Date: Feb 21, 2013 7:14 AM
Subject: Puzzle in the history of maths
I think I remember reading that it was difficult for the early pioneers of set theory such as Cantor to prove or accept that R has the same cardinality has R ^ 2. I don't understand why this was difficult for them to prove. Surely, if you provide many bright undergrads with only the knowledge that existed at the time, it's not a particularly difficult exercise. I've heard it said that it seemed counterintuitive that set A can have a greater geometric dimension than set B but still have the same cardinality. But I don't understand why that is counterintuitive. Surely the fact that the rationals have the same cardinality as the integers was known at the time people started comparing the cardinailities of R^m and R. In a geometric sense, the rationals clearly seem "much larger" than the integers. So there was already a glaring example of "geometrically larger does not imply larger cardinality."
To me, it does not seem in the least counterintuitive that all the sets of the form R^m have the same cardinality, and I don't think it ever did seem counterintuitive. However, I do seem to remember that I found it counterintuitive that the rationals have the same cardinality as the integers, and that I also found it counterintuitive that the reals have a greater cardinality than the rationals.