I'm confused. What is the countable set of hypotheses you are considering? My claim is merely that if you have hypotheses H1, H2, ..., then p(Hi) > 1/n for at most n-1 values of i. This can be thought of as a weak form of Occam's razor.

In what sense is "every statement being true" a choice of a countable set of hypotheses?

I think maybe the issue is that we are using hypothesis in a different sense. In my case a hypothesis is a complete model of the world, so it is not possible for multiple hypotheses to be true. You can marginalize out / observe a bunch of variables to talk about a subset of the world, but your hypotheses should still be mutually exclusive.