From the “weird math questions” department...

Here’s something I’ve been wondering about, in the context of Solomonoff induction and uncomputable sequences.

I have a device that is either a halting oracle, or an ordinary Turing machine which gives the correct answer to the halting problem for all programs smaller than some finite length N but always outputs “does not halt” when asked to evaluate programs larger than N. If you don’t know what N is and you don’t have infinite time, is there a way to tell the difference between the actual halting oracle (which gives correct answers for all possible programs) and a “fake” halting oracle which starts giving wrong answers for some N that just happens to be larger than any program that you’ve tested so far?

The Kolmogorov complexity of an uncomputable sequence is infinite, so Solomonoff induction assigns it a probability of zero, but there’s always a computable number with less than epsilon error, so would this ever actually matter?