In this paper we deal with the model theory of differentially closed fields of characteristic zero with finitely many commuting derivations. First we observe that the only known lower bound for the Lascar rank of types in differentially closed fields, announced in a paper of McGrail, is false. This gives us a new class of regular types which are orthogonal to fields. Then we classify the subgroups of the additive group of Lascar rank omega with differential-type 1 which are nonorthogonal to fields. The last parts consist of an analaysis of the quotients of the heat variety. We show that the generic type of such a quotient is locally modular. Finally, we answer a question of Phylliss Cassidy about the existence of certain Jordan—Hölder type series in the negative.

We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space over a finite field.

In 1984, Henson and Rubel [2] proved the following theorem: If p(x₁,…,x_n) is an exponential polynomial with coefficients in ℂ with no zeroes in ℂ, then p(x₁,…,x_n)= e^{g(x₁,…,x_n)} where g(x₁,…,x_n) is some exponential polynomial over ℂ. In this paper, I will prove the analog of this theorem for Zilber's Pseudoexponential fields directly from the axioms. Furthermore, this proof relies only on the existential closedness axiom without any reference to Schanuel's conjecture.

After showing the downwards density of nonhemimaximal degrees, Downey and Stob continued to prove that the existence of a low₂, but not low, nonhemimaximal degree, and their proof uses the fact that incomplete m-topped degrees are low₂ but not low. As commented in their paper, the construction of such a nonhemimaximal degree is actually a primitive 0''' argument. In this paper, we give another construction of such degrees, which is a standard 0''-argument, much simpler than Downey and Stob's construction mentioned above.

Recently, the Dimension Problem for effective Hausdorff dimension was solved by J. Miller in [14], where the author constructs a Turing degree of non-integral Hausdorff dimension. In this article we settle the Dimension Problem for effective packing dimension by constructing a real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one (on the other hand, it is known via [10, 3, 7] that every real of strictly positive effective Hausdorff dimension computes reals whose effective packing dimensions are arbitrarily close to, but not necessarily equal to, one).

If the Bounded Proper Forcing Axiom BPFA holds, then Mouse Reflection holds at ℵ₂ with respect to all mouse operators up to the level of Woodin cardinals in the next ZFC—model. This yields that if Woodin's ℛ_max axiom (*) holds, then BPFA implies that V is closed under the “Woodin-in-the-next-ZFC—model” operator. We also discuss stronger Mouse Reflection principles which we show to follow from strengthenings of BPFA, and we discuss the theory BPFA plus “NS_ω₁ is precipitous” and strengthenings thereof. Along the way, we answer a question of Baumgartner and Taylor, [2, Question 6.11].

In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called “uniform definability of types over finite sets” (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.

We shall show the consistency of CH+¬(+) and CH+(+)+there are no club guessing sequences on ω₁. We shall also prove that ♢⁺ does not imply the existence of a strong club guessing sequence on ω₁.

In his book on ℛ_max [7], Woodin presents a collection of partial orders whose extensions satisfy strong club guessing principles on ω₁. In this paper we employ one of the techniques from this book to produce ℛ_max variations which separate various club guessing principles. The principle (+) and its variants are weak guessing principles which were first considered by the second author [4] while studying games of length ω₁. It was shown in [1] that the Continuum Hypothesis does not imply (+) and that (+) does not imply the existence of a club guessing sequence on ω₁. In this paper we give an alternate proof of the second of these results, using Woodin's ℛ_max technology, showing that a strengthening of (+) does not imply a weakening of club guessing known as the Interval Hitting Principle. The main technique in this paper, in addition to the standard ℛ_max machinery, is the use of condensation principles to build suitable iterations.

The theory of valued difference fields (K,σ,v) depends on how the valuation v interacts with the automorphism σ. Two special cases have already been worked out - the isometric case, where v(σ(x)) = v(x) for all x∈K, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon; and the contractive case, where v(σ(x)) > n v(x) for all x∈ K^× with v(x) > 0 and n∈ℕ, has been worked out by Salih Azgin. In this paper we deal with a more general version, the multiplicative case, where v(σ(x)) = ρ· v(x), where ρ (> 0) is interpreted as an element of a real-closed field. We give an axiomatization and prove a relative quantifier elimination theorem for this theory.

Given a sequence { α_n } in (0,1) converging to a rational, we examine the model theoretic properties of structures obtained as limits of Shelah—Spencer graphs G(m, m^-α_n). We show that in most cases the model theory is either extremely well-behaved or extremely wild, and characterize when each occurs.

Rationals and countable ordinals are important examples of structures with decidable monadic second-order theories. A chain is an expansion of a linear order by monadic predicates. We show that if the monadic second-order theory of a countable chain C is decidable then C has a non-trivial expansion with decidable monadic second-order theory.

We prove that RCA₀+RT₂² ⇸ WKL₀ by showing that for any set C not of PA-degree and any set A, there exists an infinite subset G of A or A̅, such that G⊕C is also not of PA-degree.

We introduce a very weak language ℒ_M on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language ℒ_M are trivial functions. We also give a definitional expansion ℒ_M' of ℒ_M in which K has quantifier elimination, and we obtain a cell decomposition result for ℒ_M-definable sets.

Our language ℒ_M can serve as a p-adic analogue of the very weak language (<) on the real numbers, to define a notion of minimality on the field of p-adic numbers and on related valued fields. These fields are not necessarily Henselian and may have positive characteristic.

We introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy—Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable set can be partitioned into “basic rectifiable sets”, and that the Whitney arc property holds for basic rectifiable sets.

We prove that:

• if there is a model of IΔ₀ + ¬exp with cofinal Σ₁-definable elements and a Σ₁ truth definition for Σ₁ sentences, then IΔ₀ + ¬exp + ¬ BΣ₁ is consistent,

• there is a model of IΔ₀ + Ω₁ + ¬exp with cofinal Σ₁-definable elements, both a Σ₂ and a Π₂ truth definition for Σ₁ sentences, and for each n ≥ 2, a Σ_n truth definition for Σ_n sentences.

We also present an old but previously unpublished proof of the consistency of IΔ₀ + ¬exp + ¬ BΣ₁ under the assumption that the size parameter in Lessan's Δ₀ universal formula is optimal. We then discuss a possible reason why proving the consistency of IΔ₀ + ¬exp + ¬ BΣ₁ unconditionally has turned out to be so difficult.

We present sharpened lower bounds on the size of cut free proofs for first-order logic. Prior lower bounds for eliminating cuts from a proof established superexponential lower bounds as a stack of exponentials, with the height of the stack proportional to the maximum depth d of the formulas in the original proof. Our results remove the constant of proportionality, giving an exponential stack of height equal to d-O(1). The proof method is based on more efficiently expressing the Gentzen—Solovay cut formulas as low depth formulas.

We consider the following problem: Given a proof of the Skolemization of a formula F, what is the length of the shortest proof of F? For the restriction of this question to cut-free proofs we prove corresponding exponential upper and lower bounds.

We propose a probabilistic interpretation of Propositional Dynamic Logic (PDL). We show that logical and behavioral equivalence are equivalent over general measurable spaces. This is done first for the fragment of straight line programs and then extended to cater for the nondeterministic nature of choice and iteration, expanded to PDL as a whole. Bisimilarity is also discussed and shown to be equivalent to logical and behavioral equivalence, provided the base spaces are Polish spaces. We adapt techniques from coalgebraic stochastic logic and point out some connections to Souslin's operation 𝒜 from descriptive set theory. This leads to a discussion of complete stochastic Kripke models and model completion, which permits an adequate treatment of the test operator.

We address the following question: Can we expand an NIP theory by adding a linear order such that the expansion is still NIP? Easily, if acl(A)=A for all A, then this is true. Otherwise, we give counterexamples. More precisely, there is a totally categorical theory for which every expansion by a linear order has IP. There is also an ω-stable NDOP theory for which every expansion by a linear order interprets pseudofinite arithmetic.