We show that the spectrum of a sentence φ in Counting Monadic Second Order Logic (CMSOL) using one binary relation symbol and finitely many unary relation symbols, is ultimately periodic, provided all the models of φ are of clique width at most k, for some fixed k. We prove a similar statement for arbitrary finite relational vocabularies τ and a variant of clique width for τ-structures. This includes the cases where the models of φ are of tree width at most k. For the case of bounded tree-width, the ultimate periodicity is even proved for Guarded Second Order Logic GSOL. We also generalize this result to many-sorted spectra, which can be viewed as an analogue of Parikh’s Theorem on context-free languages, and its analogues for context-free graph grammars due to Habel and Courcelle. Our work was inspired by Gurevich and Shelah (2003), who showed ultimate periodicity of the spectrum for sentences of Monadic Second Order Logic where only finitely many unary predicates and one unary function are allowed. This restriction implies that the models are all of tree width at most 2, and hence it follows from our result.

We prove that for every simple theory T (or even simple thick compact abstract theory) there is a (unique) compact abstract theory T^𝔓 whose saturated models are the lovely pairs of T. Independence-theoretic results that were proved in [ppv:pairs] when T^𝔓 is a first order theory are proved for the general case: in particular T^𝔓 is simple and we characterise independence.

We show that the quasi-order of continuous embeddability between finitely branching dendrites (a natural class of fairly simple compacta) is Σ₁¹-complete. We also show that embeddability between countable linear orders with infinitely many colors is Σ₁¹-complete.

Assume that all algebras are atomless. (1) Spind(A× B)=Spind(A)∪ Spind(B). (2) Spind(∏^w_{i∈ I}A_i)={ω}∪⋃_{i∈ I} Spind (A_i). Now suppose that κ and λ are infinite cardinals, with κ uncountable and regular and with κ<λ. (3) There is an atomless Boolean algebra A such that 𝔲(A)=κ and 𝔦(A)=λ. (4) If λ is also regular, then there is an atomless Boolean algebra A such that 𝔰(A)=𝔰(A)=κ and 𝔞(A)=λ. All results are in ZFC, and answer some problems posed in Monk [Mon01] and Monk [MonInf].

A set A is m-reducible (or Karp-reducible) to B if and only if there is a polynomial-time computable function f such that, for all x, x∈ A if and only if f(x) ∈ B. Two sets are: 1-equivalent if and only if each is m-reducible to the other by one-one reductions; p-invertible equivalent if and only if each is m-reducible to the other by one-one, polynomial-time invertible reductions; and p-isomorphic if and only if there is an m-reduction from one set to the other that is one-one, onto, and polynomial-time invertible. In this paper we show the following characterization. Theorem The following are equivalent: (a) P = PSPACE. (b) Every two 1-equivalent sets are p-isomorphic. (c) Every two p-invertible equivalent sets are p-isomorphic.

We study the preservation under projective ccc forcing extensions of the property of L(ℝ) being a Solovay model. We prove that this property is preserved by every strongly-Σ₃¹ absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for δ₃¹ absolutely-ccc forcing notions. We extend these results to the higher projective classes of ccc posets, and to the class of all projective ccc posets, using definably-Mahlo cardinals. As a consequence we obtain an exact equiconsistency result for generic absoluteness under projective absolutely-ccc forcing notions.

A model is said to be Leibnizian if it has no pair of indiscernibles. Mycielski has shown that there is a first order axiom LM (the Leibniz-Mycielski axiom) such that for any completion T of Zermelo-Fraenkel set theory ZF, T has a Leibnizian model if and only if T proves LM. Here we prove: Theorem 1 Every complete theory T extending ZF+LM has 2^ℵ₀ nonisomorphic countable Leibnizian models. Theorem 2 If κ is a prescribed definable infinite cardinal of a complete theory T extending ZF+V=OD, then there are 2^ℵ₁ nonisomorphic Leibnizian models 𝔐 of T of power ℵ₁ such that (κ⁺)^𝔐 is ℵ₁-like. Theorem 3 Every complete theory T extending ZF+V=OD has 2^ℵ₁ nonisomorphic ℵ₁-like Leibnizian models.

We prove here that the intuitionistic theory T₀↾+UMID_N, or even EETJ↾+UMID_N, of Explicit Mathematics has the strength of Π₂¹-CA₀. In Section 1 we give a double-negation translation for the classical second-order μ-calculus, which was shown in [Moe02] to have the strength of Π₂¹-CA₀. In Section 2 we interpret the intuitionistic μ-calculus in the theory EETJ↾+UMID_N. The question about the strength of monotone inductive definitions in T₀ was asked by S. Feferman in 1982, and — assuming classical logic — was addressed by M. Rathjen.

We examine the question of how many Boolean algebras, distinct up to isomorphism, that are quotients of the powerset of the naturals by Borel ideals, can be proved to exist in ZFC alone. The maximum possible value is easily seen to be the cardinality of the continuum 2^ℵ₀; earlier work by Ilijas Farah had shown that this was the value in models of Martin’s Maximum or some similar forcing axiom, but it was open whether there could be fewer in models of the Continuum Hypothesis. We develop and apply a new technique for constructing many ideals whose quotients must be nonisomorphic in any model of ZFC. The technique depends on isolating a kind of ideal, called shallow, that can be distinguished from the ideal of all finite sets even after any isomorphic embedding, and then piecing together various copies of the ideal of all finite sets using distinct shallow ideals. In this way we are able to demonstrate that there are continuum-many distinct quotients by Borel ideals, indeed by analytic P-ideals, and in fact that there is in an appropriate sense a Borel embedding of the Vitali equivalence relation into the equivalence relation of isomorphism of quotients by analytic P-ideals. We also show that there is an uncountable definable wellordered collection of Borel ideals with distinct quotients.

We interpret the algebra of finite rank operators as imaginaries inside a Hilbert space. We prove that the Hilbert space enlarged with these imaginaries has built-in canonical bases.

In this paper, we prove the wellfoundedness of recursive notation systems for reflecting ordinals up to Π₃-reflection by relevant inductive definitions.

We present a comparatively simple way to construct Martin-Löf random and rec-random sets with certain additional properties, which works by diagonalizing against appropriate martingales. Reviewing the result of Gács and Kučera, for any given set X we construct a Martin-Löf random set from which X can be decoded effectively. By a variant of the basic construction we obtain a rec-random set that is weak truth-table autoreducible and we observe that there are Martin-Löf random sets that are computably enumerable self-reducible. The two latter results complement the known facts that no rec-random set is truth-table autoreducible and that no Martin-Löf random set is Turing-autoreducible [ebert-merkle-vollmer-2002,trakhtenbrot-1970].

We characterize (both from a syntactic and an algebraic point of view) the normal K4-logics for which unification is filtering. We also give a sufficient semantic criterion for existence of most general unifiers, covering natural extensions of K4.2⁺ (i.e., of the modal system obtained from K4 by adding to it, as a further axiom schemata, the modal translation of the weak excluded middle principle).

We study reals with infinitely many incompressible prefixes. Call A∈ 2^ω Kolmogorov random if (∃^∞ n) C(A↾ n)> n-(1), where C denotes plain Kolmogorov complexity. This property was suggested by Loveland and studied by Martin-Löf, Schnorr and Solovay. We prove that 2-random reals are Kolmogorov random. Together with the converse—proved by Nies, Stephan and Terwijn [NST]—this provides a natural characterization of 2-randomness in terms of plain complexity. We finish with a related characterization of 2-randomness.

A Turing degree a is said to be almost everywhere dominating if, for almost all X∈ 2^ω with respect to the “fair coin” probability measure on 2^ω, and for all g : ω→ω Turing reducible to X, there exists f : ω→ω of Turing degree a which dominates g. We study the problem of characterizing the almost everywhere dominating Turing degrees and other, similarly defined classes of Turing degrees. We relate this problem to some questions in the reverse mathematics of measure theory.