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VoiceOfRa comments on Probabilities Small Enough To Ignore: An attack on Pascal's Mugging - Less Wrong
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Assuming your preferences satisfy the axioms.
No, by completeness I mean that for any two lotteries you prefer one over the other.
So why not reject it in the finite case as well?
Care to assign a probability to that statement.
Actually, I would, but that's digressing from the subject of infinite lotteries. As I have been pointing out, infinite lotteries are outside the scope of the VNM axioms and need additional axioms to be defined. It seems no more reasonable to me to require completeness of the preference ordering over St. Petersburg lotteries than to require that all sequences of real numbers converge.
"True." At some point, probability always becomes subordinate to logic, which knows only 0 and 1. If you can come up with a system in which it's probabilities all the way down, write it up for a mathematics journal.
If you're going to cite this (which makes a valid point, but people usually repeat the password in place of understanding the idea), tell me what probability you assign to A conditional on A, to 1+1=2, and to an omnipotent God being able to make a weight so heavy he can't lift it.
Ok, so care to present an a priori pure logic argument for why St. Petersburg lottery-like situations can't exist.
FInite approximations to the St. Petersburg lottery have unbounded values. The sequence does not converge to a limit.
In contrast, a sequence of individual gambles with expectations 1, 1/2, 1/4, etc. does have a limit, and it is reasonable to allow the idealised infinite sequence of them a place in the set of lotteries.
You might as well ask why the sum of an infinite number of ones doesn't exist. There are ways of extending the real numbers with various sorts of infinite numbers, but they are extensions. The real numbers do not include them. The difficulty of devising an extension that allows for the convergence of all infinite sums is not an argument that the real numbers should be bounded.
They have unbounded expected values, that doesn't mean the St. Petersburg lottery can't exist, only that its expected value doesn't.