proving there is no proof of existence

Is it possible to give a proof there is no "explanation" of existence?

That is, what is the widest class of "explanations for existence" that can be ruled out?

Here are some thoughts.

1. you can prove
(there exists x) (x = y)
in usual 1st-order logic. This asserts something exists. Of course this is not helpful, because the only reason we can prove it is that we assumed the existence of something in the model of the language in the first place.

Still, this is not exactly a proof there is no sort of explanation whatsoever.

For example, call the language in which we can prove (there exists x) (x = y) L1, and call its model M1. Now take both (some unspecified representation of) L1 together with M1 as a new model, call it M2. Construct the theory of M2, L2. Let M3 be M2 together with L2, etc... Each stage in the sequence is justified by the one after it. The problem is, what justifies the sequence as a whole? But this is an odd problem: it seems to assume the existence of a completed infinity *before* it allows the conclusion there is no point along this sequence at which the existence of something may be said to be justified. But why would we need the existence, in whatever sense, of a completed infinity, to prove there no no explanation for existence? That seems like a it would be an obscure reason for the phenomena.

2. Suppose, for the sake of argument, that the universe is mathematical structure "all the way down". In this scenario, the stuff that the physical universe is ultimately composed of is mathematical structure. Mathematical structure can effectively be modeled by the interrelations of sentences of a formal language. So suppose what the universe is, ultimately, is (the sentences of) some formal language T. Now, some formal languages can refer to themselves. Suppose T is one of these languages. Suppose also that T incorporates the necessity modal operator.

The point is this. If T is such that it proves the theorem
T implies necessarily-exists-T
then it would be a *physical* circumstance that the universe necessarily exists.

The problem is, this interesting property of T does not actually get us anywhere in justifying T's existence in the first place. C. pointed out to me there are many things that necessarily exist, if only they exist to begin with.

But again, the fact that this attempt at an explanation for existence fails, is not the same a proof there is no explanation for existence, for a sufficiently defined class of "explanations".