Yes, the utility function is the “one dimension”. Of course, it can be as complicated as you’d like, taking into account multiple aspects of reality. But ultimately, it has to give a weight to those aspects; “this 5-year-old’s life is worth exactly XX.XXX times more/less than this 80-year-olds’ life” or whatever. It is a map from some complicated (effectively infinite-dimensional) Omega to a simple one-dimensional utility.
Yes, the utility function is the “one dimension”. Of course, it can be as complicated as you’d like, taking into account multiple aspects of reality. But ultimately, it has to give a weight to those aspects; “this 5-year-old’s life is worth exactly XX.XXX times more/less than this 80-year-olds’ life” or whatever. It is a map from some complicated (effectively infinite-dimensional) Omega to a simple one-dimensional utility.