In article <email@example.com>, firstname.lastname@example.org wrote:
> 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.
Perhaps not difficult to prove, but just difficult to believe. It's hard to put your mind back to an earlier period, with the attitudes and experiences of people at that time.
> .... > 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....
It's hard to judge such a thing without a _lot_ of historical reading. In the 19th century it was commonly said that the plane has (infinity)^2 points; meaning that its points are usually coordinatized using two real numbers, so in an obvious sense have two degrees of freedom. Likewise for higher dimensions. If we can get our heads into that way of thinking, then it really is surprising that R and R^2 have the same cardinality.