I see. I was confused for a while, but in the hypothetical examples I was considering, a link between MMR and autism might be missed (a false negative with 5% probability) but isn't going to found unless it was there (low false positive). Then Vanviver explains, above, that the canonical null-hypothesis framework assumes that random chance will make it look like there is an effect with some probability -- so it is the false positive rate you can tune with your sample size.

I marginally understand this. For example, I can't really zoom out and see why you can't define your test so that the false positive rate is low instead. That's OK. I do understand your example and see that it is relevant for the null-hypothesis framework. (My background in statistics is not strong and I do not have much time to dedicate to this right now.)