I'd like to chime in here as someone who is math-interested, but never went beyond high school maths (analytical trig, sadly).
IIRC, I first got my head around different orders of magnitude regarding infinities with George Gamow's popular "1, 2, 3, infinity" [1].
However -- I think, it's been a long long time since I read it -- that book presented different infinities in exactly the same way that someone who could only count to 20 would compare sets of hundreds of items.
That is: assume that space is infinitely bounded, with an infinite amount of matter. You might say that there are infinitely many galaxies, and there are infinitely many stars. However, there are also infinitely many more stars than there are galaxies, because each galaxy contains more than one star.
That's a fine trick for developing a simple understanding of infinities, but then apparently in later maths it completely falls over.
For example, I'm one of the many people who have a rather primal loathing for the notion that .999 repeating is really equal to 1. (I literally do have a powerful emotional response to this. I hate it. I know it's been proven. I know it's a fact. But I hate it.)
IIRC, I first got my head around different orders of magnitude regarding infinities with George Gamow's popular "1, 2, 3, infinity" [1].
However -- I think, it's been a long long time since I read it -- that book presented different infinities in exactly the same way that someone who could only count to 20 would compare sets of hundreds of items.
That is: assume that space is infinitely bounded, with an infinite amount of matter. You might say that there are infinitely many galaxies, and there are infinitely many stars. However, there are also infinitely many more stars than there are galaxies, because each galaxy contains more than one star.
That's a fine trick for developing a simple understanding of infinities, but then apparently in later maths it completely falls over.
For example, I'm one of the many people who have a rather primal loathing for the notion that .999 repeating is really equal to 1. (I literally do have a powerful emotional response to this. I hate it. I know it's been proven. I know it's a fact. But I hate it.)
[1]: http://www.amazon.com/One-Two-Three-Infinity-Speculations/dp...