How or in what ways I (or humans) cannot comprehend the concept of infinity? Like, I don't have any sort of difficulty in thinking about the universe having no beginning or end and having no dimensional bounds (i.e. I think you could theoretically move in one direction indefinitely and never repeat or hit any sort of bound.) I can't think of any way that infinity makes my mind explode or something...

... because it's not the definition or concept of infinity, but the actual use of it.
You can not possibly perceive something that has any kind of infinite quality.

You can say infinity and your head won't explode. If you try to actually completely understand it, you'll fail because it's.. well.. infinite. Basically, what IP said.

It's especially difficult when you start talking about different degrees of infinity and realize that while ₁∫^∞ 1/x is infinite, ₁∫^∞ 1/x² is finite, even though both of them consider the area under a curve that is greater than zero for an infinite distance in the positive x direction, but approaching zero as x goes to infinity. Can you picture that?

I've thought of three things related to infinity that might be puzzling to someone
- 0.999... = 1
- The concept of infinitely small values.
- The way you can compare infinite cardinal numbers.

What does a repeating decimal have to do with infinity??

Well, it's repeating indefinitely. Just a loose connection. The other two are more relevant.

Or you can't think of it at all in the first place that prevents your mind from asploding.

[edit]It seems the answers I get about "what can't I comprehend about infinity" are missing the actual 'what' that I am looking for... Is that itself what cannot be comprehended?[/edit]

Completely understand how? In the topic where the beginning/end of the universe was discussed where I first wondered about this, I was thinking about going forward and backward in the universe's history... Nothing in my mind was telling me that it can't be possible or able to imagine that before the big bang (or 'beginning') there was just a super-dense amount of matter that grew unstable or something. Maybe that came about from several other expanding areas of matter that started growing into each other, thus having portions being pulled together by gravity, which brings about the existence of these super-dense masses, which makes the process happen even more and multiplies the amount going on in my imagination... And then I can just imagine the reverse going indefinitely forward.
Most of that is probably based on my lack of understanding about the interaction of matter, but that isn't the subject of my question. If I can't imagine it being forever, am I just defining it as infinite (but not just indefinite) when I think about this?

There is an infinite possible number of values between x and x+1, maybe.. (Though I don't understand why someone would say one infinity (that you can always add 1) is greater than another (infinite numbers between 0 and 1)... If it is truly infinite then there is no value and thus you cannot compare it... I think my math teacher last year was dumb. I also think those 'types' are only distinct by definition... Numbers are completely arbitrary.[/from something a teacher once told me])

If the atoms have so much empty space, and our universe has so much empty space... I wonder if all the stars and matter make up something greater. Just like I've always wondered if within the nucleus of an atom is another amount of space... How can I not imagine this keep recurring? Who says that when you draw or observe a Sierpinski triangle you're not just looking at a little leg of another larger one that is, too, somewhere within in an infinitely large fractal?
Humans have the horrible habit of defining a specific frame of reference... Is the edges of this 'frame' what gives infinity bounds?

Post has been edited 3 time(s), last time on Jan 11 2009, 8:37 pm by FaRTy1billion.

http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
Learn it, love it.

What exactly is that trying to say? I just see a bunch of arbitrary strings of 1s and 0s, then taking a line of them and inverting them? Is this saying that the result at the bottom will always be unique because comparing each digit to a corresponding digit in the sets at least one will always be different? And how does it relate to my ramblings about something a teacher told me?
(I'm having trouble reading articles right now, by the way. I might go through that later when it is possible for me to be more attentive)

Post has been edited 1 time(s), last time on Jan 11 2009, 9:38 pm by FaRTy1billion.

Infinity is really a weird thing. I had nightmares about it when I was younger and it is really something that is scary. I hate thinking about it. I've had nightmares when I was sleep deprived about me holding up planets forever and it I kept thinking if growing numbers. How would it not make your mind explode?

I'll try to give you an explanation. Consider two sets, one that contains all integer numbers and one that contains all real numbers between 0 and 1. We can both see that the sets' size is infinite as there are an infinite number of integers and numbers between 0 and 1. Well, how about the size of these infinite sets (answering your question)?

Before I talk about the comparison, we'll need to define a way to show if a set is larger than another set even if both have infinite elements. Let's consider the set of all integers and the set of only even numbers. It's obvious that the set of even numbers is a subset of the set of all integers. But what about the size? This is where it gets a little weird: we define two sets to be equal if we can show there exists a bijection between the two sets. From this definition, we can see that the size of the set of integers is actually the same as the set of even numbers. Proof: f(x) = 2x. For any x in the set of integers, we have 2x which is an element in the set of even numbers. We also know f(x) is a bijection, but that proof is trivial.

Now let's go to the real numbers. Since finding a bijection relation between the set of integers to the set of reals between 0 and 1 is very difficult (actually impossible, but we don't know that yet ), we can do the next best thing: just assume one exists and see if we can't find a contradiction. I define some arbitrary f(x) as follows:
f(-1) = .1002234...
f(0) = .0000000...
f(1) = .1234567...
f(2) = .5098234...
f(3) = .1415926...
f(4) = .2398474...
f(5) = .7172890
...
So we've created a bijection right? The left side has every single integer and the right side has every single number between 0 and 1. This is the cool part: by creating this so-called "bijection", we've missed some real numbers (actually, an infinite number of them) on the right side. We create a new number between 0 and 1 that is guaranteed to not be in the above bijection: the first digit cannot be a 1 (ie, 5), the second digit cannot be a 0 (ie, 2), the third digit cannot be a 3 (ie, 8). We continue this and it is guaranteed that this number is not in the above right side since it differs in at least one spot. In this example, our number could be something like: .5281491...

So now we've found a contradiction which proves that the set of of reals between 0 and 1 is larger than the set of integers. D;

What exactly is a bijection? Wikipedia just showed a ton of arrows pointing to random letters.
So you are associating arbitrary values with integers? If there is no real function, what is stopping me from putting this new value as the next set?
Currently, this is still looking like greater just by definition, and not the actual quantity of values. Where can I learn more maths to know more of what I am talking about..?

Post has been edited 3 time(s), last time on Jan 12 2009, 1:11 am by FaRTy1billion.

A bijection is simply the relation between the domain and co-domain with the property that for any value in the domain, there is exactly one number in the co-domain. Unlike y=x^2, which has -2 and 2 going to 4. A bijection function could be y=x.

You can't just place a new value as the next number because whatever position that number is in, there is a new number that differs in that digit's place. Basically, the very act of trying to create a bijection destroys the bijection.

It doesn't look like "greater by definition" because of some strange bijections that do exist. For example, there is a bijection relation between the set of integers and the set of rational numbers.

Okay, I'm gonna try to write some set theory for dummies

Lets see... what mathematicians wanted was a method to compare the sizes of different sets. Lets say you got the set A = { a, b, c } and the set B = { apple, orange, pear }. Both have three elements so both are the same size. But how could you decide whether they are the same size or not without using numbers? Comparing then one to one:

a - apple
b - orange
c - pear

Same size! The same can be applied to infinity sets. Let N = { 1, 2, 3, 4, 5, ... } be natural numbers set and 2N = { 2, 4, 6, 8, ... } be the odd numbers set. You can also compare them one to one:

1 - 2
2 - 4
3 - 6
4 - 8
...
n - 2n
...

Every number in N appears exactly once on the left. The same for every number in 2N. We've compared them one to one: they're the same size!
It's not hard to notice that if the elements of a set can be writen in an infinite list then it has the same size N does. All you have to do is pair the first one with 1, the second one with 2 and so on and you'll have a biunivoc correspondence between those two sets.
When trying to list all the real numbers inbetween 0 and 1 however arises a problem: lets suppose we have all those numbers in an infinite list:

0.100000...
0.200000...
0.300000...
...
0.a1a2a3a4a5...
...

Then lets build the number b = 0.b1b2b3b4b5... such as that a1 differs from the first decimal digit of the first number on the list, a2 differs from the second decimal digit of the second number on the list, and so on. b is then different from the first number of the list, but it is also different from the second, from the third... So b is not on the list and thus for every possible list we can find another number which the list lacks: there's no list able to contain all the numbers inbetween 0 and 1 and thus the interval (0, 1) and N have different sizes despite being both infinite.

EDIT: Btw, a bijective function is just that: a one to one correspondence between two sets.

Clokr, a bijection is both one to one and onto. Otherwise, the set of reals and set of integers would be the same size. D;

..? So trying to do this destroys itself..? I don't get it.

Also, for every number that is "skipped" in the set of non-integer reals, you can still go an equal amount toward infinity in the set of integers. Both have no bounds, one is just in a fractional base.

To kibitz, I have always thought it's incredibly cool that there exists a bijection between ℝ and ℝ² (or ℝⁿ for that matter).

But back to Farty's original question, these matters in number theory show that infinity is a bit too much to simply comprehend without a formal framework of theory and definitions.

The formal definitions for anyone not understanding what is sure to follow (superscripts denote footnotes to explain general math shorthand, not exponents):

Take some function f : X → Y ¹

f is injective (an injection or one to one) iff² f(a) = f(b) ⇒ a = b ³ ∀ a,b ∈ X. ⁴

f is surjective (a surjection or onto) iff ∀ y ∈ Y ∃ x ∈ X s.t. f(x) = y. ⁵

f is bijective (a bijection) iff f is both surjective and injective. This can be proven to be equivalent to saying f is bijective iff ∀ y ∈ Y ∃ x ∈ X unique s.t. f(x) = y.

¹ This notation means a function f mapping elements of X (the domain) to elements of Y (the codomain).
² Shorthand for 'if and only if'
³ i.e. f(a) = f(b) implies a = b. (⇒ is a one-way implication)
⁴ Shorthand for 'for all a and b in X' (∀ 'for all', ∈ 'an element of')
⁵ Shorthand for 'for all y in Y, there exists an x in X such that f(x) = y' (∃ 'there exists', s.t. 'such that')

Post has been edited 1 time(s), last time on Jan 13 2009, 4:40 am by DT_Battlekruser.

I also don't see why infinity is so closely related to numbers... Infinity doesn't seem to be a number as you cannot count to it or measure it. Though I suppose it is a quantity..


If X is an integer and if A = a value representing the number of every possible real (or even real+imaginary) value in the range of X to X+1, what is preventing me from doing X + A? Unless you add some sort of bounds to the range of numbers, both will always be able to be added together making them both the same infinite...

Post has been edited 2 time(s), last time on Jan 13 2009, 5:44 am by FaRTy1billion.