Ben Karcher's question on Quantilizers

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I have a question/comment about an older video. With all this talk of maximizers, I decided to go back and rewatch the video "What's the Use of Utility Functions?" as it seems to be an important preface for these videos. At 5:00 in that video, you claim that if world states are ordered they can be mapped onto the real numbers, this is incorrect as it seems to assume the number of world states is countable or at least 1 dimensional. A good counter-example of this would be the set of all polynomials. I can order them by saying f(x)>g(x) if lim x->inf f(x)-g(x)>0. This gives us an ordering on an infinitely dimensional field that is obviously not isomorphic to R. While this may sound pedantic I actually consider it quite relevant as "world states" sound like something that may very well be multidimensional. I was wondering if this was important as a lot of these videos seem to implicitly assume a metric over world states to do things like integrating over them. Is there some way to bypass the need for a metric and abstract the idea of a satisficer/maximizer to any ordered field?

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Question Info
Asked by: Ben Karcher
OriginWhere was this question originally asked
YouTube (comment link)
On video: Quantilizers: AI That Doesn't Try Too Hard
Date: 2020-12-14T21:45
Asked on Discord? Yes