My
Oxford Dictionary of the Christian Church gives the Trinity doctrine as:
The One God exists in Three Persons and One Substance.
The
Catholic Encyclopedia says
in the
unity of the Godhead there are Three Persons, the Father, the Son, and the Holy Spirit, these Three Persons being truly distinct one from another. Thus, in the words of the Athanasian Creed : "the Father is God, the Son is God, and the Holy Spirit is God, and yet there are not three Gods but one God."
Wikipedia (the highest spiritual authority, as you know) says
God is one God, but three coeternal consubstantial persons or hypostases—the Father, the Son (Jesus Christ), and the Holy Spirit—as "one God in three Divine Persons". The three Persons are distinct, yet are one "substance, essence or nature" (homoousios).
The formulation was devised in the 4th century CE, to solve a longstanding problem in the politics of the early church, namely how to raise Jesus to god status but at the same time avoid the charge of polytheism, which was associated with paganism.
Therefore a number of earlier ideas are excluded by the formulation eg
that the one God has three manifestations, as the Father, as the Son, and as the Ghost
that the Father + the Son + and the Ghost = God as with, for example, ⅓+⅓+⅓=1
that the Father is a god, the Son is a god, and the Ghost is a god (and there are no other gods)
nor is God a corporation with a board of three,
nor a partnership of three partners
and so on.
Instead, as the
Catholic Encyclopedia entry above expressly states, "the Father is God, the Son is God, and the Holy Spirit is God, and yet there are not three Gods but one God.". That is, the Father is 100% of God, the Son is 100% of God, and the Ghost is 100% of God. Yet 100%+100%+100% = 300% = 3 Gods, and in insisting that 100%+100%+100%=100% (and
only 100%, one god) the doctrine is incoherent.
The churches do not deny this. Instead they say that the Trinity doctrine is "a mystery in the strict sense", and that means it "can neither be known by unaided human reason apart from revelation, nor cogently demonstrated by reason once it has been revealed" ─ their words, not mine. But if you unpack that wording, the only meaning is that the doctrine is incoherent, is in plain words a nonsense.
(It also leads to nonsensical consequences. In the NT Jesus never once claims to be God, and through Paul and in all four gospels expressly denies that he's God, being instead God's envoy and, implicitly, and in John expressly, having only such powers as God allows him. So if the doctrine is correct and Jesus is God, then Jesus is at all times a self-conscious and deliberate deceiver. Moreover, all versions of Jesus must be taken to be talking to themselves on each occasion when Jesus prays to God; Mark's and Matthew's Jesuses on the cross must be understood to have cried out,
Me, me, why have I forsaken me? And since Matthew's and Luke's Jesuses are the genetic offspring of God, and have his Y-chromosome, and since under the doctrine Jesus is 100% of God, and the Ghost is 100% of God, each has as good a claim to the title the Father as the Father has. And so on.)