Let T be an infinite pseudo-tree. In [2], we showed that the cellularity of the pseudo-tree algebra Treealg(T) was the maximum of four cardinals c_T, ι_T, φ_T, and μ_T: roughly, c_T is the “tallness” of T; ι_T is the “width” of T; φ is the number of “points of finite branching” in T; and μ is the number of “sections of no branching” in T. Here we ask: which inequalities among these four cardinals may be satisfied, in some sense, by a pseudo-tree? We show that the possible inequalities among c_T, ι_T, φ_T, and μ_T attained by pseudo-trees T are closely related to the existence of generalized Suslin trees.

Let $\mathcal{M} = \langle M, +, <, 0, \{\lambda\}_{\lambda \in D}\rangle$ be an ordered vector space over an ordered division ring $D$, and $G = \langle G, \oplus, e_G \rangle$ an $n$-dimensional group definable in $\mathcal{M}. We show that if $G$ is definably compact and definably connected with respect to the $t$-topology, then it is definably isomorphic to a ‘definable quotient group’ $U/L$, for some convex $\bigvee$-definable subgroup $U$ of $\langle M^n, + \rangle$ and a lattice $L$ of rank $n$. As two consequences, we derive Pillay’s conjecture for a saturated $\mathcal{M}$ as above and we show that the o-minimal fundamental group of $G$ is isomorphic to $L$.

We revisit the foundations of positive model theory, introducing h-inductive sentences. These allow a considerably simplified presentation of positive model theory, as well as a characterisation of Hausdorff cats by an amalgamation property of their $h$-inductive theory.

An amalgamation base p in a simple theory is stably definable if its canonical base is interdefinable with the set of canonical parameters for the φ-definitions of p as φ ranges through all stable formulae. A necessary condition for stably definability is given and used to produce an example of a supersimple theory with stable forking having types that are not stably definable. This answers negatively a question posed in [8]. A criterion for and example of a stably definable amalgamation base whose restriction to the canonical base is not axiomatised by stable formulae are also given. The examples involve generic relations over non CM-trivial stable theories.

By recent work on some conjectures of Pillay, each definably compact group $G$ in a saturated o-minimal expansion of an ordered field has a normal “infinitesimal subgroup” $G^{00}$ such that the quotient $G/G^{00}$, equipped with the “logic topology”, is a compact (real) Lie group. Our first result is that the functor $G \mapsto G/G^{00}$ sends exact sequences of definably compact groups into exact sequences of Lie groups. We then study the connections between the Lie group $G/G^{00}$ and the o-minimal spectrum $\widetilde{G}$ of $G$. We prove that $G/G^{00}$ is a topological quotient of $\widetilde{G}$. We thus obtain a natural homomorphism $\Psi^*$ from the cohomology of $G/G^{00}$ to the (Čech-)cohomology of $\widetilde{G}$. We show that if $G^{00}$ satisfies a suitable contractibility conjecture then $\widetilde{G^{00}}$ is acyclic in Čech cohomology and $\Psi^*$ is an isomorphism. Finally we prove the conjecture in some special cases.

This addendum to [2] shows that the set of tautological quantum logical propositional formulas for a finite dimensional vector space ℂⁿ is different for every n, affirmatively answering a question posed therein.

We consider three problems concerning alpha conversion of closed terms (combinators).

1. Given a combinator M find the an alpha convert of M with a smallest number of distinct variables.

2. Given two alpha convertible combinators M and N find a shortest alpha conversion of M to N.

3. Given two alpha convertible combinators M and N find an alpha conversion of M to N which uses the smallest number of variables possible along the way.

We obtain the following results.

1. There is a polynomial time algorithm for solving problem (1). It is reducible to vertex coloring of chordal graphs.

2. Problem (2) is co-NP complete (in recognition form). The general feedback vertex set problem for digraphs is reducible to problem (2).

3. At most one variable besides those occurring in both M and N is necessary. This appears to be the folklore but the proof is not familiar. A polynomial time algorithm for the alpha conversion of M to N using at most one extra variable is given.

There is a tradeoff between solutions to problem (2) and problem (3) which we do not fully understand.

In this paper we present two calculi for intuitionistic logic. The first one, IG, is characterized by the fact that every proof-search terminates and termination is reached without jeopardizing the subformula property. As to the second one, SIC, proof-search terminates, the subformula property is preserved and moreover proof-search is performed without any recourse to metarules, in particular there is no need to back-track. As a consequence, proof-search in the calculus SIC is accomplished by a single tree as in classical logic.

We study the problem of existence of maximal chains in the Turing degrees. We show that:

1. $ZF+DC+$“There exists no maximal chain in the Turing degrees” is equiconsistent with $ZFC+$“There exists an inaccessible cardinal”;

2. For all $a \in 2^{\omega}, (\omega_1)^{L[a]} = \omega_1$ if and only if there exists a $\Pi^1_1[a]$ maximal chain in the Turing degrees.

As a corollary, $ZFC$ + “There exists an inaccessible cardinal” is equiconsistent with $ZFC$ + “There is no (bold face) $\underset{\tilde{}}{\Sigma}^1_1$ maximal chain of Turing degrees”.

We show that in the models of 𝔲 < 𝔡 from [14] there are infinitely many near-coherence classes of ultrafilters, thus answering Banakh’s and Blass’ Question 30 of [3] negatively. By an unpublished result of Canjar, there are at least two classes in these models.

Asymptotic extremal combinatorics deals with questions that in the language of model theory can be re-stated as follows. For finite models $M, N$ of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let $p(M,N)$ be the probability that a randomly chosen sub-model of $N$ with $|M|$ elements is isomorphic to $M$. Which asymptotic relations exist between the quantities $p(M₁,N),…, p(M_h,N)$, where $M_1,…,M_h$ are fixed “template” models and $|N|$ grows to infinity? In this paper we develop a formal calculus that captures many standard arguments in the area, both previously known and apparently new. We give the first application of this formalism by presenting a new simple proof of a result by Fisher about the minimal possible density of triangles in a graph with given edge density.

We consider the notion of bounded $m$-ary patch-width defined in [9], and its very close relative $m$-constructibility defined below. We show that the notions of $m$-constructibility all coincide for $m \geq$ 3, while 1-constructibility is a weaker notion. The same holds for bounded $m$-ary patch-width. The case $m =$ 2 is left open.

A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent.

We completely characterize the logical hierarchy of subsystems of open induction introduced by Boughattas [1].

Let S be the set of those α∈ω₂ that have cofinality ω₁. It is consistent relative to a measurable that the nonempty player wins the pressing down game of length ω₁, but not the Banach-Mazur game of length ω+1 (both games starting with S).

Resolution is the most widely studied approach to propositional theorem proving. In developing efficient resolution-based algorithms, dozens of variants and refinements of resolution have been studied from both the empirical and analytic sides. The most prominent of these refinements are: DP (ordered), DLL (tree), semantic, negative, linear and regular resolution. In this paper, we characterize and study these six refinements of resolution. We give a nearly complete characterization of the relative complexities of all six refinements. While many of the important separations and simulations were already known, many new ones are presented in this paper; in particular, we give the first separation of semantic resolution from general resolution. As a special case, we obtain the first exponential separation of negative resolution from general resolution. We also attempt to present a unifying framework for studying all of these refinements.

We prove the following results: (i) PV proves NP⊆P/poly iff PV proves coNP⊆ NP/O(1). (ii) If PV proves NP⊆P/poly then PV proves that the Polynomial Hierarchy collapses to the Boolean Hierarchy. (iii) S_2¹ proves NP⊆P/poly iff S₂¹ proves coNP⊆ NP/O(log n). (iv) If S_2¹ proves NP⊆P/poly then S₂¹ proves that the Polynomial Hierarchy collapses to P^NP[log n]. (v) If S₂² proves NP⊆P/poly then S₂² proves that the Polynomial Hierarchy collapses to P^NP. Motivated by these results we introduce a new concept in proof complexity: proof systems with advice, and we make some initial observations about them.

We isolate a forcing which increases the value of δ¹₂ while preserving ω₁ under the assumption that there is a precipitous ideal on ω₁ and a measurable cardinal.

We prove constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0,1] into ℝ is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas.

We define a higher order logic which has only a notion of sort rather than a notion of type, and which permits all terms of the untyped lambda calculus and allows the use of the Y combinator in writing recursive predicates. The consistency of the logic is maintained by a distinction between use and mention, as in Gilmore’s logics. We give a consistent model theory, a proof system which is sound with respect to the model theory, and a cut-elimination proof for the proof system. We also give examples showing what formulas can and cannot be used in the logic.

We show that every nonzero Σ⁰₂ enumeration degree bounds a nonsplitting nonzero enumeration degree.

This paper calculates, in a precise way, the complexity of the index sets for three classes of computable structures: the class $K_{\omega^{CK}_1}$ of structures of Scott rank $\omega^{CK}_1$, the class $K_{\omega^{CK}_1+1}$ of structures of Scott rank $\omega^{CK}_1+1}$, and the class $K$ of all structures of non-computable Scott rank. We show that $I(K)$ is $m$-complete $\Sigma^1_1$, $I(K_{\omega^{CK}_1})$ is $m$-complete $\Pi^0_2$ relative to Kleene’s $\mathcal{O}$, and $I(K_{\omega^{CK}_1+1})$ is $m$-complete $\Sigma^0_2 relative to $\mathcal{O}$.