After presenting a general setting in which to look at forcing axioms, we give a hierarchy of generalized bounded forcing axioms that correspond level by level, in consistency strength, with the members of a natural hierarchy of large cardinals below a Mahlo. We give a general construction of models of generalized bounded forcing axioms. Then we consider the bounded forcing axiom for a class of partially ordered sets Γ1 such that, letting Γ0 be the class of all stationary-set-preserving partially ordered sets, one can prove the following:

(a) Γ0 ⊆ Γ1,

(b) Γ0 = Γ1 if and only if NSω1 is ℵ1-dense.

(c) If P ∉ Γ1, then BFA({P}) fails.

We call the bounded forcing axiom for Γ1Maximal Bounded Forcing Axiom (MBFA). Finally we prove MBFA consistent relative to the consistency of an inaccessible Σ2-correct cardinal which is a limit of strongly compact cardinals.

Kleene [5] mentions two ways of extending the constructive ordinals. The first is by relativizing the set O of notations for the constructive ordinals, using fundamental sequences which are partial recursive in O. In this way we obtain the set OO which provides notations for the ordinals less than ω1O. Continuing the process, the sequence O, OO, , … and the corresponding ordinals are obtained. A second possibility is to define (constructive) higher number classes in which partial recursive functions are used at limit ordinals to provide an “accessibility” mapping from a previously defined number class. The relationship between the ordinals obtained by the two methods of extension has been an open problem. Methods developed in this article are used to show that the two ways of extending the constructive ordinals are equivalent, provided the sets of notations for the higher number classes satisfy certain natural conditions. Equivalence is obtained, not only with respect to ordinals, but also with respect to the forms of the sets of notations for the higher number classes. Specifically, the fundamental fact that the sets of notations for the constructive ordinals are complete Π11 sets generalizes to suitably defined higher number classes. As an application we prove that the ordinals of the Addison and Kleene [1] constructive third number class are exactly the ordinals less than ω1O and the set of notations for their third number class is recursively isomorphic to OO.

In [12], Schwichtenberg showed that the System T definable functionals are closed under a rule-like version Spector’s bar recursion of lowest type levels 0 and 1. More precisely, if the functional Y which controls the stopping condition of Spector’s bar recursor is T-definable, then the corresponding bar recursion of type levels 0 and 1 is already T-definable. Schwichtenberg’s original proof, however, relies on a detour through Tait’s infinitary terms and the correspondence between ordinal recursion for $\alpha < {\varepsilon _0}$ and primitive recursion over finite types. This detour makes it hard to calculate on given concrete system T input, what the corresponding system T output would look like. In this paper we present an alternative (more direct) proof based on an explicit construction which we prove correct via a suitably defined logical relation. We show through an example how this gives a straightforward mechanism for converting bar recursive definitions into T-definitions under the conditions of Schwichtenberg’s theorem. Finally, with the explicit construction we can also easily state a sharper result: if Y is in the fragment Ti then terms built from $BR^{\mathbb{N},\sigma } $ for this particular Y are definable in the fragment ${T_{i + {\rm{max}}\left\{ {1,{\rm{level}}\left( \sigma \right)} \right\} + 2}}$.

We show that in the language of { ≤ }. the Π3-fragment of the first order theory of the -enumeration degrees is undecidable. We then extend this result to show that the Π3-theory of any substructure of the enumeration degrees which contains the -degrees is undecidable.

We introduce the safe recursive set functions based on a Bellantoni–Cook style subclass of the primitive recursive set functions. We show that the functions computed by safe recursive set functions under a list encoding of finite strings by hereditarily finite sets are exactly the polynomial growth rate functions computed by alternating exponential time Turing machines with polynomially many alternations. We also show that the functions computed by safe recursive set functions under a more efficient binary tree encoding of finite strings by hereditarily finite sets are exactly the quasipolynomial growth rate functions computed by alternating quasipolynomial time Turing machines with polylogarithmic many alternations.

We characterize the safe recursive set functions on arbitrary sets in definability-theoretic terms. In its strongest form, we show that a function on arbitrary sets is safe recursive if and only if it is uniformly definable in some polynomial level of a refinement of Jensen's J-hierarchy, relativized to the transitive closure of the function's arguments.

We observe that safe recursive set functions on infinite binary strings are equivalent to functions computed by infinite-time Turing machines in time less than ωω. We also give a machine model for safe recursive set functions which is based on set-indexed parallel processors and the natural bound on running times.

This paper deals with locales and their spaces of points in intuitionistic analysis or, if you like, in (Grothendieck) toposes. One of the important aspects of the problem whether a certain locale has enough points is that it is directly related to the (constructive) completeness of a geometric theory. A useful exposition of this relationship may be found in [1], and we will assume that the reader is familiar with the general framework described in that paper.

We will consider four formal spaces, or locales, namely formal Cantor space C, formal Baire space B, the formal real line R, and the formal function space RR being the exponential in the category of locales (cf. [3]). The corresponding spaces of points will be denoted by pt(C), pt(B), pt(R) and pt(RR). Classically, these locales all have enough points, of course, but constructively or in sheaves this may fail in each case. Let us recall some facts from [1]: the assertion that C has enough points is equivalent to the compactness of the space of points pt(C), and is traditionally known in intuitionistic analysis as the Fan Theorem (FT). Similarly, the assertion that B has enough points is equivalent to the principle of (monotone) Bar Induction (BI). The locale R has enough points iff its space of points pt(R) is locally compact, i.e. the unit interval pt[0, 1] ⊂ pt(R) is compact, which is of course known as the Heine-Borel Theorem (HB). The statement that RR has enough points, i.e. that there are “enough” continuous functions from R to itself, does not have a well-established name. We will refer to it (not very imaginatively, I admit) as the principle (EF) of Enough Functions.

Reverse mathematics uses subsystems of second order arithmetic to determine which set existence axioms are required to prove particular theorems. Surprisingly, almost every theorem studied is either provable in RCA0 or equivalent over RCA0 to one of four other subsystems: WKL0, ACA0, ATR0 or – CA0. Of these subsystems, – CA0 has the fewest known equivalences. This article presents a new equivalence of – C0 which comes from ordered group theory.

One of the fundamental problems about ordered groups is to classify all possible orders for various classes of orderable groups. In general, this problem is extremely difficult to solve. Mal'tsev [1949] solved a related problem by showing that the order type of a countable ordered group is ℤαℚε where ℤ is the order type of the integers, ℚ is the order type of the rationals, α is a countable ordinal, and ε is either 0 or 1. The goal of this article is to prove that this theorem is equivalent over RCA0 to – CA0.

In Section 2, we give the basic definitions and notation for RCA0, ACA0 and CA0 as well as for ordered groups. For more information on reverse mathematics, see Friedman, Simpson, and Smith [1983] or Simpson [1999] and for ordered groups, see Kokorin and Kopytov [1974] or Fuchs [1963]. Our notation will follow these sources. In Section 3, we show that – CA0 suffices to prove Mal'tsev's Theorem and the reversal is done over RCA0 in Section 4.

Relevance logic began in an attempt to avoid the so-called fallacies of relevance. These fallacies can be in implicational form or in deductive form. For example, Lewis's first paradox can beset a system in implicational form, in that the system contains as a theorem the formula (A & ∼A) → B; or it can beset it in deductive form, in that the system allows one to deduce B from the premisses A, ∼A.

Relevance logic in the tradition of Anderson and Belnap has been almost exclusively concerned with characterizing a relevant conditional. Thus it has attacked the problem of relevance in its implicational form. Accordingly for a relevant conditional → one would not have as a theorem the formula (A & ∼A) → B. Other theorems even of minimal logic would also be lacking. Perhaps most important among these is the formula (A → (B → A)). It is also a well-known feature of their system R that it lacks the intuitionistically valid formula ((A ∨ B) & ∼A) → B (disjunctive syllogism).

But it is not the case that any relevance logic worth the title even has to concern itself with the conditional, and hence with the problem in its implicational form. The problem arises even for a system without the conditional primitive. It would still be an exercise in relevance logic, broadly construed, to formulate a deductive system free of the fallacies of relevance in deductive form even if this were done in a language whose only connectives were, say, &, ∨ and ∼. Solving the problem of relevance in this more basic deductive form is arguably a precondition for solving it for the conditional, if we suppose (as is reasonable) that the relevant conditional is to be governed by anything like the rule of conditional proof.

Let P be a forcing notion and $G\subseteq P$ its generic subset. Suppose that we have in $V[G]$ a $\kappa{-}$ complete ultrafilter1 , 2 W over $\kappa $ . Set $U=W\cap V$ .

The main subject of the article is the finite submodel property for ℵ0-categorical structures, in particular under the additional assumptions that the structure is simple, 1-based and has trivial dependence. Here, a structure has the finite submodel property if every sentence which is true in the structure is true in a finite substructure of it. It will be useful to consider a couple of other finiteness properties, related to the finite submodel property, which are variants of the usual concept of saturation.

For the rest of the introduction we will assume that M is an ℵ0-categorical (infinite) structure with a countable language. We also assume that there is an upper bound to the arity of the function symbols in M:s language and that, for every 0 < n < ℵ0 and R ⊆ Mn which is definable in M without parameters, there exists a relation symbol, in the language of M, which is interpreted as R; these assumptions are not necessary for most results to be presented, but it simplifies the statement of a result which I mention in this introduction.

First we will consider ‘canonically embedded’ substructures of Meq. Here, a structure N is canonically embedded in Meq if N's universe is a subset of Meq which is definable without parameters and, for every 0 < n < ℵ0 and R ⊆ Nn which is ε-definable in Meq there is a relation symbol in the language of N which is interpreted as R; we also assume that the language of N has no other relation (or function or constant) symbols.

We obtain an almost everywhere quantifier elimination for (the noncritical fragment of) the logic with probability quantifiers, introduced by the first author in [10]. This logic has quantifiers like ∃≥3/4y which says that “for at least 3/4 of all y”. These results improve upon the 0-1 law for a fragment of this logic obtained by Knyazev [11]. Our improvements are:

1. We deal with the quantifier ∃≥ry, where y is a tuple of variables.

2. We remove the closedness restriction, which requires that the variables in y occur in all atomic subformulas of the quantifier scope.

3. Instead of the unbiased measure where each model with universe n has the same probability, we work with any measure generated by independent atomic probabilities PR for each predicate symbol R.

4. We extend the results to parametric classes of finite models (for example, the classes of bipartite graphs, undirected graphs, and oriented graphs).

5. We extend the results to a natural (noncritical) fragment of the infinitary logic with probability quantifiers.

6. We allow each PR, as well as each r in the probability quantifier (∃≥ry), to depend on the size of the universe.

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 assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

Here we prove that if T and T′ are strongly minimal theories, where T′ satisfies a certain property related to triviality and T does not, and T′ is model complete, then there is no computable embedding of Mod(T) into Mod(T′). Using this, we answer a question from [4], showing that there is no computable embedding of VS into ZS, where VS is the class of infinite vector spaces over ℚ, and ZS is the class of models of Th(ℤ, S). Similarly, we show that there is no computable embedding of ACF into ZS, where ACF is the class of algebraically closed fields of characteristic 0.

Let λ be a singular cardinal of uncountable cofinality ν. Under various assumptions about the sizes of covering families for cardinals below λ, we prove upper bounds for the covering number COV(λ. λ. ν+. 2). This covering number is closely related to the cofinality of the partial order ([λ]ν. ⊆).

We prove that the following are consistent with ZFC:

1. 2ω = ℵω1 + #x039A;c = ℵω1 + ΚΒ = ΚU = ω2 (for measure and category simultaneously).

2. .

This concludes the discussion about the cofinality of Κc.

A calculus, or its formalization, is m-valued when its truth-functions are allowed to take truth-values ranging from 1 through m. It is customary to divide the m truth-values that are possible into those that are “designated” and those that are “undesignated.” Furthermore, it is usually desired of a formalization that provable formulas and only provable formulas take designated truth-values exclusively. For our present purposes, we shall suppose that the truth-values 1, …, 8 are designated and the truth-values 8+1, …, m are undesignated. When calculi differ in respect to the number of their designated truth-values, we shall consider them different calculi, even if they are otherwise similar.

In Gabbay [1] it is stated as an open problem whether or not Craig's Theorem holds for the logic of constant domains CD, i.e. for the extension of the intuitionistic predicate calculus, IPC, obtained by adding the schema; . Then in the later article, [2], Gabbay gives a proof of it. The proof given in [2] is via Robinson's (weak) consistency theorem and depends on relatively complicated (Kripke-) model-theoretical constructions developed in [1] (see p. 392 of [1] for a brief sketch of the method). The aim of this note is to show that the interpolation theorem for CD can also be obtained, by simple proof-theoretic methods, from §80 of Kleene's Introduction to Metamathematics [3].

GI is the classical formal system whose postulates are given on p. 442 of [3]. Let GD be the system obtained from GI by the following modifications: (1) the sequents of GD are to have at most two formulas in their succedents and (2) the intuitionistic restriction that Θ be empty is required for the succedent rules (→ ¬) and (→ ⊃). It is a simple matter to show that: , x not free in . It then follows that, using Theorem 46 of [3], if then .