The provability logic of a theory T captures the structural behavior of formalized provability in T as provable in T itself. Like provability, one can formalize the notion of relative interpretability giving rise to interpretability logics. Where provability logics are the same for all moderately sound theories of some minimal strength, interpretability logics do show variations.

Two structures A and B are n-equivalent if Player II has a winning strategy in the n-move Ehrenfeucht-Fraïssé game on A and B. In earlier articles we studied n-equivalence classes of ordinals and coloured ordinals. In this article we similarly treat a class of scattered order-types, focussing on monomials and sums of monomials in ω and its reverse ω*.

We simultaneously generalize Silver’s perfect set theorem for co-analytic equivalence relations and Harrington-Marker-Shelah’s Dilworth-style perfect set theorem for Borel quasi-orders, establish the analogous theorem at the next definable cardinal, and give further generalizations under weaker definability conditions.

Let L be a computable vocabulary, let XL be the space of L-structures with universe ω and let $f:{2^\omega } \to {X_L}$ be a hyperarithmetic function such that for all $x,y \in {2^\omega }$ , if $x{ \equiv _h}y$ then $f\left( x \right) \cong f\left( y \right)$ . One of the following two properties must hold. (1) The Scott rank of f (0) is $\omega _1^{CK} + 1$ . (2) For all $x \in {2^\omega },f\left(

We show that arithmetical transfinite recursion is equivalent to a suitable formalization of the following: For every ordinal α there exists an ordinal β such that $1 + \beta \cdot \left( {\beta + \alpha } \right)$ (ordinal arithmetic) admits an almost order preserving collapse into β. Arithmetical comprehension is equivalent to a statement of the same form, with $\beta \cdot \alpha$ at the place of

We answer a question of Blass, Di Nasso, and Forti [2, 7] by proving, assuming Continuum Hypothesis or Martin’s Axiom, that (1) there exists a P-point which is not interval-to-one and (2) there exists an interval-to-one P-point which is neither quasi-selective nor weakly Ramsey.

We determine the consistency strength of determinacy for projective games of length ω2. Our main theorem is that $\Pi _{n + 1}^1 $ -determinacy for games of length ω2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn (A), the canonical inner model for n Woodin cardinals

We show that there is a minimal pair in the nonuniform generic degrees, and hence also in the uniform generic degrees. This fact contrasts with Igusa’s result that there are no minimal pairs for relative generic computability and answers a basic structural question mentioned in several papers in the area.

Let ${\cal I}$ be an ideal on ω. By cov ${}_{}^{\rm{*}}({\cal I})$ we denote the least size of a family ${\cal B} \subseteq {\cal I}$ such that for every infinite $X \in {\cal I}$ there is $B \in {\cal B}$ for which $B\mathop \cap \nolimits X$ is infinite. We say that an AD family ${\cal A} \subseteq {\cal I}$ is a MAD family restricted to ${\cal I}$ if for every infinite $X \in {\cal I}$ there is

For a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove in ZF (without the axiom of choice) several results concerning this notion, among which are the following:

For a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove by the method of permutation models that the following statements are consistent with ZF:

Let ${\cal T}$ be any of the three canonical truth theories CT− (compositional truth without extra induction), FS− (Friedman–Sheard truth without extra induction), or KF− (Kripke–Feferman truth without extra induction), where the base theory of ${\cal T}$ is PA (Peano arithmetic). We establish the following theorem, which implies that ${\cal T}$ has no more than polynomial speed-up over PA.

In the first part of the article, we show that if $\omega \le \kappa < \lambda$ are cardinals, ${\kappa ^{ < \kappa }} = \kappa$ , and λ is weakly compact, then in $V\left[M {\left( {\kappa ,\lambda } \right)} \right]$ the tree property at $$\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $$ is indestructible under all ${\kappa ^ + }$ -cc forcing notions

We define the notion of a completely determined Borel code in reverse mathematics, and consider the principle $CD - PB$ , which states that every completely determined Borel set has the property of Baire. We show that this principle is strictly weaker than $AT{R_0}$ . Any ω-model of $CD - PB$ must be closed under hyperarithmetic reduction, but $CD - PB$ is not a theory of hyperarithmetic analysis.

We investigate the effects of various forcings on several forms of the Halpern– Läuchli theorem. For inaccessible κ, we show they are preserved by forcings of size less than κ. Combining this with work of Zhang in [17] yields that the polarized partition relations associated with finite products of the κ-rationals are preserved by all forcings of size less than κ over models satisfying the Halpern–

We investigate the strength of a randomness notion ${\cal R}$ as a set-existence principle in second-order arithmetic: for each Z there is an X that is ${\cal R}$ -random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in $RC{A_0}$ . We verify that $RC{A_0}$ proves the basic implications among randomness notions: 2-random $\Rightarrow$

We prove that the continuous function ${\rm{\hat \Omega }}:2^\omega \to $ that is defined via $X \mapsto \mathop \sum \limits_n 2^{ - K\left( {Xn} \right)} $ for all $X \in {2^\omega }$ is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that $\mathop \smallint \nolimits _0^1{\rm{\hat{\Omega }}}\left( X \right)\,{\rm{d}}X$ is

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices

We prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose, a specific class of countable 2-colored posets $\mathbb{P}_{emb} $ equipped with the order induced by homomorphisms is embedded into the Wadge order on the $\Delta _2^0 $ -degrees of the Scott domain. We then show

In this note, we construct a distal expansion for the structure $$\left( {; + , < ,H} \right)$$ , where $H \subseteq $ is a dense $Q$ -vector space basis of $R$ (a so-called Hamel basis). Our construction is also an expansion of the dense pair $\left( {; + , < ,} \right)$ and has full quantifier elimination in a natural language.

In set theory without the Axiom of Choice, we consider Ingleton’s axiom which is the ultrametric counterpart of the Hahn–Banach axiom. We show that in ZFA, i.e., in the set theory without the Axiom of Choice weakened to allow “atoms,” Ingleton’s axiom does not imply the Axiom of Choice (this solves in ZFA a question raised by van Rooij, [27]). We also prove that in ZFA, the “multiple choice” axiom

We examine how degrees of computably enumerable equivalence relations (ceers) under computable reduction break down into isomorphism classes. Two ceers are isomorphic if there is a computable permutation of ω which reduces one to the other. As a method of focusing on nontrivial differences in isomorphism classes, we give special attention to weakly precomplete ceers. For any degree, we consider the

Duparc introduced a two-player game for a function f between zero-dimensional Polish spaces in which Player II has a winning strategy iff f is of Baire class 1. We generalize this result by defining a game for an arbitrary function f : X → Y between arbitrary Polish spaces such that Player II has a winning strategy in this game iff f is of Baire class 1. Using the strategy of Player II, we reprove

We give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that