There are an infinite number of odd numbers, as there are even. Line up an infinite number of positive integers (2x) vs. odd numbers (1x), and there is an obvious difference in the sizes of infinity. Can infinity be measured against itself?
Well, if I'm understanding what you are saying, infinity is infinity no matter what it is multiplied by. So, in your example, 1 x infinity, or 2 x infinity, are the same thing. Now let me ask you this. In mathematical terms, if we are going to look at things in terms of being "part" of the "whole," would it/we not be the top half of a fraction? The bottom half of the fraction would be infinity, since infinity is all, and we are looking at our relationship to all. What is 1/infinity? What is 9999999999999999999999999999999999/infinity? Is there any diffference? No. Therefore, in the grand scheme of things, all things are the same thing, in terms of all, or infinity.
it is an immeasurable (unlimited) quantity or extent. you can assume it means more (or less) than any number you can imagine.
There were a couple of infinity threads brought up after I had posted this, and that Cantor guy was the exact reason I posted this thread. He's got this thing called set theory or something, where basically you can have specific sets of infinity, and one can be larger than the other. The most obvious example was the one given in my original post: If you take an infinite number of odd numbers (or even), and compare it to an infinite number of positive AND odd numbers, then there is a set difference. There's a set of odd numbers (1x) and the set of odd and even (2x). There are people that are preaching how certain infinities are bigger than others. It makes sense, but this is like the only example I can even think of where one infinity holds more potential than the other. I guess I'm just looking for more discussion on this if anybody is interested in doing so. Edit: Reading over, Postal made great points
People really suck at trying to grasp infinity. I think most people look at infinity as if it's going "1,2,3,4,5,6,7,8,9" forever, as if it's constantly expanding. However for something to expand constantly it is not infinity. Infinity does not expand, it merely is already. It's not a balloon that has a constant supply of air being pumped into it because that would still be finite EXPANDING into infinity, not infinity itself.
Think of infinity as the apparent boundary of reality. Yet it cannot be a "boundary" in the typical sense, for if it were it would not be infinity.(what comes after that boundary?) Therefore, there is no set "thing" infinity, yet all "things" must be said to be on an equal standing in terms of there relationship to infinity, since infinity knows no bounds yet must be found within all bounds.
One of my teachers back in high school actually gave a lecture about infinity and the information is a little hazy but I'll try to explain what I can. (Warning: long post) There are indeed different sized infinities. In set theory two sets are the same size if there exists a functional relationship between the sets that is one-to-one (any value plugged into the function puts out only one value). So if you can create a one-to-one function that encompasses all elements in both sets they are the same size. If no function exists that can encompass all elements in one of the sets then they are different sizes. Let's demonstrate this with the infinite set of natural (counting) numbers (1,2,3,4,5...) and the infinite set of positive integers (0,1,2,3,4,5...). The positive integer set has one more term, but are they same size? Let's see. This one is fairly clear to see a function for. It's simply f(x)= x-1. Every term in the natural number set is matched with exactly one term in the positive integer set, and no positive integers will be left out, so these infinities are the same size. Now let's do the same thing and see if the infinite set of natural numbers is the same size as the infinite set of all integers. To help us see a function we're going to visualize the sets like this: The left column will be the natural numbers, and the right will be the integers, and I have them listed the way they are for a reason as you'll see. 1 0 2 -1 3 1 4 -2 5 2 6 -3 7 3 This one is not nearly as apparent how you could create a function, but you can thanks to something called a piecewise function. A piecewise function is one which treats different types of x values different ways based on certain parameters which you can set. For this example, the right column will be our x value and our function will be based on whether x is positive or negative. So we get a function like this: If x is positive, f(x) = 2x + 1 If x is negative, f(x) = -2x As you can see, every x value (from the right hand column) will match up with exactly one value from the left hand column and therefore these sets are the same size. This can take some real work to comprehend because while there are terms that do not exist in the natural numbers that do exist in the integers, they are still the same size. This is where things get hazy. You can try to do the same thing to see if the natural numbers and real numbers are the same, but things get more difficult here. In short, these sets are not the same size as there exist more terms in the real number set than can be produced by a one to one function with the natural numbers. This is proved by Cantor's Diagonal Argument, which you can feel free to google and attempt to comprehend. Finally, there is actually a way to prove that an infinite number of different sized infinities exist, but this proof is one that I've forgotten almost entirely and it involves power sets and takes a good bit of explaining. I have to leave to go somewhere but maybe tonight I'll work out how to explain Cantor's argument and maybe the last proof I mentioned, but I'm not sure. tl;dr version: The infinite sets of natural number, positive integers, and all integers are all the same size. The infinite set of real numbers is actually larger than the infinite set of natural numbers. There are an infinite number of infinities of different sizes.
Oh and to Kardredor, you are correct about infinities of different sizes, but the example you gave of the infinite set of odd numbers, and the infinite set of positive integers isn't correct as they are actually the same size. I explain in my last posts how sets are determined to be the same size and here are the functions for your examples. 0 1 1 3 2 5 3 7 4 9 5 11 and so on... The function for these sets would be, using the left column for x, f(x) = 2x + 1 So you are correct, but the example you gave to prove that you are correct is incorrect. Look up Cantor's Diagonal Argument for an actual proof.
he's talking about one infinity being bigger. In order for something to be bigger, there has to be a limit in the size of at least 1 thing, aka not infinity.
he might be right about the theorem which would place more numbers in one set than another under finite circumstances, which then ought to allow one infinity to be larger than another... so I can see why he's asking the question. I don't see what it has to do with "growing" anything, hence my confusion with your post. The problem is that both sets taken to infinity produce equally immeasurable sets of numbers, and the number of set components of both sets is undefined (did someone divide by zero!?!@#). To say that one function adding two outputs to a set is a larger infinity than a function adding only one is similar to saying that you'd get to infinity faster by reciting two numbers per second rather than one. Either way, you'd spend the rest of your life reciting numbers (maybe having to invent numbers as you go) and fail in the end.
I don't see the point in dividing up odd and even numbers to infinity. How is that even relevant to infinity? Infinity is no number. Though numbers can go on infinitely.
I found two good papers about the mathematical aspect of infinity and they explain the proofs of different size infinities better than I could. They're fairly long reads and it's a lot to wrap your head around, but once you understand it... it actually still makes very little sense and is mind blowing. http://emp.byui.edu/howardd/Math301\ToInfinityAndBeyond.pdf This one is shorter and not as in depth. http://www.math.jmu.edu/~rosenhjd/discrete/chapter6.pdf This one is a bit longer and more in depth. If you read the first one through then the second one you should be able to get a decent grasp on things. It's very relevant to infinity actually because it shows how strange the entire notion of infinity is and the problems that arise when our minds which almost always handle finite things try to comprehend the infinite. If you take the infinitely large set of odd numbers and compare it to the infinitely large set of all natural numbers, of course you'd be tempted to say that the odd number set is half the size of the set of all natural numbers, and to some extent you are correct. The odd number set contains only part of the elements of the natural number set, but the sets are actually the same size because they are infinite in size. The part is the same size as the whole.
Didn't read the first link but the second one was great. I (could) spend infinity thinking about infinity (and I plan on it)