bluuebunny/crossref_metadata_2025 huggingface dataset:
Gap-2 morasses of height ω
Authors: Velleman Dan
Published: May 2006

In [3] we have shown how the simplification of gap-1 morasses introduced in [4] can be extended to gap-2 morasses. As in the gap-1 case, the definition of simplified (κ, 2)-morasses (gap-2 morasses of height κ) makes sense for κ = ω, although the smallest size for Jensen's original gap-2 morasses was κ = ω1. The existence of simplified gap-1 morasses of height ω is provable in ZFC (see [2]), but the same cannot be true for gap-2 morasses, since the top level of a simplified (ω, 2)-morass is a simplified (ω1, 1)-morass, and the existence of simplified (ω1, 1)-morasses is not provable in ZFC. In this paper we prove the best existence theorem for simplified (ω, 2)-morasses that we can hope for:

Theorem 1.1. There is a simplified (ω, 2)-morass iff there is a simplified (ω1, 1)-morass.

Before giving the proof we briefly summarize the definition of simplified (ω, 2)-morasses. For a more thorough treatment, and proofs of some of the basic properties of simplified gap-2 morasses which will be used below, we refer the reader to [3]. Suppose ‹‹φζ ∣ ζ ≤ ω1›, ‹ℊζξ ∣ ξ < ξ ≤ ω1›› is a simplified (ω1, 1)-morass. (We are using the “expanded” version of the definition.) We assume the morass is neat (see [4]). For each ζ < ω1 let σζ be the split point of ℊζ, ζ + 1; i.e., ℊζ, ζ + 1 is a pair {id, b}, where id is the identity function, b ↾ σζ = id and b(σζ) = φζ. A simplified (ω, 2)-morass will describe the construction of this simplified (ω, 1)-morass from finite pieces. The finite pieces will be the initial segments of the morass picked out by an increasing sequence of natural numbers ‹ni ∣ i < ω ›. We call this piece level i of the (ω, 2)-morass, and we call the full simplified (ω1, 1)-morass level ω. To make sure the pieces are really finite, we require that φx and ℊxy are finite for x < y < ω.
Monadic second order definable relations on the binary tree
Authors: Läuchli Hans , Savioz Christian
Published: May 2006

Let S2S [WS2S] respectively be the strong [weak] monadic second order theory of the binary tree T in the language of two successor functions. An S2S-formula whose free variables are just individual variables defines a relation on T (rather than on the power set of T). We show that S2S and WS2S define the same relations on T, and we give a simple characterization of these relations.
Tarski and geometry
Authors: Szczerba L. W.
Published: May 2006

Tarski published his first geometry paper, [24b], in 1924. As is well known, the area of the union of two disjoint figures is the sum of the areas of these two figures. This observation is the basis of a method for proving that two figures, say A and B, have the same area: if we can divide each of the two figures A and B into a finite number of pairwise disjoint subfigures A1,…,An and B1,…,Bn such that for every i, figures Ai and Bi are congruent (we say that two such figures are equivalent by finite decomposition), then figures A and B have the same area. The method is by no means universal. For example a disc and a rectangle can never be equivalent by finite decomposition, even if they have the same area. Hilbert [1922, Kapitel IV] proved from his axiom system the so-called De Zolt axiom:

If a polygon V is a proper subset of a polygon W then they are not equivalent by a finite decomposition.

Hilbert's proof is elementary but difficult. In [24b] Tarski gave an easy but nonelementary proof of a stronger version of the De Zolt axiom:

If a polygon V is a proper subset of a polygon W then they are not equivalent by finite decomposition into any figures.
Back and forth constructions in modal logic: An interpolation theorem for a family of modal logics
Authors: Weaver George , Welaish Jeffrey
Published: May 2006

The following is a contribution to the abstract study of the model theory of modal logics. Historically, individual modal logics have been specified deductively; and, as a result, it has seemed natural to view modal logics as sets of sentences provable in some deductive system. This proof theoretic view has influenced the abstract study of modal logics. For example, Fine [1975] defines a modal logic to be any set of sentences in the modal language L□ which contains all tautologies, all instances of the schema (□(ϕ ⊃ Ψ) ⊃ (□ϕ ⊃ □Ψ)), and which is closed under modus ponens, necessitation and substitution.

Here, however, modal logics are equated with classes of “possible world” interpretations. “Worlds” are taken to be ordered pairs (a, λ), where a is a sentential interpretation and λ is an ordinal. Properties of the accessibility relation are identified with those classes of binary relational systems closed under isomorphisms. The origin of this approach is the study of alternative Kripke semantics for the normal modal logics (cf. Weaver [1973]). It is motivated by the desire that modal logics provide accounts of both logical truth and logical consequence (cf. Corcoran and Weaver [1969]) and the realization that there are alternative Kripke semantics for S4, B and M which give identical accounts of logical truth, but different accounts of logical consequence (cf. Weaver [1973]). It is shown that the Craig interpolation theorem holds for any modal logic which has characteristic modal axioms and whose associated class of binary relational systems is closed under subsystems and finite direct products. The argument uses a back and forth construction to establish a modal analogue of Robinson's theorem. The argument for the interpolation theorem from Robinson's theorem employs modal analogues of the Ehrenfeucht-Fraïssé characterization of elementary equivalence and Hintikka's distributive normal form, and is itself a modal analogue of a first order argument (cf. Weaver [1982]).
Jensen's ⃞ principles and the Novák number of partially ordered sets
Authors: Veličković Boban
Published: May 2006

In this paper we consider various properties of Jensen's □ principles and use them to construct several examples concerning the so-called Novák number of partially ordered sets.

In §1 we give the relevant definitions and review some facts about □ principles. Apart from some simple observations most of the results in this section are known.

In §2 we consider the Novák number of partially ordered sets and, using □ principles, give counterexamples to the productivity of this cardinal function. We also formulate a principle, show by forcing that it is consistent and use it to construct an ℵ2-Suslin tree T such that forcing with T × T collapses ℵ1.

In §3 we briefly consider games played on partially ordered sets and relate them to the problems of the previous section. Using a version of □ we give an example of a proper partial order such that the game of length ω played on is undetermined.

In §4 we raise the question of whether the Novák number of a homogenous partial order can be singular, and show that in some cases the answer is no.

We assume familiarity with the basic techniques of forcing. In §1 some facts about large cardinals (e.g. weakly compact cardinals are -indescribable) and elementary properties of the constructible hierarchy are used. For this and all undefined terms we refer the reader to Jech [10].
Omitting types in -minimal theories
Authors: Marker David
Published: May 2006

Let L be a first order language containing a binary relation symbol <.

Definition. Suppose ℳ is an L-structure and < is a total ordering of the domain of ℳ. ℳ is ordered minimal (-minimal) if and only if any parametrically definable X ⊆ ℳ can be represented as a finite union of points and intervals with endpoints in ℳ.

In any ordered structure every finite union of points and intervals is definable. Thus the -minimal structures are the ones with no unnecessary definable sets. If T is a complete L-theory we say that T is strongly (-minimal if and only if every model of T is -minimal.

The theory of real closed fields is the canonical example of a strongly -minimal theory. Strongly -minimal theories were introduced (in a less general guise which we discuss in §6) by van den Dries in [1]. Extending van den Dries' work, Pillay and Steinhorn (see [3], [4] and [2]) developed an extensive structure theory for definable sets in strongly -minimal theories, generalizing the results for real closed fields. They also established several striking analogies between strongly -minimal theories and ω-stable theories (most notably the existence and uniqueness of prime models). In this paper we will examine the construction of models of strongly -minimal theories emphasizing the problems involved in realizing and omitting types. Among other things we will prove that the Hanf number for omitting types for a strongly -minimal theory T is at most (2∣T∣)+, and characterize the strongly -minimal theories with models order isomorphic to (R, <).
The natural hierarchy and quasi-hierarchy of constructibility degrees
Authors: Welch Philip
Published: May 2006

We investigate the set S2 of “quickly sharped” reals:

in the manner of [K] defining a natural hierarchy and quasi-hierarchy of constructibility degrees and identifying their termination points.
Finitely generated free Heyting algebras
Authors: Bellissima Fabio
Published: May 2006

The aim of this paper is to give, using the Kripke semantics for intuitionism, a representation of finitely generated free Heyting algebras. By means of the representation we determine in a constructive way some set of “special elements” of such algebras. Furthermore, we show that many algebraic properties which are satisfied by the free algebra on one generator are not satisfied by free algebras on more than one generator.
On the intersection of closed unbounded sets
Authors: Abraham U. , Shelah S.
Published: May 2006

Forcing extensions yield models of ZFC in which a long sequence of club subsets of ω1, has the following property: every subsequence of size ℵ1, has a finite intersection.
The geometry of weakly minimal types
Authors: Buechler Steven
Published: May 2006

Let T be superstable. We say a type p is weakly minimal if R(p, L, ∞) = 1. Let M ⊨ T be uncountable and saturated, H = p(M). We say D ⊂ H is locally modular if for all X, Y ⊂ D with X = acl(X) ∩ D, Y = acl(Y) ∩ D and X ∩ Y ≠ ∅,

Theorem 1. Let p ∈ S(A) be weakly minimal and D the realizations of stp(a/A) for some a realizing p. Then D is locally modular or p has Morley rank 1.

Theorem 2. Let H, G be definable over some finite A, weakly minimal, locally modular and nonorthogonal. Then for all a ∈ H∖acl(A), b ∈ G∖acl(A) there area′ ∈ H, b′ ∈ G such that a′ ∈ acl(abb′A)∖acl(aA). Similarly when H and G are the realizations of complete types or strong types over A.
A saturation property of ideals and weakly compact cardinals
Authors: Takahashi Joji
Published: May 2006

Suppose κ is a regular uncountable cardinal, λ is a cardinal > 1, and is a κ-complete uniform ideal on κ. This paper deals with a saturation property Sat(κ, λ, ) of , which is a weakening of usual λ-saturatedness. Roughly speaking, Sat(κ, λ, ) means that can be densely extended to λ-saturated ideals on small fields of subsets of κ. We will show that some consequences of the existence of a λ-saturated ideal on κ follow from weaker ∃: Sat(κ, λ, ), and that ∃: Sat(κ, λ, ) is connected with weak compactness and complete ineffability of κ in much the same way as the existence of a saturated ideal on κ is connected with measurability of κ.

In §2, we define Sat(κ, λ, ), mention a few results that can be proved by straightforward adaptation of known methods, and discuss generic ultrapowers of ZFC−, which will be used repeatedly in the subsequent sections as a main technical tool. A related concept Sat(κ, λ) is also defined and shown to be equivalent to ∃: Sat(κ, λ, ) under a certain condition.

In §3, we show that ∃: Sat(κ, κ, ) implies that κ is highly Mahlo, improving results in [KT] and [So].
The truth is never simple
Authors: Burgess John P.
Published: May 2006

The complexity of the set of truths of arithmetic is determined for various theories of truth deriving from Kripke and from Gupta and Herzberger.
Some properties of epistemic set theory with collection
Authors: Scedrov Andre
Published: May 2006

Myhill [12] extended the ideas of Shapiro [15], and proposed a system of epistemic set theory IST (based on modal S4 logic) in which the meaning of the necessity operator is taken to be the intuitive provability, as formalized in the system itself. In this setting one works in classical logic, and yet it is possible to make distinctions usually associated with intuitionism, e.g. a constructive existential quantifier can be expressed as (∃x) □ …. This was first confirmed when Goodman [7] proved that Shapiro's epistemic first order arithmetic is conservative over intuitionistic first order arithmetic via an extension of Gödel's modal interpretation [6] of intuitionistic logic.

Myhill showed that whenever a sentence □A ∨ □B is provable in IST, then A is provable in IST or B is provable in IST (the disjunction property), and that whenever a sentence ∃x.□A(x) is provable in IST, then so is A(t) for some closed term t (the existence property). He adapted the Friedman slash [4] to epistemic systems.

Goodman [8] used Epistemic Replacement to formulate a ZF-like strengthening of IST, and proved that it was a conservative extension of ZF and that it had the disjunction and existence properties. It was then shown in [13] that a slight extension of Goodman's system with the Epistemic Foundation (ZFER, cf. §1) suffices to interpret intuitionistic ZF set theory with Replacement (ZFIR, [10]). This is obtained by extending Gödel's modal interpretation [6] of intuitionistic logic. ZFER still had the properties of Goodman's system mentioned above.
The degrees below a 1-generic degree < 0′
Authors: Haught Christine Ann
Published: May 2006

It is shown that the nonrecursive predecessors of a 1-generic degree < 0′ are all 1-generic. As a corollary, it is shown that the 1-generic degrees are not densely ordered.
Combinatorial and recursive aspects of the automorphism group of the countable atomless Boolean algebra
Authors: Madison E. W. , Zimmermann-Huisgen B.
Published: May 2006

Given an admissible indexing φ of the countable atomless Boolean algebra ℬ, an automorphism F of ℬ is said to be recursively presented (relative to φ) if there exists a recursive function p ϵ Sym(ω) such that F ∘ φ = φ ∘ p. Our key result on recursiveness: Both the subset of Aut(ℬ) consisting of all those automorphisms which are recursively presented relative to some indexing, and its complement, the set of all “totally nonrecursive” automorphisms, are uncountable.

This arises as a consequence of the following combinatorial investigations: (1) A comparison of the cycle structures of ƒ and , where ƒ is a permutation of some free basis of ℬ and is the automorphism of ℬ induced by ƒ.(2) An explicit description of the permutations of ω whose conjugacy classes in Sym(ω) are (a) uncountable, (b) countably infinite, and (c) finite.
Maximal chains in the fundamental order
Authors: Buechler Steven
Published: May 2006

Suppose T is superstable. Let ≤ denote the fundamental order on complete types, [p] the class of the bound of p, and U(—) Lascar's foundation rank (see [LP]). We prove

Theorem 1. If q < p and there is no r such that q < r < p, then U(q) + 1 = U(p).

Theorem 2. Suppose U(p) < ω and ξ1 < … < ξk is a maximal descending chain in the fundamental order with ξk = [p]. Then k = U(p).

That the finiteness of U(p) in Theorem 2 is necessary follows from

Theorem 3. There is an ω-stable theory with a type p ϵ S1(ϕ) such that

(1) U(p) = ω + 1, and

(2) there is a maximal descending chain of proper extensions of [p] which has order type ω.
On the strength of nonstandard analysis
Authors: Henson C. Ward , Keisler H. Jerome
Published: May 2006

It is often asserted in the literature that any theorem which can be proved using nonstandard analysis can also be proved without it. The purpose of this paper is to show that this assertion is wrong, and in fact there are theorems which can be proved with nonstandard analysis but cannot be proved without it. There is currently a great deal of confusion among mathematicians because the above assertion can be interpreted in two different ways. First, there is the following correct statement: any theorem which can be proved using nonstandard analysis can be proved in Zermelo-Fraenkel set theory with choice, ZFC, and thus is acceptable by contemporary standards as a theorem in mathematics. Second, there is the erroneous conclusion drawn by skeptics: any theorem which can be proved using nonstandard analysis can be proved without it, and thus there is no need for nonstandard analysis.

The reason for this confusion is that the set of principles which are accepted by current mathematics, namely ZFC, is much stronger than the set of principles which are actually used in mathematical practice. It has been observed (see [F] and [S]) that almost all results in classical mathematics use methods available in second order arithmetic with appropriate comprehension and choice axiom schemes.
Flagg realizability in arithmetic
Authors: Goodman Nicolas D.
Published: May 2006

Epistemic arithmetic—that is, first-order arithmetic with S4 as the underlying logic—was introduced by Shapiro in [7] and independently by Reinhardt in [6]. Shapiro showed that intuitionistic first-order arithmetic HA can be embedded in epistemic arithmetic EA. Moreover he showed that some of the basic proof-theoretic facts about HA, such as the existence and disjunction properties, can be extended to EA. In [3] we showed that the interpretation of HA in EA is faithful. (G.E. Mine has independently also proved this theorem.) Finally, in [2], Flagg showed that a suitable form of Church's thesis is consistent with EA. (Carlson [1] has announced another proof of this result.) Flagg's argument involves an ingenious realizability notion for EA which, as it stands, is not very perspicuous. The purpose of the present paper is to give a more transparent treatment of Flagg realizability. We obtain a new version of Flagg's proof of the consistency of Church's thesis with EA. Our main new result is that, in a sense to be made precise below, Flagg realizability coincides on HA embedded in EA with Kleene's 1945 realizability (e.g. see [5, pp. 501–516]). Thus it turns out once more that methods and results proved for EA can be viewed as extensions or generalizations of well-known methods and results for HA.
Prime models and almost decidability
Authors: Millar Terrence
Published: May 2006

This paper introduces and investigates a notion that approximates decidability with respect to countable structures. The paper demonstrates that there exists a decidable first order theory with a prime model that is not almost decidable. On the other hand it is proved that if a decidable complete first order theory has only countably many complete types, then it has a prime model that is almost decidable. It is not true that every decidable complete theory with only countably many complete types has a decidable prime model. It is not known whether a complete decidable theory with only countably many countable models up to isomorphism must have a decidable prime model. In [1] a weaker result was proven—if every complete extension, in finitely many additional constant symbols, of a theory T fails to have a decidable prime model, then T has 2ω nonisomorphic countable models. The corresponding statement for saturated models is false, even if all the complete types are recursive, as was shown in [2]. This paper investigates a variation of the open question via a different notion of effectiveness—almost decidable.

A tree Tr will be a subset of ω<ω that is closed under predecessor. For elements f, g in ω<ω ∪ ωω, ƒ ⊲ g iffdf ∀i < lh(ƒ)[ƒ(i) = g(i)].
ω1-like recursively saturated models of Presburger's arithmetic
Authors: Harnik Victor
Published: May 2006

Let Pr be Presburger's arithmetic, i.e., the complete theory of the structure (ω, +). Lipshitz and Nadel showed in [4] that a countable model of Pr is recursively saturated iff it can be expanded to a model of Peano arithmetic, PA. As a starting point for an introductory discussion, let us mention one more fact about countable recursively saturated models of Pr. If is such a model then the following is readily seen (as explained also in §1):

Any two countable recursively saturated elementary endextensions of are isomorphic.

If we drop “countable” from the assumption of this statement, we can still say that the two models are ∞ω-equivalent. Must they be isomorphic if both have cardinality ℵ1? Certainly not, since one of the models can be ω1-like while the other is not. Once we realized this much, let us concentrate on ω1-like structures. We prove in §3:

Theorem. Any countable recursively saturated modelof Pr hasisomorphism types of ω1-like recursively saturated elementary endextensions. Only one of these is the isomorphism type of a structure that can be expanded to a model of PA.

The key technical result is proven in §2. It says that an as above has precisely two countable recursively saturated elementary endextensions which are nonisomorphic over .
The axiom of determinacy implies dependent choices in L(R)
Authors: Kechris Alexander S.
Published: May 2006

We prove the following Main Theorem: ZF + AD + V ⇒ L(R) ⇒ DC. As a corollary we have that Con(ZF + AD) ⇒ Con(ZF + AD + DC). Combined with the result of Woodin that Con(ZF + AD) ⇒ Con(ZF + AD + ¬ ACω) it follows that DC (as well as ACω) is independent relative to ZF + AD. It is finally shown (jointly with H. Woodin) that ZF + AD + ¬DCR, where DCR is DC restricted to reals, implies the consistency of ZF + AD + DC, in fact implies R# (i.e. the sharp of L(R)) exists.
Deux remarques à propos de la propriété de recouvrement fini
Authors: Poizat Bruno
Published: May 2006

Cette note est consacrée à quelques compléments à mon article [6], que le lecteur est supposé avoir sous les yeux, et qui traite de la propriété de recouvrement fini, de Keisler. Je rappelle seulement ici que:

a) La définition originale est la suivante: une théorie complète T n'a pas la p.r.f. si à toute formule f(x, ȳ) de son langage est associé un entier n, de sorte que tout ensemble (fini!) {f(x, ā1), …, f(x, ās)}, où les āi; sont des paramètres extraits d'un modèle M de T, soit consistant dès que ses parties à n éléments le sont. Une théorie sans p.r.f. satisfait done un renforcement du théorème de compacité; elle est en ce sens plus simple qu'une théorie avec p.r.f.

b) T n'a pas la p.r.f. si et seulement si elle est stable (i.e. tout type sur un modèle M de T est définissable; à toute formule f(x, ȳ) est associée une formule g(ȳ, ā) à paramètres dans M, telle que les de M pour lesquels f(x, ) est dans le type en question soient précisément ceux qui vérifient g(ȳ, ā)), et si en outre on peut exprimer par une formule sur le paramètre ā que g(ȳ, ā) définit un f-type consistant: à f1(x, ȳ), …, fs(x, ȳ), g1(ȳ, ), …, gs(ȳ, ) est associé une formule h(), telle que pour tout ā de tout modèle M de T, ā satisfasse h si et seulement si l'ensemble de tous les fi(x, ), où est dans M et satisfait gi(ȳ, ā), est consistant.
Characteristically simple ℵ0-categorical groups
Authors: Gilman Robert H.
Published: May 2006

A countable group is ℵ0-categorical if it is characterized up to isomorphism among all countable groups by its first order theory. We investigate countable ℵ0-categorical groups which are characteristically simple and have proper subgroups of finite index. We determine such groups up to isomorphism (Corollary 4 to Proposition 3), and we show that their theories are finitely axiomatizable (Proposition 2).

Let K be a finite nonabelian simple group. In §3 we construct from K two denumberable groups, K* and K#, and use them to classify countable direct limits of finite Cartesian powers of K (Proposition 1). The results of §3 are used in §4 to show that K* and K# are ℵ0-categorical and to find axioms for their theories. By Corollary 2 to Proposition 2, K* and K# are counterexamples to the conjecture of U. Feigner [3, p. 309] that ℵ0-categorical groups are FC-solvable. Also in §4 the results mentioned in the first paragraph are proved. §2 is devoted to preliminary lemmas about finite Cartesian powers of K.

A characteristic subgroup of a group G is a subgroup which is mapped onto itself by all automorphisms of G. G is characteristically simple if its only characteristic subgroups are itself and the identity subgroup. Each characteristic subgroup of a group is a union of orbits of the automorphism group of the group. If G is ℵ0-categorical, then as the automorphism group of G has only a finite number of orbits, G has only a finite number of characteristic subgroups.
Accessible sets and (Lω1ω)t-equivalence for T3 spaces
Authors: Martínez Juan Carlos
Published: May 2006

Ziegler studies in [2] the expressive power of (Lωω)t for T3 topological spaces. He defines for every natural number n the set of n-types by induction: . If A is a T3 space, the n-type of a ∈ A is defined inductively by: : in every neighborhood of a there is an a′ ≠ a with tn(a′, A) = α}. These types are (Lωω)t-definable. Then, it is shown that two T3 spaces are (Lωω)t-equivalent precisely if for every n-type α they have the same number of points with n-type α (cf. [2]).

In order to study the expressability of (Lω1ω)t, for T3 spaces, we introduce in this paper the notion of accessible set. Looking at the behaviour of convergence by means of this notion, we refine Ziegler's notion of n-type and introduce a new set Sn of n-types, which are (Lω1ω)t-definable. Then, we prove a characterization of (Lω1ω)t-equivalence for a wide class of T3 spaces. A T3 space A belongs to this class if there is a κ ∈ ω such that, for every n ∈ ω, there are at most κ n-types in Sn which are satisfiable in A. Such a space is said to be of a-finite type. Some relations between these spaces and the spaces of finite type in the sense of [2] are shown in the last section.

The contents of the present paper are treated in more detail in [3].
An incomplete decidable modal logic
Authors: Cresswell M. J.
Published: May 2006

The most common way of proving decidability in propositional modal logic is to shew that the system in question has the finite model property. This is not however the only way. Gabbay in [4] proves the decidability of many modal systems using Rabin's result in [8] on the decidability of the second-order theory of successor functions. In particular [4, pp. 258-265] he is able to prove the decidability of a system which lacks the finite model property. Gabbay's system is however complete, in the sense of being characterized by a class of frames, and the question arises whether there is a decidable modal logic which is not complete. Since no incomplete modal logic has the finite model property [9, p. 33], any proof of decidability must employ some such method as Gabbay's. In this paper I use the Gabbay/Rabin technique to prove the decidability of a finitely axiomatized normal modal propositional logic which is not characterized by any class of frames. I am grateful to the referee for suggesting improvements in substance and presentation.

The terminology I am using is standard in modal logic. By aframeis understood a pair 〈W, R〉 in whichWis a class (of “possible worlds”) andR⊆W2. To avoid confusion in what follows, a frame will henceforth be referred to as aKripkeframe. By contrast, ageneral frameis a pair 〈,Π〉 in whichis a Kripke frame andΠis a collection of subsets ofWclosed under the Boolean operations and satisfying the condition that ifAis inΠthen so isR−1“A. A model on a frame (of either kind) is obtained by adding a functionVwhich assigns sets of worlds to propositional variables. In the case of a general frame we require thatV(p) ∈Π.
Arithmetic and the theory of types
Authors: Boffa M.
Published: May 2006

A hundred years ago, Frege proposed a logical definition of the natural numbers based on the following idea:

He replaced this circular definition by the following one:

He tried afterwards to found his theory over a notion of class satisfying a general comprehension principle:

Russell quickly derived a contradiction from this principle (the famous Russell's paradox) but saved Frege's arithmetic with his theory of types based on the following comprehension principle:

In 1979, talking at the Claude Bernard University in Lyon, I remarked that 3 types suffice to provide Frege's arithmetic, showing in fact that PA2 (second order Peano arithmetic) holds in TT3 + AI (theory of types 0, 1, 2 plus a suitable axiom of infinity). I asked whether TT3 + AI was a conservative extension of PA2. Pabion [3] gave a positive answer by a subtle use of the Fraenkel-Moskowski method. This result will be improved in the present paper, with a view to getting models of NF3 + AI in which Frege's arithmetic forms a model isomorphic to a given countable model of PA2.
Replacement and collection in intuitionistic set theory
Authors: Goodman Nicolas D.
Published: May 2006

Intuitionistic Zermelo-Fraenkel set theory, which we call ZFI, was introduced by Friedman and Myhill in [3] in 1970. The idea was to have a theory with the same axioms as ordinary classical ZF, but with Heyting's predicate calculus HPC as the underlying logic. Since some classically equivalent statements are intuitionistically inequivalent, however, it was not always obvious which form of a classical axiom to take. For example, the usual formulation of the axiom of foundation had to be replaced with a principle of transfinite induction on the membership relation in order to avoid having excluded middle provable and thus making the logic classical. In [3] the replacement axiom is taken in its most common classical form:

In the presence of the separation axiom,

this is equivalent to the axiom

It is this schema Rep that we shall call the replacement axiom.

Friedman and Myhill were able to show in [3] that ZFI has a number of desirable “constructive” properties, including the existence property for both numbers and sets. They were not able to determine, however, whether ZFI is proof-theoretically as strong as ZF. This is still open.

Three years later, in [2], Friedman introduced a theory ZF− which is like ZFI except that the replacement axiom is changed to the classically equivalent collection axiom:

He showed that ZF− is proof-theoretically of the same strength as ZF, and he asked whether ZF− is equivalent to ZFI.
Simplified morasses with linear limits
Authors: Velleman Dan
Published: May 2006

In a recent series of papers Kanamori ([4], [5], and [6]) defines generalizations of several combinatorial principles known to follow from the existence of morasses. Kanamori proves the consistency of his generalizations by forcing arguments which come close to satisfying the hypotheses of the Martin's Axiom-type characterizations of morasses developed independently by Shelah and Stanley [9] and the author [12]. A similar “almost application” of morasses appears in [11], in which Todorčević uses forcing to prove the consistency of the existence of Kurepa trees with no Aronszajn or Cantor subtrees. In all cases the attempted proofs using morasses fail for the same reason: the partial orders involved do not have strong enough closure properties.

In an attempt to solve this problem Shelah and Stanley strengthened their characterization of morasses to allow applications to what they called “good canonical limit” partial orders. However, for rather subtle reasons even this strengthened forcing axiom is not good enough for the proposed applications. The problem this time is that Shelah and Stanley's “weak commutativity of Lim and restriction” requirement (see [9, 3.9(iv)]) is not satisfied. Furthermore, there is reason to believe that an ordinary morass is just not good enough for these applications, since in L morasses exist at all regular uncountable cardinals, but even a weak form of Todorčević's conclusion cannot hold at ineffable cardinals (see the end of §4).

A possible solution to this problem is suggested by the fact that □κ is equivalent to a forcing axiom which applies to partial orders satisfying precisely the kind of weak closure conditions involved in the examples described above (see [13]). What is needed to make the proposed morass applications work is something which will do for morass constructions what □κ does for ordinary transfinite recursion constructions. In this paper we show how extra structure can be built into a morass to accomplish this goal.
Diamonds, uniformization
Authors: Shelah Saharon
Published: May 2006

Assume G.C.H. We prove that for singular λ, □λ implies the diamonds hold for many S ⊆ λ+ (including S ⊆ {δ:δ ∈ λ+, cf δ = cf λ}.) We also have complementary consistency results.
A normal form theorem for first order formulas and its application to Gaifman's splitting theorem
Authors: Motohashi Nobuyoshi
Published: May 2006

Let L be a first order predicate calculus with equality which has a fixed binary predicate symbol <. In this paper, we shall deal with quantifiers Cx, ∀x ≦ y, ∃x ≦ y defined as follows: CxA(x) is ∀y∃x(y ≦ x ∧ A(x)), ∀x ≦ yA{x) is ∀x(x ≦ y ⊃ A(x)), and ∃x ≦ yA(x) is ∃x(x ≦ y ∧ 4(x)). The expressions x̄, ȳ, … will be used to denote sequences of variables. In particular, x̄ stands for 〈x1, …, xn〉 and ȳ stands for 〈y1,…, ym〉 for some n, m. Also ∃x̄, ∀x̄ ≦ ȳ, … will be used to denote ∃x1 ∃x2 … ∃xn, ∀x1 ≦ y1 ∀x2 ≦ y2 … ∀xn ≦ yn, …, respectively. Let X be a set of formulas in L such that X contains every atomic formula and is closed under substitution of free variables and applications of propositional connectives ¬(not), ∧(and), ∨(or). Then, ∑(X) is the set of formulas of the form ∃x̄B(x̄), where B ∈ X, and Φ(X) is the set of formulas of the form

.

Since X is closed under ∧, ∨, the two sets Σ(X) and Φ(X) are closed under ∧, ∨ in the following sense: for any formulas A and B in Σ(X) [Φ(X)], there are formulas in Σ(X)[ Φ(X)] which are obtained from A ∧ B and A ∨ B by bringing some quantifiers forward in the usual manner.
Functional interpretation of the β-rule
Authors: Koletsos George
Published: May 2006

Since the invention of β-logic by J. Y. Girard, a lot of work has been done concerning different aspects of this logic, β-completeness (classical and intuitionistic), interpolation theorems, and completeness of Lβω are some examples of this (see [2], [4] and [5]).

In this paper we examine the question, posed by J. Y. Girard, of the Gödel-functional interpretation for this new logic.

The central notion in β-logic is the notion of the β-rule. The β-rule is a functor which appropriately groups x-proofs, for every ordinal x. An x-proof is like a proof in ω-logic but instead of the ω-rule, with premises indexed by ω, we use the x-rule, with premises indexed by x.

In order to obtain the Gödel-functional interpretation of the β-rule, we need, first, a functional interpretation of the x-proofs, which require functionals using the x-rule for their construction (the x-functionals) and, second, an appropriate grouping of these x-functionals by means of a functor (the β-functionals).

We use the letters x,y,z, … and sometimes the Greek letters α and γ to denote ordinals. ON is the category of ordinals. The objects are the ordinals, and the morphisms from x to y are the members of I(x, y), which is the set of all strictly increasing functions from x to y. ON < ω denotes the restriction of ON to ω, the set of finite ordinals. We denote direct systems, in ON or in more general categories, by (xi, fij) where fij is the morphism from xi, to xj. If the direct limit exists we denote it by (x, fi), where fi is the morphism from xi to x. We write (x, fi) =lim(xifij). In ON the direct limits are unique.
Proper classes via the iterative conception of set
Authors: Sharlow Mark F.
Published: May 2006

We describe a first-order theory of generalized sets intended to allow a similar treatment of sets and proper classes. The theory is motivated by the iterative conception of set. It has a ternary membership symbol interpreted as membership relative to a set-building step. Set and proper class are defined notions. We prove that sets and proper classes with a defined membership form an inner model of Bernays-Morse class theory. We extend ordinal and cardinal notions to generalized sets and prove ordinal and cardinal results in the theory. We prove that the theory is consistent relative to ZFC + (∃x) [x is a strongly inaccessible cardinal].
Natural deduction and sequent calculus for intuitionistic relevant logic
Authors: Tennant Neil
Published: May 2006

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.
Von Mises' definition of random sequences reconsidered
Authors: van Lambalgen Michiel
Published: May 2006

We review briefly the attempts to define random sequences (§0). These attempts suggest two theorems: one concerning the number of subsequence selection procedures that transform a random sequence into a random sequence (§§1–3 and 5); the other concerning the relationship between definitions of randomness based on subsequence selection and those based on statistical tests (§4).
Alfred Tarski and decidable theories
Authors: Doner John , Hodges Wilfrid
Published: May 2006

Any list of Alfred Tarski's achievements would mention his decision procedure for real-closed fields. He proved a number of other less publicized decidability results too. We shall survey these results. After surveying them we shall ask what Tarski had in mind when he proved them. Today our emphases and concepts are sometimes different from those of Tarski in the early 1930s. Some of these changes are the direct result of Tarski's own fundamental work in model theory during the intervening years.

Tarski's work on decidable theories is important not just for the individual decidability theorems themselves. His method for all these decidability results was elimination of quantifiers, and he systematically used this method to prove a range of related theorems about completeness and definability. He also led several of his students to do important work using this same method. Tarski's use of quantifier elimination has had a deep and cumulative influence on model theory and the logical treatment of algebraic theories.

We thank Solomon Feferman, Steven Givant, Haragauri Gupta, Yuri Gurevich. Angus Macintyre, Gregory Moore, Robert Vaught and the referee for helpful discussions and comments. Also we thank Madame Maria Mostowska and Roman Murawski for sending us material from Polish libraries.
Notions of relative ubiquity for invariant sets of relational structures
Authors: Bankston Paul , Ruitenburg Wim
Published: May 2006

Given a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers ω as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on ω. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on ω is ubiquitous in the set of linear orderings on ω.
Maximal r.e. equivalence relations
Authors: Carroll Jeffrey S.
Published: May 2006

The lattice of r.e. equivalence relations has not been carefully examined even though r.e. equivalence relations have proved useful in logic. A maximal r.e. equivalence relation has the expected lattice theoretic definition. It is proved that, in every pair of r.e. nonrecursive Turing degrees, there exist maximal r.e. equivalence relations which intersect trivially. This is, so far, unique among r.e. submodel lattices.
The axiomatization of randomness
Authors: van Lambalgen Michiel
Published: May 2006

We present a faithful axiomatization of von Mises' notion of a random sequence, using an abstract independence relation. A byproduct is a quantifier elimination theorem for Friedman's “almost all” quantifier in terms of this independence relation.
On infinite series of infinite isols
Authors: Barback Joseph
Published: May 2006

We are interested in regressive isols, recursive functions, and the extensions of recursive functions to the isols. One of the nicest concepts that has been applied to the study of these notions is of an infinite series of isols. J. C. E. Dekker introduced infinite series of isols in [3]. With this concept one may associate with each number theoretic function u and regressive isol B a value in the isols to correspond to the series

When u is chosen as a recursive function, or as a recursive combinatorial function, many of the sums that one associates with familiar finite series may be generalized to infinite series. For example, if B is any regressive isol, then

The results presented in our paper were motivated by an interest in extending infinite series to a setting where the terms being summed may be infinite isols. In our paper, we do this in a special way, as will be described below. We would first like to briefly comment on some facts about the concept of defining an infinite series in the isols.
Formal languages defined by the underlying structure of their words
Authors: Ressayre J. P.
Published: May 2006

i) We show for each context-free language L that by considering each word of L as a structure in a natural way, one turns L into a finite union of classes which satisfy a finitary analog of the characteristic properties of complete universal first order classes of structures equipped with elementary embeddings. We show this to hold for a much larger class of languages which we call free local languages, ii) We define local languages, a class of languages between free local and context-sensitive languages. Each local language L has a natural extension L∞ to infinite words, and we prove a series of “pumping lemmas”, analogs for each local language L of the “uvxyz theorem” of context free languages: they relate the existence of large words in L or L∞ to the existence of infinite “progressions” of words included in L, and they imply the decidability of various questions about L or L∞. iii) We show that the pumping lemmas of ii) are independent from strong axioms, ranging from Peano arithmetic to ZF + Mahlo cardinals.

We hope that these results are useful for a model-theoretic approach to the theory of formal languages.
Constructive validity is nonarithmetic
Authors: McCarty Charles
Published: May 2006

It follows constructively from weak versions of Markov's principle (MP) and of Church's thesis (WCT) that logical validity for single sentences is not arithmetically definable. This is a direct consequence of the constructive model-theoretic fact that, using MP and WCT, one can prove the categoricity of first-order Heyting arithmetic. By a straightforward refinement of this proof, we then obtain generalizations of the incompleteness theorem of Kreisel [K 2] as presented by van Dalen [Da] and improved by Leivant [L 1]. An analogous result for classical validity in r.e. models appears in [V].

Our methods of proof are new in that they rely not upon variants of the ideas of [K2] but upon a constructive proof idea inspired by the classical result of Tennenbaum ([Te], [E&K], [S]) on recursive models of arithmetic. This line of reasoning was suggested by the applications of readability to recursive mathematics described in [McC1], [McC2] and [McC3].
Souslin forcing
Authors: Ihoda Jaime I. , Shelah Saharon
Published: May 2006

We define the notion of Souslin forcing, and we prove that some properties are preserved under iteration. We define a weaker form of Martin's axiom, namely , and using the results on Souslin forcing we show that is consistent with the existence of a Souslin tree and with the splitting number s = ℵ1. We prove that proves the additivity of measure. Also we introduce the notion of proper Souslin forcing, and we prove that this property is preserved under countable support iterated forcing. We use these results to show that ZFC + there is an inaccessible cardinal is equiconsistent with ZFC + the Borel conjecture + -measurability.
A basis result in combinatory logic
Authors: Legrand Remi
Published: May 2006

The aim of this article is to show that a basis for combinatory logic [2] must contain at least one combinator with rank strictly greater than two. We use notation of [1].

Let Q be a primitive combinator given by its reduction rule Qx1 … xn → C, where C is a pure combination of the variables x1,…, xn. n is called the rank of the combinator.

A set {Q1,…,Qn} of combinators is a basis for combinatory logic if for every finite set {x1,…,xm} of variables and every pure combination C of these variables, there exists a pure combinator Q of Q1,…,Qn such that Qx1…xm↠C.

Property. The Church-Rosser theorem and the (quasi-)normalization theorem are valid for the combinatory reduction system under consideration.

Proof. See [4], [5], and [6].

Any basis must contain at least one combinator with rank strictly greater than two.

Let us assume a basis B with combinators of rank strictly less than 3; then there exists a pure combination X of the combinators in B such that: XABC ↠ CAB.

(*) First of all, we remark that if XABC ↠ M ↠ CAB, M must contain at least one occurrence of each of the variables A, B and C.

Notation. We denote by E[X1,…,Xn] expressions that contain at least one occurrence of every term X1,…,Xn.