Journal of Mathematical Logic

A ternary relation between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be closely related to modular pairs in the lattice of algebraically closed sets.

We introduce the notion of a preindependence relation between subsets of the big model of a complete first-order theory, an abstraction of the properties which numerous concrete notions such as forking, dividing, thorn-forking, thorn-dividing, splitting or finite satisfiability share in all complete theories. We examine the relation between four additional axioms (extension, local character, full existence and symmetry) that one expects of a good notion of independence.

We show that thorn-forking can be described in terms of local forking if we localize the number k in Kim's notion of "dividing with respect to k" (using Ben-Yaacov's "k-inconsistency witnesses") rather than the forking formulas. It follows that every theory with an M-symmetric lattice of algebraically closed sets (in T^eq) is rosy, with a simple lattice theoretical interpretation of thorn-forking.

Let ξ ≥ 1 be a countable ordinal. We study the Borel subsets of the plane that can be made by refining the Polish topology on the real line. These sets are called potentially . We give a Hurewicz-like test to recognize potentially sets.

We characterize rings over which every cotorsion module is pure injective (Xu rings) in terms of certain descending chain conditions and the Ziegler spectrum, which renders the classes of von Neumann regular rings and of pure semisimple rings as two possible extremes. As preparation, descriptions of pure projective and Mittag–Leffler preenvelopes with respect to so-called definable subcategories and of pure generation for such are derived, which may be of interest on their own. Infinitary axiomatizations lead to coherence results previously known for the special case of flat modules. Along with pseudoflat modules we introduce quasiflat modules, which arise naturally in the model-theoretic and the category-theoretic contexts.

The complexity class of -polynomial local search (PLS) problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k + 1, the -definable functions of are characterized in terms of -PLS problems. These -PLS problems can be defined in a weak base theory such as , and proved to be total in . Furthermore, the -PLS definitions can be skolemized with simple polynomial time functions, and the witnessing theorem itself can be formalized, and skolemized, in a weak base theory. We introduce a new -principle that is conjectured to separate and .

The tree property at κ⁺ states that there are no Aronszajn trees on κ⁺, or, equivalently, that every κ⁺ tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ⁺ and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper, we reconcile the two. We prove from large cardinals that the tree property at κ⁺ is consistent with failure of the singular cardinal hypothesis at κ.