- #1
tahayassen
- 270
- 1
leroyjenkens said:I used to have a problem with infinity. I kept using it like it was a number.
For example, I couldn't understand that the amount of numbers between both 0 and 1 and 0 and 2 were both the same.
tahayassen said:Aren't some infinities larger than other infinities?
Jack21222 said:In a way, yes, that's what Cantor demonstrated in the late 1800s.
If you took the number of numbers in between 0-1 and divided it by the number of numbers between 0-2, you should get 1/2.
Let x be the number of numbers between 0-1. There are an equal number of numbers between 0-1 and between 1-2, so the number of numbers between 0-2 is x + x, or 2x. So you have x/2x, and even if x is infinity, they cancel (they're the same infinity).
I'm sure mathematicians will murder me for doing it that way, since I probably did all kinds of things wrong, but I think that's the general idea.
leroyjenkens said:The problem is you're using infinity as if it's a number. You added infinity with infinity. That makes no sense if infinity isn't a number.
Jack21222 said:It makes plenty of sense. For every number in the 0-1 set, there is a corresponding number in the 1-2 set. In my example, x is not necessarily infinity, it's the number of numbers in between 0-1.
The concept of infinities cancelling out, and one infinity being "bigger" than the other, is used ALL THE TIME in calculus when dealing with limits. For example, consider (2^x)/(x!) As x goes to infinity, the top and bottom are both infinity. However, the bottom infinity is "larger" so the limit as it goes to infinity is zero.
EricVT said:For x equal to infinity, both the numerator and denominator are infinitely large, but their ratio is not zero.
For x approaching infinity -- but still finite -- the numerator and denominator also have finite values and their ratio is close to zero, but not zero.
Taking the limits of functions like this is not the same as dividing infinity by infinity.
This is a discussion we had in another part of the forum and I'm wondering who is correct. The discussion is becoming increasingly confusing and annoyingly (regardless of the posts in between the discussion), no one with a "Science Advisor", "Homework Helper", or "PF Mentor" title is stepping into end the argument, so I would appreciate it if you would end the argument.