Good so far…
No… can you explain how you go from the first step to the second?
The Trinity isn’t a mathematical equation, such that you can do algebraic substitutions on the equations and have something that makes sense result. It seems that you’re essentially saying:
F = G
So = G
Sp = G
G = HT
HT = {F, So, Sp}
Therefore,
G = {F, So, Sp}
and
F = {F, So, Sp}
So = {F, So, Sp}
Sp = {F, So, Sp}
Doesn’t work that way.
If you want to use an algebra of sets, you might get valid results, but you’re not going to get what you want:
F ∈ G
So ∈ G
Sp ∈ G
G = HT
HT = {F, So, Sp}
But this only gives you
F ∈ {F, So, Sp}
So ∈ {F, So, Sp}
Sp ∈ {F, So, Sp}
Even so, we’re still missing an essential property of the Trinity, when we reduce the nature of a person in the Trinity to an element of a set.
A couple of problems here. First, in terms of your definitions: ‘physically’ doesn’t hold, since there aren’t physical bodies for the Trinity – the Father is pure spirit, the Holy Spirit is pure spirit, and the Word is pure spirit (although there is also the Incarnation of the Word, that is, Jesus). Second, ‘actions of the past’ has no meaning in the context of the Trinity – the Godhead exists in an eternal timelessness. Third, to say that one member of the Trinity is an ‘image’ of any other member, would deny that they have the same nature – instead, it would be saying that they image the nature (which isn’t what the Church teaches). Fourth, to say that one person of the Trinity ‘is’ another member, does harm to the notion of and distinct person-ness of each of the persons of the Trinity.
Blessings,
G.