Yes, it seems that the generated sequence will always be outside the space of considered predictors. When the predictors are polynomial, the sequence can be exponential. When the predictors are algorithms in general, the sequence is uncomputable.
Would it be possible to make a space of all possible predictors? No, because the way the sequence is generated is… kinda like a probabilistic version of Cantor’s diagonal argument—only instead of contradicting the n-th prediction in n-th step, it tries to contradict the greatest mass of remaining predictions at each step. But that still means that each individual prediction will be contradicted at some moment, because if its prior probability is p, then at each step either this specific prediction is contradicted, or some other set of predictions with total probability at least p is contradicted, so we arrive at this specific prediction after at most 1/p steps.
Also, if we want to assign a nonzero prior probability to each prediction, we can only have a countable number of predictions. But there is an uncountable number of possible sequences. Therefore some of them (actually most of them) are outside the set of considered predictions.
Yes, it seems that the generated sequence will always be outside the space of considered predictors. When the predictors are polynomial, the sequence can be exponential. When the predictors are algorithms in general, the sequence is uncomputable.
Would it be possible to make a space of all possible predictors? No, because the way the sequence is generated is… kinda like a probabilistic version of Cantor’s diagonal argument—only instead of contradicting the n-th prediction in n-th step, it tries to contradict the greatest mass of remaining predictions at each step. But that still means that each individual prediction will be contradicted at some moment, because if its prior probability is p, then at each step either this specific prediction is contradicted, or some other set of predictions with total probability at least p is contradicted, so we arrive at this specific prediction after at most 1/p steps.
Also, if we want to assign a nonzero prior probability to each prediction, we can only have a countable number of predictions. But there is an uncountable number of possible sequences. Therefore some of them (actually most of them) are outside the set of considered predictions.