more worrying is real errors that may have slipped in because of simplifying the formal proofs. "Proper proofs" are dead hard to read, but are very carfefully written to ensure nothing dodgy.Theresa knott 20:25, 19 Jan 2004 (UTC)

[edit] I don't understand something in the closed forms section

Copied from the page :So the closed form of

1 + x + x2 + x3 + ...

is

\frac{1}{1 - x}

We can equate them:

1 + x + x^2 + x^3 + ... = \frac{1}{1 - x}

We are definitely not saying the two expressions are equal in the numerical sense, for example if we substitute x = 2 into the equation we get on the right hand side -1, but we know the left hand side equals infinity. Instead we use the closed-forms for their nice algebraic properties.

It doesn't make sense to equal two things that are not equal. The series converges for |x| less than 1 so shouldn't we say something along these lines ? (I don't know anything about closed ofrms unfortunately) Theresa knott 14:21, 20 Jan 2004 (UTC)

This is tricky. They are not equal numerically but are equal algebraicly. Maybe things will clear up when the whole thing is written up. Xiaodai 14:29, 20 Jan 2004 (UTC)#

Okey dokey I'll wait until it's finished Theresa knott 14:32, 20 Jan 2004 (UTC)

[edit] Limits in relation to infinity

My original intention was to introduce the more rigorous definition of limits in relation to infinity in this chapter. The explanation of limits already present is intuitively very clear, but to be mean to the readers I would really want to see it expanded upon. I was thinking of covering (lim x -> L = inf) kind of thing as well. Diagrams would be nice. What do you think? Xiaodai 14:40, 20 Jan 2004 (UTC)

I think it's a good idea. Being mean is good as long as the section gets prgressively harder as it goes along so that the readers can choose to stop at a point and still get some good info up to that point. Theresa knott 15:03, 20 Jan 2004 (UTC)

[edit] Eh?

(Is very jetlagged, so veracity of comment may not be 100% ;) - didn't Godel go about showing that it really didn't matter whether you took the continuum to be aleph_1 or something "in between" or greater than aleph_1 or something, that basically things would work out either way? Should check later... Dysprosia 01:25, 27 Jan 2004 (UTC)

It rings a bell, I'll try to check too. Theresa knott 19:40, 27 Jan 2004 (UTC)
OK I've found a good website - here. Basically the continium hypothesis cannot be proved or disproved using the standard set of axioms of set theory. Godel proved that CH being true was consistant with standard set theory and Cohen prove that CH being false was also consistant. So it is not possible to prove CH without adding some new axioms. Theresa knott 13:18, 29 Jan 2004 (UTC)
Personally I think that it now looks very convoluted, perhaps it could just be removed? r3m0t (cont) (talk)
What does? Its transfinite arithmetic, it's supposed to be convoluted ;) Dysprosia 12:13, 30 Jan 2004 (UTC)
It's infinitely convoluted and nothing can be done about it for however long you work on it, except for just removing it. QED. r3m0t (cont) (talk) 12:27, 30 Jan 2004 (UTC)
Just removing what ? The question, the whole section on CH, the whole page ? Theresa knott 12:50, 30 Jan 2004 (UTC)
Whatever it is, it can't be that bad... Let us know what exactly you have an issue with and we may be able to sort it out... Dysprosia 23:10, 30 Jan 2004 (UTC)
I meant the question. No, it isn't that bad ;) r3m0t (cont) (talk) 04:58, 31 Jan 2004 (UTC)
OK I'll take the question out. Theresa knott 11:49, 2 Feb 2004 (UTC)

[edit] Need feed back

I have finished writing the first example in the Linear Recurrence Relations section. I think it can be improved upon, but i do not know how to go about doing it. That's why i need your feed back. Do you think it's easy enough to understand? But hard enough that they'll need to have a good think about it. If you dont already know the work, did your understanding of generating functions increase? So many questions i want to ask. I want to know what you guys think before I make the next example. Xiaodai 01:12, 5 Feb 2004 (UTC)

I've got lost with this whole module recently, but I'll try to read up on it soon. Wish me good luck, for I am soon to do the Maths Challenge! I'll try memorising a few puzzles and gbringing them over :) r3m0t (cont) (talk) 06:55, 5 Feb 2004 (UTC)

[edit] Question on "Are there even bigger infinities?"

One way is to do it is to take

   a1 = x1
   a2 = y1
   a3 = x2
   a4 = y2
   .
   .
   .

This defines a one to one correspondance between the points in the plane and the points in the line. (Actually, for the sharp amongst you, not quite one to one. Can you spot the problem and how to cure it?)

I've found the problem. This doesn't work for a number that "approach zero" -- A number that is as close to zero as possible but not exactly zero, i.e. 0.000000......(after infinite 0s) 001. For this number, we don't know if the "1" belongs to x or y, so it can be (0.000...01,0) or (0,0.000...01) But I've no ideas on how to cure this. Can anyone help?


To get to it, the problem is that you are looking at the plane R2 and not something like Z2. R is uncountable, Z is. Do you see the problem? R2 is not countable then. Dysprosia 13:34, 26 Oct 2004 (UTC)
"(after infinite 0s)" ? If we're still talking about real numbers, that would be identical to the real number zero. I don't see what the problem is. (I don't want to get into transfinite numbers just yet).

[edit] Natural number definition

Since set theory is being referenced early in the book, shouldn't we use the set-theory notion of Natural numbers, and call the first natural number 0? I know that in various parts of math and science, people begin the "natural numbers" at 1, but here it seems innappropriate. Siroxo 10:20, 15 Mar 2005 (UTC)

Good point. Xiaodai 8 July 2005 11:25 (UTC)

[edit] Why not add the 0.99999=1 proof

Why not add the interesting 0.99999... = 1. That kind of has stuff to do with infinity, and it is an interesting result.

Yes i think it shall be done when someone gets to it.Xiaodai 8 July 2005 11:25 (UTC)

[edit] zeno's paradox

there is something wrong, or i dont get the paradox, like its common sense the fast guy will overtake the turtle after some time, just think about it, try it with a friend - you run he walks, see if you can overtake him

That's why it's a paradox, because in the explaination your dont pass that guy but in reality you do. There is a contradiction, so it's a paradox. Xiaodai 8 July 2005 11:27 (UTC)
Just a note, the key is to realise that the timeframe that the explanation cover is limited. As the distance it descibe decreases indefinitely, the time it takes also decreases indefinitely. So the explanation didn't consider after that timeframe passed. --Lemontea 8 July 2005 14:19 (UTC)

opps, dunno what i was thinking. sorry for wasting your time

well i think that this "zeno's paradox" is pretty weird, but it's like the only thing on this entire page that i actually get, so :). by the way, can anyone give me a numerical example that demonstrates it? [morandi]

WOW, does ((infinty)^2+infinty) / (infinty)^2 = 1?

Kind of. But we shall expand on what we mean by infinity later on in the chapter to try and clear everything up. Xiaodai 8 July 2005 11:27 (UTC)

[edit] a kind freshman's question

well im a freshman at highschool and i had to look up this infinity concept for extra credit and your website has proven to be useful a bit confusing for me but i get the bigger concept but the one concept that has croosed my mind is that infinity cna be limited when you say x>1 and x<2. just in there the answers are infinite but yet they are limited. you can chose any number between 1 and 2 and the decimal number between 1 and 2 are infinite so you have an infinite number of solutions but yet you are still being limited how can that be? what type of infinity is that?

It is true that all the real numbers between 1 and 2 are quite small in magnitude (they are all less than 2). Also all the numbers fit into this line of no more than 1 unit in length. Yet there are infinitely many real numbers in there. It is strange, no doubt. I think you have already noted that "infinitely many" does not mean "very large". In set theory, we have this concept of "countably infinite" and "uncountably infinite". The set of natural number are "countable" in the sense that you can list them one by one in a meaningful order, and there are infinitely many natural numbers. So we call the natural numbers "countably infinite". All the real numbers between 1 and 2 are not "uncountably infinite". That's because you can not count them in a meaningful way. A mathematical way of saying that is: there is no one to one correspondence between the natural numbers and the real numbers. It's like the natural numbers and the real numbers come from two type of infinity, one countable, the other not. Xiaodai 23:19, 7 January 2006 (UTC)

Hi, the point here is not so much concerning the uncountability (viz. just take the rational numbers between 1 and 2, they are countably infinite, and yet your kind of doubt remains to hold). The point here is about the two notions of numbers, one being those which count the elements in a set (here that can be an infinity), and the other, being those on the number line. In this case the infinity is the furthest imaginary point on the right end. If you keep these two interpretations as different in your mind, the problem disappears! Really, the size of the set of rational/real numbers between 1 and 2 is infinite, however, the numerical stickers on them which place them on the number line are bounded (or as you say, "limited"); hence, there is no mathematical inconsistency here in deed. So far as the size of the set is concerned, the numerical identification of its elements does not matter much as you can easily see that the number of points on any two line segments are the same (take a triangle ABC, let R and S be the midpoints of AB and AC resp. for any point P on BC, the line AP cuts the line RS at a unique point Q, giving a one-to-one correspondence between the points on BC, and those on RS -- although the lengths are different). Thus you can keep stretching one of these segments to have bigger and bigger limits, and the number of points on them do not change! In fact you can stretch it all the way to infinity (the "infinity" point on the number line). Try the function tan(π(x − 1) / 2) on the numbers between 1 and 2 for instance. Hope this helps. Suman

[edit] Answers to Exercise

Where can I find answers to exercises?

sorry no exercise answers as of yet. Xiaodai 02:01, 15 June 2006 (UTC)

[edit] Suggest a Title Change

It seems like the article doesn't really deal with set theory in more than a basic way. Since the bulk of material is concerned with infinity, maybe the title should just be something like "Infinity". Well, maybe not so vague, but at least strip the "Set Theory" from the title, as there is no real discussion of it. --Darkxxxxillusion

Thanks. Your observation is correct, but the module will be rewritten to incorporate set theory. That is the plan. Xiaodai 16:51, 17 July 2006 (UTC)

[edit] Errata

In "Limits Infinity got rid of, Examples", it is correctly said, that "Sin(x)is a function that you should already be familiar with (or you soon will be) its value oscillates between 1 and -1 depending on x. This means that the absolute value of sin(x) (the value ignoring the plus or minus sign) is always less than or equal to 1:", but then the formula states "|x|<=1". There's no reason for x to behave so: shouldn't that read "|sin(x)|<=1"?

Gulliveig 18:16, 8 January 2007 (UTC)

Retrieved from "http://en.wikibooks.org/wiki/Talk:High_School_Mathematics_Extensions/Set_Theory_and_Infinite_Processes"