To extend Expansionism to worlds with less structure, you could try to come up with a very general kind of distance metric between locations of value (whatever they may be), and use that instead of distance in spacetime to define your “spheres”. I’m not sure this can cover all possibilities we’d want to consider while also ensuring the spheres only contain finitely many locations of value with nonzero value at a time, for finite partial sums over each sphere.
Here’s one idea, although I’m skeptical that it works. If there are only countably many binary predicates about locations of value (or you only consider countably many of them), you can order them P1,P2,…, and define a metric by something like
d(x,y)=∞∑i=1si|Pi(x)−Pi(y)|, for some s,0<s<1,
or just map f(x)=∑∞i=13−iPi(x)∈R and use d(x,y)=|x−y|. (I use 3 instead of 2 to avoid 0.01111...=0.1000… in binary.)
With this, you can define the ball of radius r around x as Br(x)={y∈X|d(x,y)<r}, and expand using these nested balls instead of the spacetime spheres. This basically represents the location of value x as the binary sequence P1(x),P2(x),…, (or a real number) so you can only uniquely represent a number of locations of value with at most the cardinality of the real numbers this way. That’s still a lot of possible locations, though: we could potentially represent a continuum of different universes each with a continuum of possible locations, as long as only countably many locations bear any value at all in a given outcome, or we can take well-defined integrals over locations.
To extend Expansionism to worlds with less structure, you could try to come up with a very general kind of distance metric between locations of value (whatever they may be), and use that instead of distance in spacetime to define your “spheres”. I’m not sure this can cover all possibilities we’d want to consider while also ensuring the spheres only contain finitely many locations of value with nonzero value at a time, for finite partial sums over each sphere.
Here’s one idea, although I’m skeptical that it works. If there are only countably many binary predicates about locations of value (or you only consider countably many of them), you can order them P1,P2,…, and define a metric by something like
d(x,y)=∞∑i=1si|Pi(x)−Pi(y)|, for some s, 0<s<1,or just map f(x)=∑∞i=13−iPi(x)∈R and use d(x,y)=|x−y|. (I use 3 instead of 2 to avoid 0.01111...=0.1000… in binary.)
With this, you can define the ball of radius r around x as Br(x)={y∈X|d(x,y)<r}, and expand using these nested balls instead of the spacetime spheres. This basically represents the location of value x as the binary sequence P1(x),P2(x),…, (or a real number) so you can only uniquely represent a number of locations of value with at most the cardinality of the real numbers this way. That’s still a lot of possible locations, though: we could potentially represent a continuum of different universes each with a continuum of possible locations, as long as only countably many locations bear any value at all in a given outcome, or we can take well-defined integrals over locations.