# 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? Posted on