How do we measure infinity? How do we compare infinite sets? If someone
tells you that “it is equally likely for a prime number to have residue +1 as
it is likely for it to have residue −1 modulo 4”—forgetting any proof of this
claim for the moment—in how many ways might one formulate (correct)
rigorous mathematical statements that are in accord with that claim?
I will elaborate on such questions, and on heuristics that sometimes guide
us to make plausible conjectures about them, in hopes of launching a general
discussion.
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