I'm looking for a development of (alternate?) set theory to model a space that contains an infinite number of points (urelements) and their properties. Instead of treating subsets as objects, I wish to treat them as properties. Hence, 1) a collection can only contain urelements, 2) the power set of a given collection defines all possible properties of the collection. I'm interested in the minimal axioms required and what can be worked out from there. My skill at set theory is not yet advanced enough for me to do this independenly nor do I wish to replicate previous work. Any references or ideas for how to proceed would be greatly appreciated.
As an example: Urelements a, b, c. Possible Properties: {}, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}. This essentially "turns around" the principle of abstraction by defining a property by enumerating all elements that have that property. Then {a} becomes something like "being a", {a,b,c} becomes "exists as an object", ...