This is awesome. How would one describe this formal system? There are no cells essentially
I wonder if it's the same for reality. Everytime we've been looking at some particles we deemed fundamental they appear to be made of even smaller ones.
Not everytime. There is a list of particles that we currently deem fundamental, because no one has found any structure in them. But maybe we haven't looked closely enough yet. Some entertain the idea that they consist of vibrating strings instead of structurless, pointlike particles, but then those would be fundamental.
But they can also be seen as quantized excitations of some underlying quantum field.
Each cell in level N is the stable pattern of gliders arranged into a square shape in the lower level N-1. If you think about it, this is how "objects" in the real world exist - not as an essence of the object, but as a pattern of simpler elements arranged in space-time that we recognize as an object.