names of my friends here, but my students would be pleased if i used them: albert, bilbert, and cuthbert. >> all right. >> and i'd like to invite some, perhaps all, of my friends over for a dinner party. now, i can invite all three, i can invite none, maybe just albert alone, or maybe just bilbert and cuthbert. if you look at it, there's eight different possibilities, eight subsets from those original three. >> yep, either zero, one, two, or all three people. >> and if you notice, i had three original friends -- now, i don't have eight digits on this hand, but imagine i had eight fingers here. eight is definitely bigger than three; i cannot set up a correspondence. now, the remarkable thing that cantor did here was that he could start with an infinite number of friends. start with any set, take the set of subsets of those friends, it will be a bigger set indeed. you will not be able to set up a correspondence. >> one friend at a time, two at a time, even infinite -- all infinite subsets at a time. >> the set of all subsets is a bigger set even if the original set was infinite in its own