I did some infinite series calculations while studying Fourier analysis and the concept of infinity really bugs me. I haven't read or heard not one sensible explanation yet (for me), what infinity really means? It's like a magic trick.
For example the Hilbert's hotel paradox bugs me and especially the solution of it. We have a hotel with countably infinite number of hotel rooms with all the rooms taken. First of all I have no idea, what that actually even means?! How is it possible to have infinite number of rooms? I can see my reflection going to infinity if I look at myself from a mirror, with mirror behind me, but infinite number of rooms..c'mon ;D Only in video games.
Secondly the notion that something of infinite quantity being full sounds to me like saying that: "Blue is not blue, blue is red"...you go like "huh?!" ;)
The solution for it is also bizarre: By "pushing" the guests into the next room we make room for new guests. Now wait a minute...I thought the hotel was full? If you have a glass full of water, meaning no single extra atom of water will fit into it, then how can you make more room into it for new guests? This also seems like a magic trick :D
Another example of infinity that bugs me is the series:
It seems this series is supposed to be divergent...why? What if we forget algebra for a while and do a practical example: There's a food basket on the table. I keep putting apples into it and Mike keeps eating them after every time I add an apple into it. I put an apple into it, he eats it etc, and we keep doing this "infinitely" x) It's really difficult to comprehend this kind of a scenario, which must be a result of the whole concept of "infinity".
Can someone give a layman's explanation on what infinity is. Does anyone actually even really really know what it is? Where did idea of infinity come from? The only physical phenomena, where I have met "infinity" is in the mirror...is this a good example of infinity or for the whole source of the idea?
Thnx for any help =) Please note my point is not to offend anyone, this notion of infinity in math and series calculations is just sometimes driving me nuts x)
I know my question sounds like a newbie question, but I can bet many many more people are wondering the same thing x)