On the strength of the finite intersection principle

We study the logical content of several maximality principles related to the finite intersection principle (F\IP) in set theory. Classically, these are all equivalent to the axiom of choice, but in the context of reverse mathematics their strengths vary: some are equivalent to \ACA over \RCA, while others are strictly weaker, and incomparable with \WKL. We show that there is a computable instance of F\IP all of whose solutions have hyperimmune degree, and that every computable instance has a solution in every nonzero c.e.\ degree. In terms of other weak principles previously studied in the literature, the former result translates to F\IP implying the omitting partial types principle ($\mathsf{OPT}$). We also show that, modulo $\Sigma^0_2$ induction, F\IP lies strictly below the atomic model theorem ($\mathsf{AMT}$).Comment: This paper corresponds to section 3 of arXiv:1009.3242, "Reverse mathematics and equivalents of the axiom of choice", which has been abbreviated and divided into two pieces for publicatio

Depth, Highness and DNR Degrees

A sequence is Bennett deep [5] if every recursive approximation of the Kolmogorov complexity of its initial segments from above satisfies that the difference between the approximation and the actual value of the Kolmogorov complexity of the initial segments dominates every constant function. We study for different lower bounds r of this difference between approximation and actual value of the initial segment complexity, which properties the corresponding r(n)-deep sets have. We prove that for r(n) = εn, depth coincides with highness on the Turing degrees. For smaller choices of r, i.e., r is any recursive order function, we show that depth implies either highness or diagonally-non-recursiveness (DNR). In particular, for left-r.e. sets, order depth already implies highness. As a corollary, we obtain that weakly-useful sets are either high or DNR. We prove that not all deep sets are high by constructing a low order-deep set. Bennett's depth is defined using prefix-free Kolmogorov complexity. We show that if one replaces prefix-free by plain Kolmogorov complexity in Bennett's depth definition, one obtains a notion which no longer satisfies the slow growth law (which stipulates that no shallow set truth-table computes a deep set); however, under this notion, random sets are not deep (at the unbounded recursive order magnitude). We improve Bennett's result that recursive sets are shallow by proving all K-trivial sets are shallow; our result is close to optimal. For Bennett's depth, the magnitude of compression improvement has to be achieved almost everywhere on the set. Bennett observed that relaxing to infinitely often is meaningless because every recursive set is infinitely often deep. We propose an alternative infinitely often depth notion that doesn't suffer this limitation (called i.o. depth).We show that every hyperimmune degree contains a i.o. deep set of magnitude εn, and construct a π01- class where every member is an i.o. deep set of magnitude εn. We prove that every non-recursive, non-DNR hyperimmune-free set is i.o. deep of constant magnitude, and that every nonrecursive many-one degree contains such a set

Reduction games, provability and compactness

Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between [Formula: see text] principles over [Formula: see text]-models of [Formula: see text]. They also introduced a version of this game that similarly captures provability over [Formula: see text]. We generalize and extend this game-theoretic framework to other formal systems, and establish a certain compactness result that shows that if an implication [Formula: see text] between two principles holds, then there exists a winning strategy that achieves victory in a number of moves bounded by a number independent of the specific run of the game. This compactness result generalizes an old proof-theoretic fact noted by H. Wang (1981), and has applications to the reverse mathematics of combinatorial principles. We also demonstrate how this framework leads to a new kind of analysis of the logical strength of mathematical problems that refines both that of reverse mathematics and that of computability-theoretic notions such as Weihrauch reducibility, allowing for a kind of fine-structural comparison between [Formula: see text] principles that has both computability-theoretic and proof-theoretic aspects, and can help us distinguish between these, for example by showing that a certain use of a principle in a proof is “purely proof-theoretic”, as opposed to relying on its computability-theoretic strength. We give examples of this analysis to a number of principles at the level of [Formula: see text], uncovering new differences between their logical strengths. </jats:p