Leonard Bernstein & Bernstein’s Theorem on Infinite Cardinals
Leonard Bernstein ranks among the brightest musical talents of all time. Here’s a two-part video of him conducting Gershwin’s masterpiece “Rhapsody in Blue” …while also tearing through the piano part at the same time!!
Now that I’m inspired from that great music, I’d like to think a little bit about this theorem that came up a couple posts down, due to a very different Bernstein - Felix Bernstein, as in the famous Cantor-Schroeder-Bernstein Theorem (which is not the one we’ll look at here). Here’s the best picture of him I could find:
This little theorem of his says that for two infinite cardinal numbers $$\aleph_n$$ and $$\aleph_{\beta},$$ we have
I don’t know the proof of this theorem, but I’d like to understand its content intuitively. I’ve read before that the continuum hypothesis (which asserts that there are no ‘sizes of infinity’ strictly in between those of the rational and real numbers) implies that $$2^{\aleph_n} = \aleph_{n+1}.$$This seems to be a way of saying that there are $$\aleph_{n+1}$$ different ways to use only 2 colors to paint $$\aleph_n$$ elements in a set (using only one color per element, of course!). I’m willing to take this result for granted for the moment and see where it takes us.
Then the $$2^{\aleph_{\beta}}$$ in Bernstein’s theorem above can be replaced with $$\aleph_{\beta + 1}.$$Doing this we get $$\aleph_n^{\aleph_\beta} = \aleph_{\beta + 1}*\aleph_n.$$Now let’s recall how multiplication works for infinite cardinals: the larger of the two cardinals in the product ‘swallows up’ the other. Putting all this together we can rewrite Bernstein’s theorem this way: $$\aleph_n^{\aleph_{\beta}} = Max{(\aleph_n, \aleph_{\beta+1})}.$$
So when we exponentiate one infinite cardinal by another, the result is whichever is the larger between the base cardinal and one cardinal higher than the exponent cardinal. This doesn’t seem entirely crazy. If you take the cardinality of the continuum and raise it to the cardinality of the rationals, you shouldn’t expect to get something even bigger, because the real numbers are so much more numerous than the rationals.
