I should define super-exponentially large, since I’m using it as a hand-wavy term. Let’s call a number super-exponentially large if it’s bigger than anything you could write down using scientific notation using the matter available on earth. Numbers like a googol (=10^100) or 1,000,000,000^1,000,000,000 (=10^000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000) are not super-exponentially large, since I can write them as a power of ten in a reasonable amount of space. A googolplex (10^10^100, or 1 followed by a googol zeros) is super-exponentially large because there is not enough matter on earth to write a googol zeros.
A super-exponentially small probability is a probability less than 1 over a super-exponentially large number.
Speaking loosely, if you express a probability in scientific notation, a given amount of evidence lets you add or remove a constant number of digits from the exponent. For most things, that’s pretty powerful. But if you’re dealing with super-exponentially small probabilities, to hold real sway you would need more evidence than you could write down with the matter available on earth.
I should define super-exponentially large, since I’m using it as a hand-wavy term. Let’s call a number super-exponentially large if it’s bigger than anything you could write down using scientific notation using the matter available on earth. Numbers like a googol (=10^100) or 1,000,000,000^1,000,000,000 (=10^000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000) are not super-exponentially large, since I can write them as a power of ten in a reasonable amount of space. A googolplex (10^10^100, or 1 followed by a googol zeros) is super-exponentially large because there is not enough matter on earth to write a googol zeros.
A super-exponentially small probability is a probability less than 1 over a super-exponentially large number.
Speaking loosely, if you express a probability in scientific notation, a given amount of evidence lets you add or remove a constant number of digits from the exponent. For most things, that’s pretty powerful. But if you’re dealing with super-exponentially small probabilities, to hold real sway you would need more evidence than you could write down with the matter available on earth.