>! Let $X$ be the set of complete theories that satisfy "everything
is countable" and have unboundedly many $\alpha<\omega_1^L$
with $L_\alpha$ realising them. The theory of $L_{\omega_1^L}$ is one such theory, and we will be done if we prove that there are some others. Now $X$ is a definable class in $L_{\omega_1^L}$, and so it must have some other elements or else $L_{\omega_1^L}$ would admit a truth defintion ($\varphi$ is true in $L_{\omega_1^L}$ iff the unique element of $X$ contains $\varphi$).
I have before me a copy of "The Indian Mathematician Ramanujan", by G. H. Hardy (not actually related to me, as far as I know), which appeared in volume 44, number 3 (March 1937) of The American Mathematical Monthly, on pages 137‒155.
One of the theorems of Ramanujan stated there is this:
If...
Here are two beliefs. I think everybody will agree that one of them, at least, is false. I adhere to the second one.
Belief 1. The simplest way to compute the exponential $e^A$ of a complex square matrix $A$ is to use the Jordan decomposition.
Belief 2. It's simpler and more efficient to use t...
