For every natural number m, the existentially closed models of the theory of fields with m commuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential fields has a model-companion. The axioms are that certain differential varieties determined by certain ordinary varieties are nonempty. There is no restriction on the characteristic of the underlying field.

By the complexity of a finite sequence of 0’s and 1’s we mean the Kolmogorov complexity, that is the length of the shortest input to a universal recursive function which returns the given sequence as output. By initial segment complexity of an infinite sequence of 0’s and 1’s we mean the asymptotic behavior of the complexity of its finite initial segments. In this paper, we construct infinite sequences of 0’s and 1’s with given recursive lower bounds on initial segment complexity which do not compute any infinite sequences of 0’s and 1’s with a significantly larger recursive lower bound on initial segment complexity. This improves several known results about randomness extraction and separates many natural degrees in the lattice of Muchnik degrees.

Let ${\cal N}\left( G \right)$ be the maximal normal definable torsion-free subgroup of a group G definable in an o-minimal structure M. We prove that the quotient $G/{\cal N}\left( G \right)$ has a maximal definably compact subgroup K, which is definably connected and unique up to conjugation. Moreover, we show that K has a definable torsion-free complement, i.e., there is a definable torsion-free subgroup H such that $G/{\cal N}\left( G \right) = K \cdot H$ and $K\mathop \cap \nolimits^ \,H = \left\{ e \right\}$. It follows that G is definably homeomorphic to $K \times {M^s}$ (with $s = {\rm{dim}}\,G - {\rm{dim}}\,K$), and homotopy equivalent to K. This gives a (definably) topological reduction to the compact case, in analogy with Lie groups.

Fix a language extending the language of ordered fields by at least one new predicate or function symbol. Call an L-structure R pseudo-o-minimal if it is (elementarily equivalent to) an ultraproduct of o-minimal structures. We show that for any recursive list of L-sentences , there is a real closed field satisfying which is not pseudo-o-minimal. This shows that the theory To−min consisting of those -sentences true in all o-minimal -structures, also called the theory of o-minimality (for L), is not recursively axiomatizable. And, in particular, there are locally o-minimal, definably complete expansions of real closed fields which are not pseudo-o-minimal.

We study computably enumerable equivalence relations (ceers), under the reducibility $R \le S$ if there exists a computable function f such that $x\,R\,y$ if and only if $f\left( x \right)\,\,S\,f\left( y \right)$, for every $x,y$. We show that the degrees of ceers under the equivalence relation generated by $\le$ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first-order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if $R\prime \le R$, where $R\prime$ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are ${\rm{\Sigma }}_3^0$-complete (the former answering an open question of Gao and Gerdes).

We characterize the effective content and the proof-theoretic strength of a Ramsey-type theorem for bi-colorings of so-called exactly large sets. An exactly large set is a set $X \subset {\bf{N}}$ such that ${\rm{card}}\left( X \right) = {\rm{min}}\left( X \right) + 1$. The theorem we analyze is as follows. For every infinite subset M of N, for every coloring C of the exactly large subsets of M in two colors, there exists and infinite subset L of M such that C is constant on all exactly large subsets of L. This theorem is essentially due to Pudlák and Rödl and independently to Farmaki. We prove that—over RCA0 —this theorem is equivalent to closure under the ωth Turing jump (i.e., under arithmetical truth). Natural combinatorial theorems at this level of complexity are rare. In terms of Reverse Mathematics we give the first Ramsey-theoretic characterization of ${\rm{ACA}}_0^ +$. Our results give a complete characterization of the theorem from the point of view of Computability Theory and of the Proof Theory of Arithmetic. This nicely extends the current knowledge about the strength of Ramsey’s Theorem. We also show that analogous results hold for a related principle based on the Regressive Ramsey’s Theorem. We conjecture that analogous results hold for larger ordinals.

Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, thus saturating any stable theory. We then prove directly that a “bottleneck” in the inductive construction of a regular ultrafilter on λ (i.e., a point after which all antichains of ${\cal P}\left( \lambda \right)/{\cal D}$ have cardinality less than λ) essentially prevents any subsequent ultrafilter from being flexible, thus from saturating any nonlow theory. The constructions are as follows. First, we construct a regular filter ${\cal D}$ on λ so that any ultrafilter extending ${\cal D}$ fails to ${\lambda ^ + }$-saturate ultrapowers of the random graph, thus of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a measurable cardinal κ, we construct a regular ultrafilter on $\lambda > \kappa$ which is λ-flexible but not ${\kappa ^{ + + }}$-good, improving our previous answer to a question raised in Dow (1985). Third, assuming a weakly compact cardinal κ, we construct an ultrafilter to show that ${\rm{lcf}}\left( {{\aleph _0}} \right)$ may be small while all symmetric cuts of cofinality κ are realized. Thus certain families of precuts may be realized while still failing to saturate any unstable theory.

We establish several results regarding dividing and forking in NTP2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking ideal is S1, in Hrushovski’s terminology). Using it we prove an independence theorem over extension bases (which, in the case of simple theories, specializes to the ordinary independence theorem). As an application we show that Lascar strong type and compact strong type coincide over extension bases in an NTP2 theory.

We also define the dividing order of a theory—a generalization of Poizat’s fundamental order from stable theories—and give some equivalent characterizations under the assumption of NTP2. The last section is devoted to a refinement of the class of strong theories and its place in the classification hierarchy.

We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.

If X0 and X1 are both generic, the theories of the degrees below X0 and X1 are the same. The same is true if both are random. We show that the n-genericity or n-randomness of X do not suffice to guarantee that the degrees below X have these common theories. We also show that these two theories (for generics and randoms) are different. These results answer questions of Jockusch as well as Barmpalias, Day and Lewis.

We prove two results about generically stable types p in arbitrary theories. The first, on existence of strong germs, generalizes results from [2] on stably dominated types. The second is an equivalence of forking and dividing, assuming generic stability of p(m) for all m. We use the latter result to answer in full generality a question posed by Hasson and Onshuus: If P(x) ε S(B) is stable and does not fork over A then prestrictionA is stable. (They had solved some special cases.)

We characterize the ω-stable theories all of whose countable models admit decidable presentations. In particular, we show that for a countable ω-stable T, every countable model of T admits a decidable presentation if and only if all n-types in T are recursive and T has only countably many countable models. We further characterize the decidable models of ω-stable theories with countably many countable models as those which realize only recursive types.

An inaccessible cardinal is strongly compact if, and only if, it satisfies thestrong tree property. We prove that if there is a model of ZFC with infinitelymany supercompact cardinals, then there is a model of ZFC where ${\aleph _{\omega + 1}}$ has the strong tree property. Moreover, we prove that everysuccessor of a singular limit of strongly compact cardinals has the strong treeproperty.

Hirsch and Hodkinson proved, for $3 \le m < \omega $ and any $k < \omega $, that the class $SNr_m {\bf{CA}}_{m + k + 1} $ is strictly contained in $SNr_m {\bf{CA}}_{m + k} $ and if $k \ge 1$ then the former class cannot be defined by any finite set offirst-order formulas, within the latter class. We generalize this result to thefollowing algebras of m-ary relations for which the neat reductoperator $_m $ is meaningful: polyadic algebras with or without equality andsubstitution algebras. We also generalize this result to allow the case wherem is an infinite ordinal, using quasipolyadic algebras inplace of polyadic algebras (with or without equality).

Criteria are obtained for a filter ${\cal F}$ of subsets of a set I to be an intersection of finitely many ultrafilters, respectively, finitely many κ-complete ultrafilters for a given uncountable cardinal $\kappa .$ From these, general results are deduced concerning homomorphisms on infinite direct product groups, which yield quick proofs of some results in the literature: the Łoś–Eda theorem (characterizing homomorphisms from a not-necessarily-countable direct product of modules to a slender module), and some results of Nahlus and the author on homomorphisms on infinite direct products of not-necessarily-associative k-algebras. The same tools allow other results of Nahlus and the author to be nontrivially strengthened, and yield an analog to one of their results, with nonabelian groups taking the place of k-algebras.

We briefly examine the question of how the common technique used in applying the general results of this note to k-algebras on the one hand, and to nonabelian groups on the other, might be extended to more general varieties of algebras in the sense of universal algebra.

In a final section, the Erdős–Kaplansky theorem on dimensions of vector spaces ${D^I}$$(D$ a division ring) is extended to reduced products ${D^I}/{\cal F},$ and an application is noted.

We examine the computable part of the differentiability hierarchy defined by Kechris and Woodin. In that hierarchy, the rank of a differentiable function is an ordinal less than ${\omega _1}$ which measures how complex it is to verify differentiability for that function. We show that for each recursive ordinal $\alpha > 0$, the set of Turing indices of $C[0,1]$ functions that are differentiable with rank at most α is ${{\rm{\Pi }}_{2\alpha + 1}}$-complete. This result is expressed in the notation of Ash and Knight.

An uncountable cardinal κ is called ${\omega _1}$-strongly compact if every κ-complete ultrafilter on any set I can be extended to an ${\omega _1}$-complete ultrafilter on I. We show that the first ${\omega _1}$-strongly compact cardinal, ${\kappa _0}$, cannot be a successor cardinal, and that its cofinality is at least the first measurable cardinal. We prove that the Singular Cardinal Hypothesis holds above ${\kappa _0}$. We show that the product of Lindelöf spaces is κ-Lindelöf if and only if $\kappa \ge {\kappa _0}$. Finally, we characterize ${\kappa _0}$ in terms of second order reflection for relational structures and we give some applications. For instance, we show that every first-countable nonmetrizable space has a nonmetrizable subspace of size less than ${\kappa _0}$.

We investigate the partial orderings of the form P(X),⊂〉, where X is a relational structure and P(X) the set of the domains of its isomorphic substructures. A rough classification of countable binary structures corresponding to the forcing-related properties of the posets of their copies is obtained.

We describe the Ziegler spectrum of a Bézout domain B=D+XQ[X] where D is a principal ideal domain and Q is its field of fractions; in particular we compute the Cantor–Bendixson rank of this space. Using this, we prove the decidability of the theory of B-modules when D is “sufficiently” recursive.

We define a functional interpretation of KPω using Howard’s primitive recursive tree functionals of finite type and associated terms. We prove that the Σ-ordinal of KPω is the least ordinal not given by a closed term of the ground type of the trees (the Bachmann-Howard ordinal). We also extend KPω to a second-order theory with Δ1-comprehension and strict-${\rm{\Pi }}_1^1$ reflection and show that the Σ-ordinal of this theory is still the Bachmann-Howard ordinal. It is also argued that the second-order theory is Σ1-conservative over KPω.