## The Trinity and Coherency

When I hear things like this (taken from the Catechism of the Catholic Church, but could well be from any Christian Church):

The Trinity is One. We do not confess three Gods, but one God in three persons, the “consubstantial Trinity”.83 The divine persons do not share the one divinity among themselves but each of them is God whole and entire: “The Father is that which the Son is, the Son that which the Father is, the Father and the Son that which the Holy Spirit is, i.e. by nature one God.”84 In the words of the Fourth Lateran Council (1215), “Each of the persons is that supreme reality, viz., the divine substance, essence or nature.”85

254 The divine persons are really distinct from one another. “God is one but not solitary.”86 “Father”, “Son”, “Holy Spirit” are not simply names designating modalities of the divine being, for they are really distinct from one another: “He is not the Father who is the Son, nor is the Son he who is the Father, nor is the Holy Spirit he who is the Father or the Son.”87 They are distinct from one another in their relations of origin: “It is the Father who generates, the Son who is begotten, and the Holy Spirit who proceeds.”88 The divine Unity is Triune.

This is what I hear:

$a=b=c\\ a\neq b\\ a\neq c\\ c\neq b$

And then when I hear a Christian say something like this, which was on the Unbelievable facebook forum,

We […] stress the Trinity a lot, not only because we think it is true but because it explains some things.

For one thing, it means that the ideas of personal relationship and love are built into existence and have always been. For another it helps us understand how God can be completely transcendent (the Father) and yet reveal Himself to us (via the Word and Spirit).

What I hear is:

Not only do I think
$a=b=c\\ a\neq b\\ a\neq c\\ c\neq b$
is true, but it explains a number of things. For one thing, it means that
$a+x=b+x$
is built into the system. For another it helps us understand how
$a-x\neq b-x$
and yet
$b-x=c-x$

So where am I going wrong here? Where does the analogy break? If you say that the Trinity is a “mystery”, that is simply an admission that the Trinity is incoherent. If you say that I am not using the word “is” correctly, then please enlighten me as to the proper definition, but make sure it is well defined.