Certainly there is no proof that A can be explained; what I am doing in my presentation is laying out a road map of the best we can possibly do (from a statistical perspective) given what we know (whatever that might be)!
You are assuming that a particular element of A is referable and this, I think, is an incorrect assumption. From a statistical perspective, you haven't given a shred of evidence that you can predict new general features of A that we have yet to understand, so I have no clue as to why you think you have mastered A.
When I do that, I discover that, no matter if the given information is utterly meaningless random drivel, I produce a mechanism which will yield expectations exactly conforming to that "meaningless random drivel". And, I don't care what the demon is providing (lies, truth, meaningless drivel or whatever), given any information to go on, your best bet is to expect to see patterns in the new information similar to what you have already seen. My model reproduces the statistical patterns in the information available and provides one with rational expectations (that makes it an explanation under the definition given).
If that were true then you should be able to predict new information about A that is not currently accessible, but you do none of those things.
1) I am asking you to believe that mathematics is a logical structure: i.e., that any results of any mathematical deduction are as true as is the start position of the deduction. If you don't believe in that then you don't believe in the validity of mathematical deduction.
I won't say they are 'true' since they may be inventions (e.g., Tide detergent, yellow smiley faces, and those sticky notes that you take messages on). However, most scientists I think (and myself included, if that means anything) believe they are 'true' but scientists are not paid for their beliefs per pound as if this were a deli. The most you can say about mathematical deduction and the axioms they are based on is that it is necessary for science and that the results have been fruitful to say the least. They look 'true'. If you want to base a model on what looks true must be true, then you are only speculating from that point or you must ask us to suspend our disbelief until we see your results. In case of the latter, you need to show people why you want someone to suspend disbelief. That is, the results must be so overwhelming that we just have to take the 'leap of faith' and believe your model.
If you honestly think that my disagreement is being illogical, then no wonder we cannot see eye to eye.
2) That there exists something to explain which we do not understand! I call it A. If you don't believe in A then you have nothing which needs explanation.
I have no problem with this other than an explanation might be another thing we invent (like Tide and sticky notes and that candy that pops in your mouth when you eat it, I forgot the name though).
3) That there exists some information about A which is available to us. I call it C. If you don't believe in C then you have nothing on which to build an explanation of A.
As some perspective of where I'm coming from and why your views conflict with it, I believe A to require an infinite number of 'true' explanations to be properly understood, and by your definition, C is finite. A finite explanation of an infinite thing is going to be technically false. This is what you do not seem to grasp (which I tried to summons a demon to make you see... ah, to no avail).
Look at it like this. Let's say I have am holding an object and I'll give you 20 questions to find out what the object is. After 20 questions are finished you have a pretty good idea what the object is and you say "based on my 20 questions I think you are holding a book of crayons". I say "that's somewhat right, but no, it's not correct, I'll now give you 200 questions to find out what it is". You ask the 200 questions and then based on new terminology, new science you couldn't even imagine, you are ready to guess again. I say "no that's still not quite correct, look I'll give you a few million questions", and so on. You won't get an exact answer to the question on what object I am holding until you ask an infinite number of questions, but, this is something you cannot do since you can only ask a finite number of questions no matter how many times you ask. Hence, you will never know what I am holding, and any guess that you give will always be an 'approximate truth', or in the case of a demon who wants you to guess, he's just not providing you with set A so that you can guess, you are stuck with set 666.
Hence, the technical answer to your assumption is 'no', you cannot have any information in C which is about A. That is, the infinite number of questions that you never get to ask reduces your real knowledge of A for each question you never got to ask, hence that's an infinite amount of knowledge that you lack - or if you prefer the vanacular you no jack about A. All you know is what is for all practical purposes (FAPP) on what your finite number of questions gave you. But, knowing is not what I would call it. You have a sense or gut feeling of A, nothing more.
4) That there exists some part of C which we can use to evaluate our explanation. I call it B. If you don't believe in B then you have no way to defend an explanation of A.
B is also finite, so it suffers the same problem as C. It is a finite number of questions that are trying to explain something that requires an infinite number of explanations to be understood.
5) That there exists a way to refer to significant aspects of A, B, and C. If you believe there exist aspects of the problem which cannot be referred to, then I hold that you believe you cannot think about the problem.
You can only refer to some general properties of A that allow you to re-construct what you think are members in the set. That is, after your 20 questions you might have some general properties about A in which you feel you can construct some of the elements of A, however you'd be wrong in thinking you are actually referring to elements of A. Let me try yet another example. You might gather that A is an infinite set of integers (for example), and from that you deduce that A has inside a number of integers which you can refer to (1, 2, 3, etc). So, your finite set B is an integer set that maps a number of integers of A to one of your integers in B in some rough and approximate way. However, what you don't know is that A is actually a set of complex reals and that your finite collection of B elements is not matching with anything in A. In fact, you can't even identify one member in A since all your 20 questions allowed you to understand is that A is an integer set. You didn't get to the point in your understanding where you saw that A is a set of complex reals. And, of course, this is just an example. In reality, it would take an infinite number of questions to say that A is "some a set of complex reals" (or some other 'exact' property).
So, how can we really know anything about anything if A is not accessible? Well, fortunately for us we can view A not as it is, but as to how it appears. So, even without asking any questions, set A has an appearance to us. It looks like a number line for example. If we look at the numbered line a little closely, well, it looks like a line of integers, and so on. As we continue to investigate A we see the world slightly differently, but in a sense our main understanding of the world doesn't much change because the most important questions to ask come first (e.g., is it a numbered line?, etc). Even though we can never know A, we still feel a sense of comfort in our universe because it seems comprehensible for all practical purposes. If it weren't then we wouldn't be here in the first place.
But, occasionally someone will come along and talk about set A as something that can be understand in terms of some model, and oh boy, that's what you cannot do.