bluuebunny/crossref_metadata_2025_split huggingface dataset:
A remark on Scott's interpolation theorem for Lω1ω
Authors: Motoháshi Nobuyoshi 
Published: May 2006

In [1], H. Africk proved that Scott's interpolation theorem does not hold in the infinitary logic Lω1ω. In this paper we shall show that there is an interpolation theorem in Lω1ω which can be considered as an extension of Scott's interpolation theorem in Lω1ω by using a technique developed in Motohashi [2] and [3]. We use the terminology in [1]. Therefore {Ri; i ∈ J} is the set of predicate symbols in our language. Now let us divide the set of all the free variables into mutually disjoint infinite sets {VI; I ⊆ J}. Suppose that ℱ ⊆ (J). Then a formula in Lω1ω is said to be an ℱ′-formula if it is obtained from atomic formula of the form Ri(X1, …, Xn) for some I ∈ i ∈ I and X1, …, Xn in V1,, by applying ¬ (negation), ∧ (countable conjunction), ∨ (countable disjunction), → (implication), ∀ (universal quantification), and ∃ (existential quantification). Notice that every ℱ-sentence in [1] is an ℱ′. sentence (ℱ′-closed formula) in our sense.Then we have the following theorem which is an immediate consequence of the interpolation theorem in [2].Theorem. Let A and ? be sentences. There is an ℱ′-sentence C such that A→C and C→B are provable iff whenever and are ℱ-isomorphic structures and satisfies A, then satisfies B.
Index sets in Ershov's hierarchy
Authors: Grassin Jacques 
Published: May 2006

This work is an attempt to characterize the index sets of classes of recursively enumerable sets which are expressible in terms of open sets in the Baire topology on the power set of the set N of natural numbers, usual in recursion theory. Let be a class of subsets of N and be the set of indices of recursively enumerable sets Wх belonging to .A well-known theorem of Rice and Myhill (cf. [5, p. 324, Rice-Shapiro Theorem]) states that is recursively enumerable if and only ifis a r.e. open set. In this case, note that if is not empty and does not contain all recursively enumerable sets, is a complete set. This theorem will be partially extended to classes which are boolean combinations of open sets by the following:(i) There is a canonical boolean combination which represents, namely the shortest among boolean combinations which represent.(ii) The recursive isomorphism type of depends on the length n of this canonical boolean combination (and trivial properties of ); for instance, is recursively isomorphic (in the particular case where is a boolean combination of recursive open sets) to an elementary set combination Yn or Un, constructed from {х ∣ х Wх) and depending on the length n. We can say also that is a complete set in the sense of Ershov's hierarchy [1] (in this particular case).
Lω1ω is enough: a reduction theorem for some infinitary languages
Authors: Reyes Gonzalo E. 
Published: May 2006

In this paper we deal with infinitary languages which allow arbitrary infinite disjunctions and conjunctions, but finite strings of quantifiers only. Furthermore, we shall assume that the primitive predicate symbols are finitary. (Cf. [K] and §2 for further information and unexplained notations.)Our main result shows that every “reasonable” language of this type is, in a certain sense, reducible to one of the same type which allows countable disjunctions and conjunctions only. More precisely if we let θ be the first measurable cardinal and μ ≤ θ, we have the following result partially announced in [R1]:Theorem. Every theory in a Lμω language is equivalent to some theory in a Lω1ω language in the sense that atomic formulas of one language can be mapped into formulas of the other in such a way that, for every set, these maps establish a bijective correspondence between their models having that set as a universe.The paper is divided into three sections. In the first, we derive the main result from a theorem of Sikorski on ω-homomorphisms of μ-complete boolean algebras. In the second, we give a refined version of Sikorski's theorem from which cardinality bounds can be given for the Lω1ω theories obtained by our reduction. In the last, we use a Skolem ultrapower construction to give examples for which these bounds are actually attained. As a corollary, we obtain the converse of our main result.
Post completeness in modal logic
Authors: Segerberg Krister 
Published: May 2006

Let ⊥, →, and □ be primitive, and let us have a countable supply of propositional letters. By a (modal) logic we understand a proper subset of the set of all formulas containing every tautology and being closed under modus ponens and substitution. A logic is regular if it contains every instance of □A ∧ □B ↔ □(A ∧ B) and is closed under the ruleA regular logic is normal if it contains □⊤. The smallest regular logic we denote by C (the same as Lemmon's C2), the smallest normal one by K. If L and L' are logics and L ⊆ L′, then L is a sublogic of L', and L' is an extension of L; properly so if L ≠ L'. A logic is quasi-regular (respectively, quasi-normal) if it is an extension of C (respectively, K).A logic is Post complete if it has no proper extension. The Post number, denoted by p(L), is the number of Post complete extensions of L. Thanks to Lindenbaum, we know thatThere is an obvious upper bound, too:Furthermore,.
A new proof of a theorem of Shelah
Authors: Rosenthal John W. 
Published: May 2006

In [10, §0, E), 5)] Shelah states using the proofs of 7.9 and 6.9 in [9] it is possible to prove that if a countable first-order theory T is ℵ0-stable (totally transcendental) and not ℵ1-categorical, then it has at least ∣1 + α∣ models of power ℵα.In this note we will give a new proof of this theorem using the work of Baldwin and Lachlan [1]. Our original proof used the generalized continuum hypothesis (GCH). We are indebted to G. E. Sacks for suggesting that the notions of ℵ0-stability and ℵ1-categoricity are absolute, and that consequently our use of GCH was eliminable [8]. Routine results from model theory may be found, e.g. in [2].Proof (with GCH). In the proof of Theorem 3 of [1] Baldwin and Lachlin show of power ℵα such that there is a countable definable subset in . Let B0 be such a subset. Say . We will give by transfinite induction an elementary chain of models of T of power ℵα such that B[i1 … in] has power ℵβ and such that every infinite definable subset of has power ≥ℵβ. This clearly suffices.
Systems of notations and the ramified analytical hierarchy
Authors: Lukas Joan D. , Putnam Hilary 
Published: May 2006

The purpose of this paper is to show that arithmetically minimal systems of notations can be constructed which provide notations for all ramified analytical ordinals (all the ordinals in the minimum β-model for analysis). This is a much larger section of the second number class than the Church-Kleene constructive ordinals (although still only an initial segment of the ordinals). Arithmetic minimality means that if H is an “H-set” associated with an ordinal α in our system and H′ is an H-set associated with the same ordinal α in an arbitrary system of notations S, then H is arithmetical in H′. Thus the arithmetical degrees associated with ordinals in our system are as low as possible.In order to clarify the structure of degrees of unsolvability and, more generally, to gain a deeper insight into the power set of the integers, coarser but neater classifications than the structure of Turing degrees have been sought. Several hierarchies of sets of integers have been studied, each of which organizes a certain class of sets (or their degrees of unsolvability) into a well-ordering of levels with increasing complexity of nonrecursiveness appearing at each new level. The best known of these hierarchies is the Kleene hierarchy of arithmetical sets.
Analytic inductive definitions
Authors: Cenzer Douglas 
Published: May 2006

An operator Γ mapping P(ω) to itself is inductive if Γ(A) ⊇ A for all A. For such an inductive operator Γ we define {Γα: α ∈ ORD} by letting Γ = ⌀, Γα + 1 = Γ(Γα) for all α, and Γβ = ⋃{Γα: α < β} for limit ordinals β. The closure ordinal ∣Γ∣ of Γ is the least ordinal α such that Γα+1 = Γα and the closure is Γ∣Γ∣.Let be a class of operators over the natural numbers. The closure ordinal ∣∣ of is the supremum of {∣Γ∣: Γ is an inductive operator and }. The closure algebra generated by is {A ⊆ ω: A is 1-1 reducible to for some inductive operator }. The inductive algebra generated by is {Γα: Γ is an inductive operator in and α < ∣Γ∣}.For A ∈ P(ω), is the supremum of the ordinals of well-orderings recursive in A. For , let be the supremum of . For example, it is well known that ω() = ω(∆10 = ω1, ω(Π11) = ω1 and ω(Δn1) = δn1 for n > 1 (the latter by definition).
Infinitary formulas preserved under unions of models
Authors: Nebres Bienvenido F. 
Published: May 2006

So-called “preservation theorems” relate the (possible) syntactic form of the axioms of a theory to certain closure conditions on its class of models. Such results are well known for the first-order predicate calculus, Lω, ω, and there are various expositions; e.g., Keisler [14], [15]. For the language , the first results were the theorems of Lopez-Escobar on sentences preserved under homomorphic images and of Malitz on formulas preserved under substructures. More recently, Feferman added a result on formulas preserved under (or persistent for) ∈-extensions. Some of these theorems will be considered in subsequent sections. A more thorough treatment may be found in Makkai [17]. The main new preservation result obtained here characterizes the sentences preserved under ω-unions. This notion and the statement of the theorem will be explained shortly.It is a familiar experience in mathematical research that concepts which are equivalent in a special case diverge in general. In the case at hand, one must expect to consider different possible statements for , which generalize a known result for Lω, ω. Moreover, diverse proofs may yield the same result in the special case, not all of which can be extended to the general case. Again, since the compactness theorem fails for , one cannot expect to extend the arguments from Lω, ω which use this in an essential way.
Almost strongly minimal theories. I
Authors: Baldwin John T. 
Published: May 2006

In [1] the notions of strongly minimal formula and algebraic closure were applied to the study of ℵ1-categorical theories. In this paper we study a particularly simple class of ℵ1-categorical theories. We characterize this class in terms of the analysis of the Stone space of models of T given by Morley [3].We assume familiarity with [1] and [3], but for convenience we list the principal results and definitions from those papers which are used here. Our notation is the same as in [1] with the following exceptions.We deal with a countable first order language L. We may extend the language L in several ways. If is an L-structure, there is a natural extension of L obtained by adjoining to L a constant a for each (the universe of ). For each sentence A(a1, …, an) ∈ L(A) we say satisfies A(a1, …, an) and write if in Shoenfield's notation If is an L-structure and X is a subset of , then L(X) is the language obtained by adjoining to L a name x for each is the natural expansion of to an L(X)-structure. A structure is an inessential expansion [4, p. 141] of an L-structure if for some .
Omitting quantifier-free types in generic structures
Authors: Macintyre Angus 
Published: May 2006

The central result of this paper was proved in order to settle a problem arising from B. H. Neumann's paper [10].In [10] Neumann proved that if a finitely generated group H is recursively absolutely presentable then H is embeddable in all nontrivial algebraically-closed groups. Harry Simmons [14] clarified this by showing that a finitely generated group H is recursively absolutely presentable if and only if H can be recursively presented with solvable word-problem. Therefore, if a finitely generated group H can be recursively presented with solvable word-problem then H is embeddable in all nontrivial algebraically-closed groups.The problem arises of characterizing those finitely generated groups which are embeddable in all nontrivial algebraically-closed groups. In this paper we prove, by a forcing argument, that if a finitely generated group H is embeddable in all non-trivial algebraically-closed groups then H can be recursively presented with solvable word-problem. Thus Neumann's result is sharp.Our results are obtained by the method of forcing in model-theory, as developed in [1], [12]. Our method of proof has nothing to do with group-theory. We prove general results, Theorems 1 and 2 below, about constructing generic structures without certain isomorphism-types of finitely generated substructures. The formulation of these results requires the notion of Turing degree. As an application of the central result we prove Theorem 3 which gives information about the number of countable K-generic structures.We gratefully acknowledge many helpful conversations with Harry Simmons.
Reduction of tense logic to modal logic. I
Authors: Thomason S. K. 
Published: May 2006

It will be shown that propositional tense logic (with the Kripke relational semantics) may be regarded as a fragment of propositional modal logic (again with the Kripke semantics). This paper deals only with model theory. The interpretation of formal systems of tense logic as formal systems of modal logic will be discussed in [6].The languages M and T, of modal and tense logic respectively, each have a countable infinity of propositional variables and the Boolean connectives; in addition, M has the unary operator ⋄ and T has the unary operators F and P. A structure is a pair <W, ≺<, where W is a nonempty set and ≺ is a binary relation on W. An assignment V assigns to each propositional variable p a subset V(p) of W. Then V(α)£ W is defined for all formulas α of M or T by induction:We say that α is valid in <W, ≺>, or <W, ≺>, ⊨ α, if ⊩ (α) = W for every assignment V for <W, ≺>. If Γ is a set of formulas of M [T] and α is a formula of M [T], then α is a logical consequence of Γ, or Γ ⊩ α,if α is valid in every model of Γ i.e., in every structure in which all γ ∈ Γ are valid.
A minimal degree not realizing least possible jump
Authors: Sasso Leonard P. 
Published: May 2006

The least possible jump for a degree of unsolvability a is its join a ∪ 0′ with 0′. Friedberg [1] showed that each degree b ≥ 0′ is the jump of a degree a realizing least possible jump (i.e., satisfying the equation a′ = a ∪ 0′). Sacks (cf. Stillwell [8]) showed that most (in the sense of Lebesgue measure) degrees realize least possible jump. Nevertheless, degrees not realizing least possible jump are easily found (e.g., any degree b ≥ 0′) even among the degrees <0′ (cf. Shoenfield [5]) and the recursively enumerable (r.e.) degrees (cf. Sacks [3]).A degree is called minimal if it is minimal in the natural partial ordering of degrees excluding least element 0. The existence of minimal degrees <0” was first shown by Spector [7]; Sacks [3] succeeded in replacing 0” by 0′ using a priority argument. Yates [9] asked whether all minimal degrees <0′ realize least possible jump after showing that some do by exhibiting minimal degrees below each r.e. degree. Cooper [2] subsequently showed that each degree b > 0′ is the jump of a minimal degree which, as corollary to his method of proof, realizes least possible jump. We show with the aid of a simple combinatorial device applied to a minimal degree construction in the manner of Spector [7] that not all minimal degrees realize least possible jump. We have observed in conjunction with S. B. Cooper and R. Epstein that the new combinatorial device may also be applied to minimal degree constructions in the manner of Sacks [3], Shoenfield [6] or [4] in order to construct minimal degrees <0′ not realizing least possible jump. This answers Yates' question in the negative. Yates [10], however, has been able to draw this as an immediate corollary of the weaker result by carrying out the proof in his new system of prioric games.
The consistency of a system of combinatory restricted generality
Authors: Curry Haskell B. 
Published: May 2006

The system , proposed without a claim as to its consistency in [CLg, §15C], has turned out to entail a consequence which is extremely suspicious. For from the axiom ⊦LH it follows thatholds for arbitrary X; hence for X ≡ YH, where Y is a fixed-point combinator, we haveWhether or not this leads to an actual contradiction it would be interesting to know; but, no matter whether it does or not, (2) seems highly counterintuitive.The aim of this note is to point out that a related system, here called , is demonstrably consistent and sufficient for all practical purposes served by , and at the same time to correct an error in [CLg, §15D2], which was discovered too late to be corrected on the proofs. This is formed from by dropping H from the list of θ's, and hence deleting ⊦LH and (1). The new system does not allow inferential rules to be converted into formulas with the same ease as does ; but if one is content with stating the results as rules, is adequate for all the main results deduced for in [CLg, §15C–D].
On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem Löwenheim theorems and compactness of related quantifiers
Authors: Rubin Matatyahu , Shelah Saharon 
Published: May 2006

AbstractTheorem 1. (◊ℵ1,) If B is an infinite Boolean algebra (BA), then there is B1, such that ∣ Aut (B1) ≤∣B1∣ = ℵ1 and 〈B1, Aut (B1)〉 ≡ 〈B, Aut(B)〉.Theorem 2. (◊ℵ1) There is a countably compact logic stronger than first-order logic even on finite models.This partially answers a question of H. Friedman. These theorems appear in §§1 and 2.Theorem 3. (a) (◊ℵ1) If B is an atomic ℵ-saturated infinite BA, Ψ Є Lω1ω and 〈B, Aut (B)〉 ⊨Ψ then there is B1, Such that ∣Aut(B1)∣ ≤ ∣B1∣ =ℵ1, and 〈B1, Aut(B1)〉⊨Ψ. In particular if B is 1-homogeneous so is B1. (b) (a) holds for B = P(ω) even if we assume only CH.
Theories with exactly three countable models and theories with algebraic prime models
Authors: Pillay Anand 
Published: May 2006

We prove first that if T is a countable complete theory with n(T), the number of countable models of T, equal to three, then T is similar to the Ehrenfeucht example of such a theory. Woodrow [4] showed that if T is in the same language as the Ehrenfeucht example, T has elimination of quantifiers, and n(T) = 3 then T is very much like this example. All known examples of theories T with n(T) finite and greater than one are based on the Ehrenfeucht example. We feel that such theories are a pathological case. Our second theorem strengthens the main result of [2]. The theorem in the present paper says that if T is a countable theory which has a model in which all the elements of some infinite definable set are algebraic of uniformly bounded degree, then n(T) ≥ 4. It is known [3] that if n(T) > 1, then n(T) > 3, so our result is the first nontrivial step towards proving that n(T) ≥ ℵ0. We would also like, of course, to prove the result without the uniform bound on the finite degrees of the elements in the subset.Theorem 2.1 is included in the author's Ph. D. thesis, as is a weaker version of Theorem 3.7. Thanks are due to Harry Simmons for his suggestions concerning the presentation of the material, and to Wilfrid Hodges for his advice while I was a Ph. D. student.
Upper bounds on locally countable admissible initial segments of a Turing degree hierarchy
Authors: Hodes Harold T. 
Published: May 2006

AbstractWhere AR is the set of arithmetic Turing degrees, 0(ω) is the least member of {a(2) ∣ a is an upper bound on AR}. This situation is quite different if we examine HYP, the set of hyperarithmetic degrees. We shall prove (Corollary 1) that there is an a, an upper bound on HYP, whose hyperjump is the degree of Kleene's . This paper generalizes this example, using an iteration of the jump operation into the transfinite which is based on results of Jensen and is detailed in [3] and [4]. In § 1 we review the basic definitions from [3] which are needed to state the general results.
Modal semantics without possible worlds
Authors: Kearns John T. 
Published: May 2006

In this paper I will develop a semantic account for modal logic by considering only the values of sentences (and formulas). This account makes no use of possible worlds. To develop such an account, we must recognize four values. These are obtained by subdividing (plain) truth into necessary truth (T) and contingent truth (t); and by subdividing falsity into contingent falsity (f) and necessary falsity (impossibility: F). The semantic account results from reflecting on these concepts and on the meanings of the logical operators.To begin with, we shall consider the propositional language L0. The language L0 has (1) infinitely many atomic sentences, (2) the two truth-functional connectives ∼, ∨, and the modal operator □. (Square brackets are used for punctuation.) The other logical expressions are defined as follows:D1 [A & B] = (def)∼[∼A ∨ ∼ B],D2 [A ⊃ B] = (def)[∼A ∨ B],D3 ◊ A =(def)∼□∼A.I shall use matrices to give partial characterizations of the significance of logical expressions in L0. For negation, this matrix is wholly adequate:Upon reflection, it should be clear that this matrix is the obviously correct matrix for negation.
On generic extensions without the axiom of choice
Authors: Monro G. P. 
Published: May 2006

AbstractLet ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let M be a countable transitive model of ZF. The method of forcing extends M to another model M[G] of ZF (a “generic extension”). If the axiom of choice holds in M it also holds in M[G], that is, the axiom of choice is preserved by generic extensions. We show that this is not true for many weak forms of the axiom of choice, and we derive an application to Boolean toposes.
Countable models of nonmultidimensional ℵ0-stable theories
Authors: Bouscaren Elisabeth , Lascar Daniel 
Published: May 2006

In this paper T will always be a countable ℵ0-stable theory, and in this introduction a model of T will mean a countable model.One of the main notions we introduce is that of almost homogeneous model: we say that a model M of T is almost homogeneous if for all ā and finite sequences of elements in M, if the strong type of ā is the same as the strong type of (i.e. for all equivalence relations E, definable over the empty set and with a finite number of equivalence classes, ā and are in the same equivalence class), then there is an automorphism of M taking ā to . Although this is a weaker notion than homogeneity, these models have strong properties, and it can be seen easily that the Scott formula of any almost homogeneous model is in L1. In fact, Pillay [Pi.] has shown that almost homogeneous models are characterized by the set of types they realize.The motivation of this research is to distinguish two classes of ℵ0-Stable theories:(1) theories such that all models are almost homogeneous;(2) theories with 2ℵ0 nonalmost homogeneous models.The example of theories with Skolem functions [L. 1] (almost homogeneous is then equivalent to homogeneous) seems to indicate a link between these properties and the notion of multidimensionality, and that nonmultidimensional theories are in the first case.