In Ockhamist branching-time logic [Prior 67], formulas are meant to be evaluated on a specified branch, or history, passing through the moment at hand. The linguistic counterpart of the manifoldness of future is a possibility operator which is read as ‘at some branch, or history (passing through the moment at hand)’. Both the bundled-trees semantics [Burgess 79] and the 〈moment, history〉 semantics [Thomason 84] for the possibility operator involve a quantification over sets of moments. The Ockhamist frames are (3-modal) Kripke structures in which this second-order quantification is represented by a first-order quantification. The aim of the present paper is to investigate the notions of modal definability, validity, and axiomatizability concerning 3-modal frames which can be viewed as generalizations of Ockhamist frames.

Assuming determinacy, the model L[T2] does not depend on the choice of T2.

Inspired by Pohlers' proof-theoretic analysis of KPω we give a straightforward non-metamathematical proof of the (well-known) classification of the provably total functions of PA, PA + TI(⊰ ↾) (where it is assumed that the well-ordering ⊰ has some reasonable closure properties) and KPω. Our method relies on a new approach to subrecursion due to Buchholz, Cichon and the author.

The purpose of this note is to clarify two points about the topos Lif, introduced in [13] as a generalization of Lifschitz' realizability ([9, 12]). Lif is a subtopos of Hyland's Effective topos ([1]). The points I want to make are:

Remark 1. Lif is the largest subtopos of satisfying the axiom (O):

where denotes partial recursive application, and “∈ Tot” means that e and f range over codes for total recursive functions. One may read (O) as the statement “The union of two -sets is again a -set”. That is, let be a subtopos of . Then (O) is true in for the standard interpretation (the variables range over the natural numbers object of , etc.) if and only if the inclusion ↣ factors through the inclusion Lif ↣ .

Remark 2. Like , Lif contains at least two weakly complete internal full subcategories, thus providing us with more models of polymorphism and other impredicative type theories.

The principle (O) has some standing in the history of constructive mathematics:

- H. Friedman has proved that (O) is equivalent to a formulation of intuitionistic completeness of the intuitionistic predicate calculus for Tarskian semantics; see [8]. This is not to imply that this result is of immediate relevance to Lif: Friedman works in a system of analysis, a theory of lawless sequences with an axiom of “open data” for arithmetical formulas, which at least for the domain of all functions from N to N fails in Lif. However, there might exist a “nonstandard model” of arithmetic and a corresponding system of analysis, in which we may be able to carry out his proof.

- Moreover, Remark 2 entails that Lif should provide us with models of synthetic domain theory (for an exposition, see [3]), and one with the nice property that the dual of one of the axioms (axiom 7 in [3]) comes for free, by (O).

We study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions may be large while the covering for measure is small.

We characterize the bimodal provability logics for certain natural (classes of) pairs of recursively enumerable theories, mostly related to fragments of arithmetic. For example, we shall give axiomatizations, decision procedures, and introduce natural Kripke semantics for the provability logics of (IΔ0 + EXP, PRA); (PRA, IΣn); (IΣm, IΣn) for 1 ≤ m < n; (PA, ACA0); (ZFC, ZFC + CH); (ZFC, ZFC + ¬CH) etc. For the case of finitely axiomatized extensions of theories these results are extended to modal logics with propositional constants.

The context for this paper is Feferman's theory of explicit mathematics, T0. We address a problem that was posed in [6]. Let MID be the principle stating that any monotone operation on classifications has a least fixed point. The main objective of this paper is to show that T0 + MID, when based on classical logic, also proves the existence of non-monotone inductive definitions that arise from arbitrary extensional operations on classifications. From the latter we deduce that MID, when adjoined to classical T0, leads to a much stronger theory than T0.

We study the expressive power in the finite of the logic Fixed-Point+Counting, the extension of first-order logic which is obtained through adding both the fixed-point constructor and the ability to count.

To this end an isomorphism preserving (‘generic’) model of computation is introduced whose PTime restriction exactly corresponds to this level of expressive power, while its PSpace restriction corresponds to While+Counting. From this model we obtain a normal form which shows a rather clear separation of the relational vs. the arithmetical side of the algorithms involved.

In parallel, we study the relations of Fixed-Point+Counting with the infinitary logics and the corresponding pebble games.

The main result, however, involves the concept of an arithmetical invariant. By this we mean a functor taking every finite relational structure to an expansion of (an initial segment of) the standard arithmetical structure. In particular its values are linearly ordered structures. We establish the existence of a family of arithmetical invariants with the following properties:

• The invariants themselves can be evaluated in polynomial time.

• A class of finite relational structures is definable in Fixed-Point+Counting if and only if membership can be decided in polynomial time on the basis of the values of one of the invariants.

• The invariant r classifies all finite relational structures exactly up to equivalence with respect to the logic

We also give a characterization of Fixed-Point+Counting in terms of sequences of formulae in the : It corresponds exactly to the polynomial time computable families (φn)n ∈ ω in these logics.

Towards a positive assessment of the expressive power of Fixed-Point+Counting, it is shown that the natural extension of fixed-point logic by Lindström quantifiers, which capture all the PTime computable properties of cardinalities of definable predicates, is strictly weaker than what we get here. This implies in particular that every extension of fixed-point logic by means of monadic Lindström quantifiers, which stays within PTime, must be strictly contained in Fixed-Point+Counting.

In this article we give a forcing characterization for the Ramsey property of -Sets of reals. This research was motivated by the well-known forcing characterizations for Lebesgue measurability and the Baire property of -sets of reals. Further we will show the relationship between higher degrees of forcing absoluteness and the Ramsey property of projective sets of reals.

The syntactic structure of the system of pure implicational relevant logic P − W is investigated. This system is defined by the axioms B = (b → c) → (a → b) → a → c, B′ = (a → b) → (b → c)→ a → c, I = a → a, and the rules of substitution and modus ponens. A class of λ-terms, the closed hereditary right-maximal linearλ-terms, and a translation of such λ-terms M to BB′ I-combinators M+ is introduced. It is shown that a formula a is provable in P − W if and only if α is a type of some λ-term in this class. Hence these λ-terms represent proof figures in the Natural Deduction version of P − W.

Errol Martin (1982) proved that no formula with form α → α is provable in P − W without using the axiom I. We show that a β-normal form λ-term M in the class is η reducible to λx.x if the translated BB′ I-combinator M+ contains I. Using this theorem and Martin's result, we prove that a λ-term in the class is βη-reducible to λx.x if the λ-term has a type α → α. Hence the structure of proofs of α → α in P − W is determined.

We characterize the generalized quantifiers Q which satisfy the scheme QxQyф ↔ QyQxф, the so-called self-commuting quantifiers, or quantifiers with the Fubini property.

Using Steel's recent result that assuming AD, in L[ℝ] below Θ, κ is regular iff κ is measurable, we mimic below Θ certain earlier results of Gitik. In particular, we construct via forcing a model in which all uncountable cardinals below Θ are singular and a model in which the only regular uncountable cardinal below Θ is ℵ1.

We characterize omissibility of a type, or a family of types, in a countable theory in terms of non-existence of a certain tree of formulas. We extend results of L. Newelski on omitting < covK non-isolated types. As a consequence we prove that omissibility of a family of < covK types is equivalent to omissibility of each countable subfamily.

Let denote the ideal of Lebesgue measure zero subsets of the real line. Then add() denotes the minimal cardinality of a subset of whose union is not an element of . In [1] Bartoszynski gave an elegant combinatorial characterization of add(), namely: add() is the least cardinal number κ for which the following assertion fails:

For every family of at mostκ functions from ω to ω there is a function F from ω to the finite subsets of ω such that:

1. For each m, F(m) has at most m + 1 elements, and

2. for each f inthere are only finitely many m such that f(m) is not an element of F(m).

The symbol A(κ) will denote the assertion above about κ. In the course of his proof, Bartoszynski also shows that the cardinality restriction in 1 is not sharp. Indeed, let (Rm: m < ω) be any sequence of integers such that for each m Rm, ≤ Rm+1, and such that limm→∞Rm = ∞. Then the truth of the assertion A(κ) is preserved if in 1 we say instead that

1′. For each m, F(m) has at most Rm elements.

We shall use this observation later on. We now define three more statements, denoted B(κ), C(κ) and D(κ), about cardinal number κ.

Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1, …, N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kiĭ [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kiĭ also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable.

We prove that there are 2χ0 pairwise non elementarily equivalent existentially closed ordered groups, which solve the main open problem in this area (cf. [3, 10]).

A simple direct proof is given of the weaker fact that the theory of ordered groups has no model companion; the case of the ordered division rings over a field k is also investigated.

Our main result uses constructible sets and can be put in an abstract general framework.

Comparison with the standard methods which use forcing (cf. [4]) is sketched.

For a primitive recursive consistent and strong enough theory T we construct an independent statement which has some clear metamathematical meaning.

We follow [8] in asking when a set of ordinals X ⊆ α is a countable union of sets in K, the core model. We show that, analogously to L, an X closed under the canonical Σ1 Skolem function for Kα can be so decomposed provided K is such that no ω-closed filters are put on its measure sequence, but not otherwise. This proviso holds if there is no inner model of a weak Erdős-type property.

There exists a family of sets of countable ordinals such that:

(1) max Bα = α,

(2) if α ∈ Bβ then Bα ⊆ Bβ,

(3) if λ ≤ α and λ is a limit ordinal then Bα ∩ λ is not in the ideal generated by the Bβ, β < α, and by the bounded subsets of λ,

(4) there is a partition of ω1 such that for every α and every n, Bα ∩ An is finite.

Assume T is stable, small and Φ(x) is a formula of L(T). We study the impact on T⌈Φ of naming finitely many elements of a model of T. We consider the cases of T⌈Φ which is ω-stable or superstable of finite rank. In these cases we prove that if T has countable models and Q = Φ(M) is countable and atomic or saturated, then any good type in S(Q) is τ-stable. If T⌈Φ is ω-stable and (bounded, 1-based or of finite rank) with , then we prove that every good p ∈ S(Q) is τ-stable for any countable Q. The proofs of these results lead to several new properties of small stable theories, particularly of types of finite weight in such theories.