L is not equal to infinity; that’s a type error. L is equal to 1⁄2 A_0 + 1⁄4 A_1 + 1⁄8 A_2 …
ℓ1 is a bona fide vector space—addition behaves as you expect. The points are infinite sequences (x_i) such that ∑i|xi| is finite. This sum is a norm and the space is Banach with respect to that norm.
Concretely, our interpretation is that x_i is the probability of being in world A_i.
A utility function is a linear functional, i.e. a map from points to real numbers such that the map commutes with addition. The space of continuous linear functionals on ℓ1 is ℓ∞, which is the space of bounded sequences. A special case of this post is that unbounded linear functionals are not continuous. I say ‘special case’ because the class of “preference between points” is richer than the class of utility functions. You get a preference order from a utility function via “map to real numbers and use the order there.” The utility function framework e.g. forces every pair of worlds to be comparable, but the more general framework doesn’t require this—Paul’s theorem follows from weaker assumptions.
The presentation tries to deal with unbounded utilities. Assuming ∑i|x1| to be finite exludes the target of investigation from the scope.
Supposedly there are multiple text input methods but atleast on the website I can highlight text and use a f(x) button to get math rendering.
I don’t know enough about the fancy spaces whether a version where the norm can take on transfinite or infinidesimal values makes sense or that the elements are just sequences without a condition to converge. Either (real number times a outcome) is a type for which finiteness check doesn’t make sense or the allowable conversions from outcomes to real numbers forces the sum to be bigger than any real number.
Requiring ∑i|xi| to be finite is just part of assuming the xi form a probability distribution over worlds. I think you’re confused about the type difference between theAi and the utility of Ai. (Where in the context of this post, the utility is just represented by an element of a poset.)
I’m not advocating for or making arguments about any fanciness related to infinitesimals or different infinite values or anything like that.
L is not equal to infinity; that’s a type error. L is equal to 1⁄2 A_0 + 1⁄4 A_1 + 1⁄8 A_2 …
ℓ1 is a bona fide vector space—addition behaves as you expect. The points are infinite sequences (x_i) such that ∑i|xi| is finite. This sum is a norm and the space is Banach with respect to that norm.
Concretely, our interpretation is that x_i is the probability of being in world A_i.
A utility function is a linear functional, i.e. a map from points to real numbers such that the map commutes with addition. The space of continuous linear functionals on ℓ1 is ℓ∞, which is the space of bounded sequences. A special case of this post is that unbounded linear functionals are not continuous. I say ‘special case’ because the class of “preference between points” is richer than the class of utility functions. You get a preference order from a utility function via “map to real numbers and use the order there.” The utility function framework e.g. forces every pair of worlds to be comparable, but the more general framework doesn’t require this—Paul’s theorem follows from weaker assumptions.
The presentation tries to deal with unbounded utilities. Assuming ∑i|x1| to be finite exludes the target of investigation from the scope.
Supposedly there are multiple text input methods but atleast on the website I can highlight text and use a f(x) button to get math rendering.
I don’t know enough about the fancy spaces whether a version where the norm can take on transfinite or infinidesimal values makes sense or that the elements are just sequences without a condition to converge. Either (real number times a outcome) is a type for which finiteness check doesn’t make sense or the allowable conversions from outcomes to real numbers forces the sum to be bigger than any real number.
Requiring ∑i|xi| to be finite is just part of assuming the xi form a probability distribution over worlds. I think you’re confused about the type difference between theAi and the utility of Ai. (Where in the context of this post, the utility is just represented by an element of a poset.)
I’m not advocating for or making arguments about any fanciness related to infinitesimals or different infinite values or anything like that.