### Different sizes of infinities

I know the following piece is a bit long, but I REALLY hope you will take the time to read and try to understand the thoughts (if you don't know them already). This is one of the areas where YOU can show your students something about mathematics that is FASCINATING, AMAZING, awesome, mind-boggling!!! ... show them that mathematics is NOT just mundane, mind-numbing manipulation of numbers in calculations. This topic is not extra-curricular; it IS taught in pre-algebra / algebra 1 (real numbers), BUT you can explain some parts of it even to 4-6th graders. This sounds really strange. How could there be different kinds or sizes of infinities? I first encountered this idea in my university studies, and I thought it was absolutely FASCINATING! The video below goes through the basics and explains them well, though really fast. I recommend you pause the video at those points where it goes too fast for you, and "rewind" or rewatch those parts. I will explain some of the