The concept GOD is the concept of a being that cannot be constituted in consciousness in Husserl's sense of 'constitution,' a being that cannot be a transcendence-in-immanence, but must be absolutely transcendent, transcendent in itself, not merely for us.  It follows that there cannot be a phenomenology of God. At best, there can be a phenomenology of such of our experiences as purport to be of or about God.

We know that the concept GOD is the concept of something absolutely transcendent, and we know this by purely conceptual means. We have the concept GOD and we analyze it: we simply unpack its meaning. Whatever the origin of this concept, it is there in us and available for analysis. Of course, we cannot learn by conceptual analysis that God exists, but we can know something about what God cannot be like, if he does exist.  We can know, for example, that God, if he exists, is not a concept.  No surprise here, and nothing that distinguishes God from my chair, since my chair is not a concept either.  (One cannot sit on a concept.)  The difference between the concept CHAIR and the concept GOD is the difference between an ordinary concept and a limit concept (Grenzbegriff).* 

This is the distinction between those concepts that can capture (mirror, represent) the essences or natures of the things of which they are the concepts, and those concepts that cannot. Call the first type ordinary concepts and the second limit concepts. Thus the concept CUBE captures the essence of every cube, which is to be a three-dimensional solid bounded by six square faces or sides with three meeting at each vertex, and it captures this essence fully.   The concept HELIOTROPIC PLANT captures, partially,  the essence of those plants that exhibit diurnal or seasonal motion of plant parts in response to the direction of the sun.  Concepts are mental representations.  Essences are extra-mental.

Now the concept GOD cannot be ordinary since this concept cannot capture the essence of God. For (i) in God essence and existence are one, and (ii) there is no ordinary concept of existence.

Ad (i). That in God essence and existence are one follows from the fact that nothing could count as the Absolute if it were a composite of essence and existence.  And we know by conceptual analysis that God is the Absolute: the concept GOD is the concept of 'something' absolute.  This is the case whether or not God exists.

Ad (ii).  When I say that there is no ordinary concept of existence, I mean that there is no ordinary (non-limit) concept that is adequate to existence. (There are bogus concepts of existence such as Quine's.) There is no ordinary (non-limit) concept of existence because the existence of a thing, as other than its essence, cannot be conceptualized.  Why not? 

This is because each existing thing has its own existence.  Thus the existence of Al is Al's existence, the existence of Bob is Bob's, and the existence of Carla is Carla's.  For the existence of a thing is that which makes that very thing exist.  Existence cannot be a property like being human, being sentient, being sunburned.  These properties are multiply instantiable; existence, however, is not multiply instantiable. There are no instances of existence.  

Now if each thing has its own existence, then existence is implicated in the irreducible singularity of each existing thing. Irreducible singularity, in turn, cannot be conceptualized by minds like ours which trade only in the general and multiply instantiable. It's an Aristotelian point. If Aristotle wrote in Latin it would go: individuum qua individuum ineffabile est. The individual as such, the singular as such,  is ineffable and cannot be conceptualized.  The Peripatetic tells us that science is never of the particular as particular but only of the particular as exhibiting general or repeatable features. The particular as such is unrepeatable.   But of course there are no individuals (particulars) bare of properties. Every finite individual is a this-such. This is a law of metaphyica generalis. So, while the individual as individual cannot be conceptualized, the individual as bearer of properties can be conceptualized as an instance of those properties.  If  I think of Mary as an instance of lovable properties, then I abstract from the haecceity (thisness) that makes her different numerically from her indiscernible twin Sherry.  So if I love Mary precisely and only as an instance of lovable properties, then it will make no difference to my so loving her whether Mary or Sherry is its object. It will, however, make a difference to Mary. "I want that you should love me for what makes me me, and not for what I have in common with her!"  I explain this all in great detail in Do We Love the Person or Only Her Qualities?

The crucial point here is that when we think of an individual as an instance of properties, we abstract from (leave out of consideration) the individual's thisness and its existence.  I am not saying that the existence and the thisness of a concrete individual are one and the same; I am saying that that they go together as a matter of metaphysical necessity.  

Ad (i + ii).  In God there is no real distinction between existence and nature. That was the first point. The second was that no ordinary (non-limit) concept captures the individuality of the thing of which it is the concept. Therefore, since God is (identically) his nature, there can be no ordinary concept of God, whence it follows frat GOD is a limit concept.

There is, then, a  clear sense in which God is unconceptualizable or unbegreiflich: he cannot be grasped by the use of any ordinary concept. But it doesn't follow that we have no concept of God.  We do. The concept GOD is a limit concept: it is the concept of something that cannot be grasped using ordinary concepts.  Our cognitive architecture is such that we can grasp only the general, the repeatable, but never the irreducibly singular.  The concept GOD, however, is the concept of 'something' absolutely and irreducibly singular.  God is one without a second, one without even the possibility of a second. Any god that doesn't satisfy this metaphysical exigency just isn't worth his salt.

The concept GOD is the concept of something that lies at the outer limit of discursive intelligibility, and indeed just beyond that limit. We can argue up to this Infinite Object/Subject, but then discursive operations must cease. We cannot penetrate the divine essence since this essence is one with existence, and existence cannot be conceptually penetrated. We can however point to God, in a manner of speaking, using limit concepts. The concept GOD is the concept of an infinite, absolute and wholly transcendent reality whose realitas formalis so exceeds our powers of understanding that it cannot be taken up into the realitas objectiva of any of our ordinary concepts.

Now if you have followed that, then you are in a position to see that the following objection is a 'cheap shot' easily dismissed.  "You contradict yourself. You say that God cannot be conceptualized but at the same time you operate with a concept of God as unconceptualizable."  But no contradiction arises once we distinguish ordinary from limit concepts. 

If the critic accuses me of inventing a distinction ad hoc to save the ineffability and transcendence of God, then my reply will be that there are numerous other examples of limit concepts.  See the aptly appellated category, Limit Concepts. 

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*The term Grenzbegriff first enters philosophy in 1781 in Kant's Critique of Pure Reason. Curiously, he uses the term only once in the works he himself published. The term surfaces a few more times in his Nachlass.  The sole passage in the published works is at A255/B311 where Kant remarks that the concept noumenon is a Grenzbegriff.