The Holy Trinity as Incoherent

The Holy Trinity has had a problematic history, partly evidenced by point of fact that theologians still don't agree on how it works, and partly  seen from its ex post facto evolution, shoehorned into the scant evidence of the biblical texts. From Ignatius of Antioch onwards we see development of the idea in early church thinking, until it is codified at the Council of Nicaea in the 4th century CE. There will be more talk later on what was creedally set out.
Existence Properties
In simple terms, we have three aspects (in a simple philosophical/general sense) to the Godhead, such that God the Father, God the Son, and God the Holy Spirit are in some way all God. Let us look at very fundamental understandings of existence properties.  Logically, Jesus is God as well as God the Father being God. Where A = God the Father, B = Jesus and C = the Holy Spirit:
  1. A = B
  2. A = C
  3. A = D
  4. But B ≠ A
  5. B ≠ C
  6. C ≠ D
and so on.
What this means is that if the Godhead has, say, 10 properties that identify it has having the label Godhead (something like this must be the case - it must have identifiable properties or the term 'Godhead' is meaningless and has no reference) the we have some issues. If Jesus = the Godhead, then Jesus must have all of these properties. But then Jesus is exactly synonymous with the Godhead. But since Jesus is seen as in some way identifiably different to the Godhead, hence the two semantic terms, this cannot logically be the case. And vice versa. So to say Jesus is fully God is meaningless unless to say that Jesus is a perfect synonym of the Godhead. Which then must apply to all the others. The Holy Spirit to be fully God must fully have these properties too, but then they are all synonymous and not able to be differentiated.
Now, these aspects can't be fully God if they lack some Godlike/Godhead properties. But what if they had more? Perhaps Jesus had 11 properties, 10 of which belonged to the Godhead. But then the Godhead cannot be part or a subset of the Jesus properties, less than Jesus in some discernible way. Or perhaps Jesus = the Godhead + human body. This is intuitively problematic, such that Jesus becomes a different 'entity'/aspect with more than the properties of the Godhead. (I shall return to this later.)
Furthermore, The HS and Father must have different properties to Jesus in order to be differentiated as aspects/parts/persons/essences/whatever in order to be identified as such. It cannot be that the HS has exactly the same, no more no less, properties as Jesus because otherwise that would be Jesus (as mentioned before)! In order to say, "That is Jesus and that, over there, is the HS" such that they are identifiable as each person, they must have differentiated properties. But that means, if these properties are properties of being God, that neither can be fully God. The HS cannot have a godlike property that Jesus does not have, otherwise Jesus cannot be fully God. This seems to be the crux. Before I continue, let us look at some heresies.

Modalism suggests, as hinted in the previous paragraphs, that the Trinity has three aspects or manifestations of one person. Thus the Father could present himself as the Son, and in turn the Son could be manifested as the Spirit. However, this is seen as heresy (Sabellianism). But my points stand regardless of this position. Let me list a few other manifestations that the Trinity supposedly isn't.

Adoptionism was the belief that Jesus was an ordinary man, born of Joseph and Mary, who became the Christ and Son of God at his baptism.

wiki - The Arian concept of Christ is that the Son of God did not always exist, but was created by—and is therefore distinct from—God the Father. This belief is grounded in the Gospel of John (14:28)[3] passage: "You heard me say, 'I am going away and I am coming back to you.' If you loved me, you would be glad that I am going to the Father, for the Father is greater than I."

The best way of understanding this is that each part makes up a third, mathematically speaking, and they all add up to one. No, not so, apparently.

No, they're not three gods, either. Not separately so.
So where does this leave us? Well, here is one particular definition:
The Divine Name is numerically one, and yet in this One Name there are three persons distinguished: The Father, the Son and the Holy Spirit (Matthew 28:19-20).
The church receives her benediction from peculiar blessing for each of these Three (2 Corinthians 13:14).
Their different personalities are recognised (1 Corinthians 12:4-6). Christ refers to Himself with the pronoun "I," and at the same time to the Father as "He," and to the Spirit (who proceeds from the Father, and thus distinct from Him), again as "He," rather than "it," clearly meaning a person, not an influence or mere power. And yet these Three possess the one indivisible divine essence, and are constituted distinct persons by certain incommunicable properties, not common in one with the other two.
As wiki states:
The Christian doctrine of the Trinity (Lat., trinitas from the Lat. triad, "three"[1]) defines God as three consubstantial persons,[2]expressions, or hypostases:[3] the Father, the Son (Jesus), and the Holy Spirit; "one God in three persons". The three persons are distinct, yet are one "substance, essence or nature".[4] In this context, a "nature" is what one is, while a "person" is who one is.[5][6][7]
According to this central mystery of most Christian faiths,[8] there is only one God in three persons: while distinct from one another in their relations of origin (as the Fourth Lateran Council declared, "it is the Father who generates, the Son who is begotten, and the Holy Spirit who proceeds") and in their relations with one another, they are stated to be one in all else, co-equal, co-eternal and consubstantial, and "each is God, whole and entire".[9] Accordingly, the whole work of creation and grace is seen as a single operation common to all three divine persons, in which each shows forth what is proper to him in the Trinity, so that all things are "from the Father", "through the Son" and "in the Holy Spirit".[10]
In 325, the Council of Nicaea adopted the Nicene Creed which described Christ as "God of God, Light of Light, very God of very God, begotten, not made, being of one substance with the Father". The creed used the term homoousios (of one substance) to define the relationship between the Father and the Son. After more than fifty years of debate, homoousios was recognised as the hallmark of orthodoxy, and was further developed into the formula of "three persons, one being".
To me, this is just word salad (as Andy Schueler would no doubt say). Try as I might, I can't get this to make any sense. Because it doesn't. They want to get out of the logical paradoxes and problems other theories imply, and so they produce something which doesn't quite make sense and which relies on mystery! Yup, the mystery card. A bit like how some people around here have claimed contra-causal free will works! You can't sidestep logic with a mystery card.
Anyhow, one God-essence is tapped into by three persons, without there being any clarity as to what a person might be. But if Jesus can tap into the essence of God, then he must have that exact same essence as shared between all the Trinity, and the whole. But for Jesus to be identified as a different person (hypostasis) but have the same nature as the others means that Jesus has properties that the others do not have and vice versa. So whatever is claimed of Jesus, he cannot be fully God. There is equivocation here on what nature means. These different properties which each person of the Trinity must have in order to identified as different to the others must, therefore, not be properties of being God, must not be essential properties of being God, must not be the essence or nature of God.
And here is the problem, the contradiction. The Christian needs, obviously, the properties of all three persons to be godly, and yet they must be individual to each one. Again, Jesus cannot be fully God because he, in his person, has some essence of God that he shares with the others, and yet not all essential properties. Unless the other properties are not essential. But then they are contingent, and we have non-necessary elements of all persons of the Trinity. But surely all persons are necessary (in the beginning was the Logos). This is a problem which, to me, seems insurmountable.
The only way out is that each person taps into an essence but that that essence is not the totality of God. And then we get back to one of the heresies.
The idea that each of thee persons do things to each other (pray etc) adds a level of confusion. How is this best explained?
Andy Schueler, ove ron my blog at SIN where I ran this article originally in slightly truncated form, stated:

If wikipedia doesn´t lie to me, then what christian philosophers mean with with the hypostases vs ousia distinction is conceptually equivalent to the distinction between the general vs the particular. In that case, three particular cars, would be the "same" in that they are all cars, conceptually. If that would be what Christians actually meant by "trinity" - one abstract concept "God" and three particular instantiations of it - then I wouldn´t call it incoherent BS.

But Christians go beyond saying that there is one Godhood (the general) and three instantiations of it (the particulars) - they say that father, son and holy spirit are absolutely equal in every respect - " co-equal and co-eternal, one in essence, nature, power, action, and will" and "They cannot be separate from one another. Each person is understood as having the identical essence or nature, not merely similar natures" etc.pp.

And that is incoherent BS.

That is exactly like me saying that my current Skoda Fabia and my old Renault Twingo are the exact same thing in every respect but they are not the exact same thing in every respect but they are but they´re not but they are.
Presumably Jesus had a mind, and that mind did not appear to know the mind of the Father (he was, at the end of the day, praying to the other mind).
Now, since there were at least 2 minds/consciousnesses in the Trinity, then it is hard to see how Jesus could be fully God unless the nature/necessary aspects of God have nothing to do with the entirety of the properties of those 2 minds. At least the difference which those minds share must be contingent aspects of Godhood.
It is all a confused mystery, which is made worse by the expectation that we must come to love this immaterial entity which no one is able to adequately describe or account for. 
So I understand why theologians have to resort to mystery. I get it. They have no choice. It is logically incoherent. But they can't admit that because, like free will, you take that away, and they ain't got nothin'. And that doesn't even take into account the nonsense of the Atonement.