HIGHNESS PROPERTIES CLOSE TO PA COMPLETENESS

Suppose we are given a computably enumerable object. We are interested in the strength of oracles that can compute an object that approximates this c.e. object. It turns out that in many cases arising from algorithmic randomness or computable analysis, the resulting highness property is either close to, or equivalent to being PA complete. We examine, for example, majorizing a c.e. martingale by an oracle-computable martingale, computing lower bounds for two variants of Kolmogorov complexity, and computing a subtree of positive measure with no dead-ends of a given Pi(0)(1) tree of positive measure. We separate PA completeness from the latter property, called the continuous covering property. We also separate the corresponding principles in reverse mathematics.