Dear Jochen,
Yes, that is the main question. We actually have no logical support for believing in anything, except in a probabilistic way. While I agree there is theory-ladenness of data, I do not think it is an essential limit as many others I discuss. For me, in essence, it is a problem of translation. Different theories provide us with their own languages we use to interpret things. But we can always revert to the most basic description of the data if we really must. Suppose someone would propose some crazy theory with none of our fundamental particles. That would indeed make it hard to describe many basic experiments. But hard is not impossible. Instead of saying we have observed an electron, for example, we can say we have seen some patterns on a computer screen.
I do know about Solomonoff and have mentioned SI at the end of page 7. The essay size limits did not allow me to say more, however. I do discuss it in length, in a chapter by itself, in my coming book, "Arguments, Cognition, and Science" - https://rowman.com/ISBN/9781786615077/Arguments-Cognition-and-Science-Consequences-of-Probabilistic-Induction-in-Science
Hutter AIXI approach, however, is new, and I will need to check it. Thanks a lot for the reference.
I did take a look at your essay, and I found the concepts of finiteness and extensibility quite interesting. Quantum mechanics does seem to work as you describe. You also did an outstanding job of explaining the fundamental problems in measurement. I did enjoy reading it.
Cheers,
André