elementary set theory – Expressing a fusion as a set


Michael Potter in his “Set Theory and its Philosophy” separates two types of aggregations of objects:

  • Fusion is a transitive aggregation: a part of an aggregated object becomes a part of the fusion;

  • Collection is a non-transitive aggregation: a part of an aggregated object does not become a part of the collection (unless it is the object itself).

The concept of a collection is studied in naive set theories: https://en.wikipedia.org/wiki/Naive_set_theory

I am trying to better understand the concept of a fusion.

According to the same classification there are two corresponding types of parts:

Let’s denote a fusion of objects in square brackets, similar to how we denote a set (collection).

According to the definitions, we can interpret a fusion as a collection of all parts of aggregated objects:

  • if $A = {a, b}$, then $(A) = {a, b, {}, {a},{b},{a,b}}$.

Thus, the powerset of an object is a subset of the fusion of the object.
Would it be a correct statement?

Now, assuming $A = {{a}}$, $a ne {a} ne {{a}} ne a$.
${a}$ is a member of $A$; $a$ is a member of ${a}$.
Membership is not transitive, therefore $a$ is not a part of $A$.

Which statement is correct in this case:

  • $({{a}}) = {{a},{},{{a}}}$, or
  • $({{a}}) = {a,{a},{},{{a}}}$?

Are there formal studies on fusions?