Lacking: A Proof there's No Explanation for Existence
Philosophers have tried to construct explanations for the existence of our universe for at least 2500 years, unsuccessfully. Yet we lack something like a proof there is no explanation for existence, for a sufficiently wide class of "explanations". Such a proof would at least indicate where not to look for an explanation, and almost certainly clarify why none have been forthcoming. I don't have such a proof, but would like to call attention to the conspicuous lack of one.
Here are four would-be explanations. (References welcome.)
1. Allow, for the sake of argument, that what physical matter is is ultimately mathematical structure. The universe is mathematical structure "all the way down". Some formal mathematical theories T can prove things about themselves. So suppose our universe is a mathematical theory T such that
T implies necessarily(there exists(T))
In this case, it would be a physical fact that the existence of the universe was logically inevitable (necessary). The laws of physics would explain existence.
The problem is, there's no reason to postulate the existence of T in the first place. So it doesn't work.
2. It could be that everything that's not inconsistent exists. Okay, but this doesn't explain why any of it exists in the first place.
3. Two unicorns don't exist, so their 1st-order properties are not instantiated. Nevertheless, it's true that they both potentially have one horn. But then something exists in virtue of which we can declare there is a sense in which the unicorns have a common property.
4. That a particular unicorn doesn't exist implies the possibility of the existence of the unicorn, so at least the possibility exists. Interestingly a possibility is in a different ontology then the objects that possibility exist.
All four fail. What's still lacking is a proof, if there is one, of why such explanations will inevitably fail. The place to start would be to make definitions such as
(1) an explanation1 for the existence of something is a formal mathematical argument whose axioms do not assume the existence of anything, but which reaches the conclusion that something exists
(2) an explanation2 for the existence of something is a mathematical argument that makes no assumptions whatsoever (has no axioms)
So far as (1) is concerned, existential-introduction of a particular a in 1st-order logic is valid, roughly, if I understand it, when there is something that's true of a, a is the value of a variable (Quine), or a has a property. None of these help, since they assume the existence of something that is "about a". But this isn't a proof there is no explanation.
So far as (2) is concerned I don't know how to get a mathematical explanation off the ground if it has no axioms. But that's not a proof, either.
Some would argue that there can be no explanation because it would require the assumption of a formal apparatus for demonstrating one. But this is unconvincing: if there is one there's nothing wrong with supposing there is also the apparatus for showing it.
Existence is not supposed to be a 1st-order property. So the nature of existence might be like this. The difference between Bubbles the cat (who does exist) and Bonzai the unicorn (who doesn't) is that there is a relation that Bubbles has to itself that Bonzai is lacking. This makes existence ontologically ineffable (maybe), in the language of the previous posts.
In conclusion, what's conspicuously lacking is a proof there is no explanation for existence, for the appropriate class(es) of "explanations".