The Jensen covering lemma says that either L has a club class of indiscernibles, or else, for every uncountable set A of ordinals, there is a set B ∈ L with A ⊆ B and card (B) = card(A). One might hope to extend Jensen's covering lemma to richer core models, which for us will mean to inner models of the form L[] where is a coherent sequence of extenders of the kind studied in Mitchell-Steel [8], The papers [8], [12], [10] and [1] show how to construct core models with Woodin cardinals and more. But, as Prikry forcing shows, one cannot expect too direct a generalization of Jensen's covering lemma to core models with measurable cardinals.

Recall from [8] that if L[] is a core model and α is an ordinal, then either Eα = ∅, or else Eα is an extender over As in [8], we assume here that if Eα is an extender, then Eα is below superstrong type in the sense that the set of generators of Eα is bounded in (crit(Eα)). Let us say that L[] is a lower-part core model iff for every ordinal α, Eα is not a total extender over L[]. In other words, if L[] is a lower-part core model, then no cardinal in L[] is measurable as witnessed by an extender on . Other than the “below superstrong” hypothesis, we impose no bounds on the large cardinal axioms true in the levels of a lower-part core model.

This paper presents a new correctness criterion for marked Danos-Reginer graphs (D-R graphs, for short) of Multiplicative Cyclic Linear Logic MCLL and Abrusci's non-commutative Linear Logic MNLL.

As a corollary we obtain an affirmative answer to the open question whether a known quadratic-time algorithm for the correctness checking of proof nets for MCLL and MNLL can be improved to linear-time.

Call a set A n-correctable if every set Turing reducible to A via a Turing machine that on any input makes at most n queries is Turing reducible to A via a Turing machine that on any input makes at most n-queries and on any input halts no matter what answers are given to its queries. We show that if a c.e. set A is n-correctable for some n ≥ 2, then it is n-correctable for all n. We show that this is the optimal such result by constructing a c.e. set that is 1-correctable but not 2-correctable. The former result is obtained by examining the logical closure properties of c.e. sets that are 2-correctable.

Pure Type Systems. PTSs, introduced as a generalisation of the type systems of Barendregt's lambda-cube, provide a foundation for actual proof assistants, aiming at the mechanic verification of formal proofs. In this paper we consider simplifications of some of the rules of PTSs. This is of independent interest for PTSs as this produces more flexible PTS-like systems, but it will also help, in a later paper, to bridge the gap between PTSs and systems of Illative Combinatory Logic.

First we consider a simplification of the start and weakening rules of PTSs. which allows contexts to be sets of statements, and a generalisation of the conversion rule. The resulting Set-modified PTSs or SPTSs, though essentially equivalent to PTSs, are closer to standard logical systems.

A simplification of the abstraction rule results in Abstraction-modified PTSs or APTSs. These turn out to be equivalent to standard PTSs if and only if a condition (*) holds. Finally we consider SAPTSs which have both modifications.

Five types of constructions are introduced for non-prioritized belief revision, i.e., belief revision in which the input sentence is not always accepted. These constructions include generalizations of entrenchment-based and sphere-based revision. Axiomatic characterizations are provided, and close interconnections are shown to hold between the different constructions.

Each Girard quantale (i.e., commutative quantale with a selected dualizing element) provides a support for a semantics for linear propositional formulas (but not for linear derivations). Several constructions of Girard quantales are known. We give two more constructions, one using an arbitrary partially ordered monoid and one using a partially ordered group (both commutative). In both cases the semantics can be controlled be a relation between pairs of elements of the support and formulas. This gives us a neat way of handling duality.

It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be “completed” conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives.

We generalize the result of Mekler and Shelah [3] that the existence of a canary tree is independent of ZFC + GCH to uncountable regular cardinals. We also correct an error from the original proof.

It is shown constructively that a strongly extensional function of bounded variation on an interval is regulated, in a sequential sense that is classically equivalent to the usual one.

This paper concerns Tarski's use of the term “model” in his 1936 paper “On the Concept of Logical Consequence.” Against several of Tarski's recent defenders. I argue that Tarski employed a non-standard conception of models in that paper. Against Tarski's detractors. I argue that this nonstandard conception is more philosophically plausible than it may appear. Finally, I make a few comments concerning the traditionally puzzling case of Tarski's ω-rule example.

The recursion theorem in abstract partially ordered algebras, such as operative spaces and others, is the most fundamental result of algebraic recursion theory. The primary aim of the present paper is to prove this theorem for iterative operative spaces in full generality. As an intermediate result, a new and rather large class of models of the combinatory logic is obtained.

We show that in the lattice of classes there are initial segments [∅, P] = (P) which are not Boolean algebras, but which have a decidable theory. In fact, we will construct for any finite distributive lattice L which satisfies the dual of the usual reduction property a class P such that L is isomorphic to the lattice (P)*, which is (P). modulo finite differences. For the 2-element lattice, we obtain a minimal class, first constructed by Cenzer, Downey, Jockusch and Shore in 1993. For the simplest new class P constructed, P has a single, non-computable limit point and (P)* has three elements, corresponding to ∅, P and a minimal class P0 ⊂ P, The element corresponding to P0 has no complement in the lattice. On the other hand, the theory of (P) is shown to be decidable.

A class P is said to be decidable if it is the set of paths through a computable tree with no dead ends. We show that if P is decidable and has only finitely many limit points, then (P)* is always a Boolean algebra. We show that if P is a decidable class and (P) is not a Boolean algebra, then the theory of (P) interprets the theory of arithmetic and is therefore undecidable.

A model = (M. E, …) of Zermelo-Fraenkel set theory ZF is said to be 0-like. where E interprets ∈ and θ is an uncountable cardinal, if ∣M∣ = θ but ∣{b ∈ M: bEa}∣ < 0 for each a ∈ M, An immediate corollary of the classical theorem of Keisler and Morley on elementary end extensions of models of set theory is that every consistent extension of ZF has an ℵ1-like model. Coupled with Chang's two cardinal theorem this implies that if θ is a regular cardinal 0 such that 2<0 = 0 then every consistent extension of ZF also has a 0+-like model. In particular, in the presence of the continuum hypothesis every consistent extension of ZF has an ℵ2-like model. Here we prove:

Theorem A. If 0 has the tree property then the following are equivalent for any completion T of ZFC:

(i) T has a 0-like model.

(ii) Ф ⊆ T. where Ф is the recursive set of axioms {∃κ (κ is n-Mahlo and “Vκis a Σn-elementary submodel of the universe”): n ∈ ω}.

(iii) T has a λ-like model for every uncountable cardinal λ.

Theorem B. The following are equiconsistent over ZFC:

(i) “There exists an ω-Mahlo cardinal”.

(ii) “For every finite language , all ℵ2-like models of ZFC() satisfy the schemeФ().

Let ℜ be an expansion of a dense linear order (R, <) without endpoints having the intermediate value property, that is, for all a, b ∈ R, every continuous (parametrically) definable function f: [a, b] → R takes on all values in R between f(a) and f(b). Every expansion of the real line (ℝ, <), as well as every o-minimal expansion of (R, <), has the intermediate value property. Conversely, some nice properties, often associated with expansions of (ℝ, <) or with o-minimal structures, hold for sets and functions definable in ℜ. For example, images of closed bounded definable sets under continuous definable maps are closed and bounded (Proposition 1.10).

Of particular interest is the case that ℜ expands an ordered group, that is, ℜ defines a binary operation * such that (R, <, *) is an ordered group. Then (R, *) is abelian and divisible (Proposition 2.2). Continuous nontrivial definable endo-morphisms of (R, *) are surjective and strictly monotone, and monotone nontrivial definable endomorphisms of (R, *) are strictly monotone, continuous and surjective (Proposition 2.4). There is a generalization of the familiar result that every proper noncyclic subgroup of (ℝ, +) is dense and codense in ℝ: If G is a proper nontrivial subgroup of (R, *) definable in ℜ, then either G is dense and codense in R, or G contains an element u such that (R, <, *, e, u, G) is elementarily equivalent to (ℚ, <, +, 0, 1, ℤ), where e denotes the identity element of (R, *) (Theorem 2.3).

Here is an outline of this paper. First, we deal with some basic topological results. We then assume that ℜ expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an explicit axiomatization (used in the proof of Theorem 2.3) is given for the complete theory of the structure (ℚ, <, +, 0, 1, ℤ).

We show that the lattice L20 is not embeddable into the lattice of ideals of computably enumerable Turing degrees (ℐ), We define a structure called a pseudolattice that generalizes the notion of a lattice, and show that there is a Π2 necessary and sufficient condition for embedding a finite pseudolattice into ℐ.

A sequent calculus is given in which the management of weakening and contraction is organized as in natural deduction. The latter has no explicit weakening or contraction, but vacuous and multiple discharges in rules that discharge assumptions. A comparison to natural deduction is given through translation of derivations between the two systems. It is proved that if a cut formula is never principal in a derivation leading to the right premiss of cut, it is a subformula of the conclusion. Therefore it is sufficient to eliminate those cuts that correspond to detour and permutation conversions in natural deduction.

We fix an arbitrary o-minimal structure (R, ω, …), where (R, <) is a dense linearly ordered set without end points. In this paper “definable” means “definable with parameters from R”, We equip R with the interval topology and Rn with the induced product topology. The main result of this paper is the following.

Theorem. Let V ⊆ Rnbe a definable open set and suppose that f: V → Rnis a continuous injective definable map. Then f is open, that is, f(U) is open whenever U is an open subset of V.

Woerheide [6] proved the above theorem for o-minimal expansions of a real closed field using ideas of homology. The case of an arbitrary o-minimal structure remained an open problem, see [4] and [1]. In this paper we will give an elementary proof of the general case.

Basic definitions and notation. A box B ⊆ Rn is a Cartesian product of n definable open intervals: B = (a1, b1) × … × (an, bn) for some ai, bi, ∈ R ∪ {−∞, +∞}, with ai < bi, Given A ⊆ Rn, cl(A) denotes the closure of A, int(A) denotes the interior of A, bd(A) ≔ cl(A) − int(A) denotes the boundary of A, and ∂A ≔ cl(A) − A denotes the frontier of A, Finally, we let π: Rn → Rn− denote the projection map onto the first n − 1 coordinates.

Background material. Without mention we will use notions and facts discussed in [5] and [3]. We will also make use of the following result, which appears in [2].

We prove weak elimination of imaginary elements for Boolean orderings with finitely many atoms. As a consequence we obtain equivalence of the two notions of o-minimality for Boolean ordered structures, introduced by C. Toffalori. We investigate atoms in Boolean algebras induced by algebraically closed subsets of Boolean ordered structures. We prove uniqueness of prime models in strongly o-minimal theories of Boolean ordered structures.