The starting point of this article is an old question asked by Feferman in his paper on Hancock’s conjecture [6] about the strength of ID₁^*. This theory is obtained from the well-known theory ID₁ by restricting fixed point induction to formulas that contain fixed point constants only positively. The techniques used to perform the proof-theoretic analysis of ID₁^* also permit to analyze its transfinitely iterated variants ID_α^*. Thus, we eventually know that |\hat{ID}_α| = |ID_α^*|.

Given a weakly o-minimal theory T, the T-height of an element of a model of T is defined as a means of classifying the order of magnitude of the element. If T satisfies some easily met technical conditions, then this classification is coarse enough for a Wilkie-type inequality: given a set of elements of a model of T, each of which has a different T-height, the cardinality of this set is at most 1 plus the minimum cardinality of a set that generates the structure.

We prove that the definition of supersimplicity in metric structures from [7] is equivalent to an a priori stronger variant. This stronger variant is then used to prove that if T is a supersimple Hausdorff cat then so is its theory of lovely pairs.

The method of analytic tableaux is employed in many introductory texts and has also been used quite extensively as a basis for automated theorem proving. In this paper, we discuss the complexity of the system as a method for refuting contradictory sets of clauses, and resolve several open questions. We discuss the three forms of analytic tableaux: clausal tableaux, generalized clausal tableaux, and binary tableaux. We resolve the relative complexity of these three forms of tableaux proofs and also resolve the relative complexity of analytic tableaux versus resolution. We show that there is a quasi-polynomial simulation of tree resolution by analytic tableaux; this simulation is close to optimal, since we give a matching lower bound that is tight to within a polynomial.

We show, first, that the normalization procedure for S4 modal logic presented by Dag Prawitz in [5] does not work. We then develop a new natural deduction system for S4 classical modal logic that is logically equivalent to that of Prawitz, and we show that every derivation in this new system can be transformed into a normal derivation.

We give some general criteria for the stable embeddedness of a definable set. We use these criteria to establish the stable embeddedness in algebraically closed valued fields of two definable sets: The set of balls of a given radius r < 1 contained in the valuation ring and the set of balls of a given multiplicative radius r < 1. We also show that in an algebraically closed valued field a 0-definable set is stably embedded if and only if its algebraic closure is stably embedded.

In the last few decades many formal systems of fuzzy logics have been developed. Since the main differences between fuzzy and classical logics lie at the propositional level, the fuzzy predicate logics have developed more slowly (compared to the propositional ones). In this text we aim to promote interest in fuzzy predicate logics by contributing to the model theory of fuzzy predicate logics. First, we generalize the completeness theorem, then we use it to get results on conservative extensions of theories and on witnessed models.

This paper is concerned with n—localization property introduced by Newelski and Rosłanowski in [10] and getting it for CS iterations of forcing notions.

We prove that for any locally countable ᶺ¹₁ partial order ℛ=〈 2^ω, ≤_P 〉, there exists a nonmeasurable antichain in ℛ. Some applications of the result are also presented.

We define elementary extension and elementary equivalence in Positive Logic.

We produce counterexamples to the unique and cofinal branches hypotheses, assuming (slightly less than) the existence of a cardinal which is strong past a Woodin cardinal.

This paper studies definability within the theory of institutions, a version of abstract model theory that emerged in computing science studies of software specification and semantics. We generalise the concept of definability to arbitrary logics, formalised as institutions, and we develop three general definability results. One generalises the classical Beth theorem by relying on the interpolation properties of the institution. Another relies on a meta Birkhoff axiomatizability property of the institution and constitutes a source for many new actual definability results, including definability in (fragments of) classical model theory. The third one gives a set of sufficient conditions for ‘borrowing’ definability properties from another institution via an ‘adequate’ encoding between institutions. The power of our general definability results is illustrated with several applications to (many-sorted) classical model theory and partial algebra, leading for example to definability results for (quasi-)varieties of models or partial algebras. Many other applications are expected for the multitude of logical systems formalised as institutions from computing science and logic.

This paper investigates when it is possible for a partial ordering ℛ to force 𝒫_κ(λ)∖ V to be stationary in V^ℛ. It follows from a result of Gitik that whenever ℛ adds a new real, then 𝒫_κ(λ)∖ V is stationary in V^ℛ for each regular uncountable cardinal κ in V^ℛ and all cardinals λ>κ in V^ℛ [4]. However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of ω₁-Erdős cardinals: If ℛ is ℵ₁-Cohen forcing, then 𝒫_κ(λ)∖ V is stationary in V^ℛ, for all regular κ≥ℵ₂ and all λ>κ. The following is equiconsistent with an ω₁-Erdős cardinal: If ℛ is ℵ₁-Cohen forcing, then 𝒫_ℵ₂(ℵ₃)∖ V is stationary in V^ℛ. The following is equiconsistent with κ measurable cardinals: If ℛ is κ-Cohen forcing, then 𝒫_κ⁺(ℵ_κ)∖ V is stationary in V^ℛ.

We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Löf randomness.

We establish a surprising connection between Menger’s classical covering property and Blass-Laflamme’s modern combinatorial notion of groupwise density. This connection implies a short proof of the groupwise density bound on the additivity number for Menger’s property.

We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the proof-theoretic strength of the regularity of Lebesgue measure for G_δ sets. Our constructions essentially settle the reverse mathematical classification of this principle.

We discuss the existence of universal spaces (either in the sense of embeddings or continuous images) for some classes of scattered Eberlein compacta. Given a cardinal κ, we consider the class 𝒮_κ of all scattered Eberlein compact spaces K of weight ≤κ and such that the second Cantor-Bendixson derivative of K is a singleton. We prove that if κ is an uncountable cardinal such that κ = 2^{< κ}, then there exists a space X in 𝒮_κ such that every member of 𝒮_κ is homeomorphic to a retract of X. We show that it is consistent that there does not exist a universal space (either by embeddings or by mappings onto) in 𝒮_ω₁. Assuming that 𝔡= ω₁, we prove that there exists a space X∈𝒮_ω₁, which is universal in the sense of embeddings. We also show that it is consistent that there exists a space X∈𝒮_ω₁, universal in the sense of embeddings, but 𝒮_ω₁ does not contain an universal element in the sense of mappings onto.