I found a lot of these questions around non-computability became a lot intuitively to clearer once I read a good deal about Busy-Beaverology (which is also fascinating, well worth climbing down the rabbit-hole for a while). They’re currently only up to about 6-state Turing Machines, and they’re already found things for which the question of whether they halt or not depends on currently-unproven-and-notoriously-hard mathematical problems (so far, generally variations on the Coltaz conjecture), and also things that provably do halt but whose runtimes require Knuth’s uparrow notation to even write down within the size of the physical Universe. Extrapolate that from N=6 to N=>infinity and that shows you how hairy non-computable problems really are. Nevertheless, the enormous majority of the state machines they find either clearly halt or clearly run forever, this is easily provable, and if they run forever they do something quire boring like loop, or head steadily in on direction. Then there are categories of trickier ones. The hard ones are rare, but they also get dramatically harder, and there is simply no limit to how hard they can get. It’s known that for a sufficiently large number of Tiring machine states there are ones where whether or not they halt is actually unprovable within widely used axiom systems (and their runtimes even if they do can easily be such that just running them to see is physically impossible before the heat death of the universe), and it’s strongly hypothesized that this may actually be the case for state counts not hugely larger than the Beaverologists have already reached.
So my intuition is that for the great majority of problems (by any “reasonable measure”), most finite approximations to Solomonoff induction (over a certain problem-dependent capacity level) will do just fine, for relatively boring reasons. But for any given capacity level, problems exist that are too hard for it — they simply become rarer and harder to find as capacity increases. Nevertheless, no matter how good an approximation to Solomonoff induction you are, there exist problems that it can solve but that you can’t — they’re just rare. That’s why you’re computable and it’s not.
I found a lot of these questions around non-computability became a lot intuitively to clearer once I read a good deal about Busy-Beaverology (which is also fascinating, well worth climbing down the rabbit-hole for a while). They’re currently only up to about 6-state Turing Machines, and they’re already found things for which the question of whether they halt or not depends on currently-unproven-and-notoriously-hard mathematical problems (so far, generally variations on the Coltaz conjecture), and also things that provably do halt but whose runtimes require Knuth’s uparrow notation to even write down within the size of the physical Universe. Extrapolate that from N=6 to N=>infinity and that shows you how hairy non-computable problems really are. Nevertheless, the enormous majority of the state machines they find either clearly halt or clearly run forever, this is easily provable, and if they run forever they do something quire boring like loop, or head steadily in on direction. Then there are categories of trickier ones. The hard ones are rare, but they also get dramatically harder, and there is simply no limit to how hard they can get. It’s known that for a sufficiently large number of Tiring machine states there are ones where whether or not they halt is actually unprovable within widely used axiom systems (and their runtimes even if they do can easily be such that just running them to see is physically impossible before the heat death of the universe), and it’s strongly hypothesized that this may actually be the case for state counts not hugely larger than the Beaverologists have already reached.
So my intuition is that for the great majority of problems (by any “reasonable measure”), most finite approximations to Solomonoff induction (over a certain problem-dependent capacity level) will do just fine, for relatively boring reasons. But for any given capacity level, problems exist that are too hard for it — they simply become rarer and harder to find as capacity increases. Nevertheless, no matter how good an approximation to Solomonoff induction you are, there exist problems that it can solve but that you can’t — they’re just rare. That’s why you’re computable and it’s not.