Yes, St. Peterburg and Petrograd (= St. Petersburg with all payouts increased by one) are given the same infinite utility. Neither is preferable to the other, despite the intuition saying that Petrograd is better. While intuition can be a guide, it is an untrustworthy one, a castle in the air that requires a foundation to be built underneath it.
The problem with comparing infinities is that if you impose conditions on the preference relation that seem reasonable for finite games, then before you know it — literally so in Savage’s case — you end up excluding all the infinities, and neither St. Peterburg nor Petrograd exist. To avoid doing that, you have to give up some of those conditions. Savage’s P2, for example, sounds perfectly reasonable if you don’t think about infinite games, but as soon as you do, you can see that it must fail. Not that there’s anything special about P2, it’s really the basic ontology of the system that is at fault.
I have to wonder how strong a mathematical background some of the people who have published on the subject had. Attempting to construct a total ordering on all functions from a probability space to the real numbers, or even just on the measurable functions, seems doomed to failure.
To my perspective that infinities are equal is a unfounded intuitiion. Reading a surreal proof on how w+1 is strictly greater than w is partly amazing how you can have claims about infinities without relying on intuition (ie can actually proove stuff). Then a law like “infinity + infinity = infinity” starts to feel like “positive + positive = positive”, “positive” is not a number but a quality. There is additional structure in “2+2=4″ than that positivity is preserved.
In the same way that if one option has negative utlity and one options has positive utility you can safely choose the positive one without regard to the actual magnitude so it is also safe to choose a transfinite positive over a finite positive.
If the theory doesn’t treat finites in a special way (is finiteness-ambivalent) then the core material should transfer for applicable parts to the transfinite domain.
Yes, St. Peterburg and Petrograd (= St. Petersburg with all payouts increased by one) are given the same infinite utility. Neither is preferable to the other, despite the intuition saying that Petrograd is better. While intuition can be a guide, it is an untrustworthy one, a castle in the air that requires a foundation to be built underneath it.
The problem with comparing infinities is that if you impose conditions on the preference relation that seem reasonable for finite games, then before you know it — literally so in Savage’s case — you end up excluding all the infinities, and neither St. Peterburg nor Petrograd exist. To avoid doing that, you have to give up some of those conditions. Savage’s P2, for example, sounds perfectly reasonable if you don’t think about infinite games, but as soon as you do, you can see that it must fail. Not that there’s anything special about P2, it’s really the basic ontology of the system that is at fault.
I have to wonder how strong a mathematical background some of the people who have published on the subject had. Attempting to construct a total ordering on all functions from a probability space to the real numbers, or even just on the measurable functions, seems doomed to failure.
To my perspective that infinities are equal is a unfounded intuitiion. Reading a surreal proof on how w+1 is strictly greater than w is partly amazing how you can have claims about infinities without relying on intuition (ie can actually proove stuff). Then a law like “infinity + infinity = infinity” starts to feel like “positive + positive = positive”, “positive” is not a number but a quality. There is additional structure in “2+2=4″ than that positivity is preserved.
In the same way that if one option has negative utlity and one options has positive utility you can safely choose the positive one without regard to the actual magnitude so it is also safe to choose a transfinite positive over a finite positive.
If the theory doesn’t treat finites in a special way (is finiteness-ambivalent) then the core material should transfer for applicable parts to the transfinite domain.