bluuebunny/crossref_metadata_2025 huggingface dataset:
Diagonalization in degree constructions
Authors: Posner D. , Epstein R.
Published: May 2006

We present two theorems whose applications are to eliminate diagonalization arguments from a variety of constructions of degrees of unsolvability.

All definitions and notations come from [1, Chapter 1]. We give a brief resumé of them here.

We identify a set with its characteristic function. (A(x)= 1 if x ∈ A and A (x) = 0 if x ∉ A.) A string σ is the restriction of a characteristic function to a finite initial segment of natural numbers, lh(σ) = length of σ = n + 1 if σ = A[n] for some set A. (A [n] is the restriction of A to {m: m ≤ n}.) If i = 0 or 1, σ * i is defined as the string of length lh(σ) + 1 such that σ * i ⊇ σ and σ * i(lh(σ)) = i. We write σ ∣ τ if σ ⊉ τ and τ ⊉ σ.

{Φn} is a listing of the partial recursive functionals. We write “A ≤TB” (“A is Turing reducible to B”) if ∃n∀xΦn(B)(x) = A(x).

A partial function, T, from strings to strings is a tree if T is order preserving and for all strings, σ, if one of T(σ * 0), T(σ * 1) is defined then T(σ), T(σ * 0), T(σ * 1) are all defined and T(σ * 0)∣(σ * 1).
Syntactic characterisations of amalgamation, convexity and related properties
Authors: Bacsich Paul D. , Hughes Dafydd Rowlands
Published: May 2006

We prove that certain syntactic conditions similar to separation principles on a theory are equivalent to semantic properties such as amalgamation and strong amalgamation, by showing that appropriate classes of structures are definable by Lω1ω-sentences. Then we characterise the elements of core models and thus give a natural proof of Rabin's characterisation of convex theories.

The notion of a syntactic characterisation of a semantic property of a theory is by now fairly well known. The earliest such were the classical preservation theorems: For example, a theorem of Lyndon characterised the theories whose models were closed under homomorphic images as those with a set of positive axioms.

Presumably the notion of syntactic characterisation can be made precise, but it is probably better at this stage to leave it vague. The general idea is that theories are “algebras” (cylindric algebras, or logical categories, with suitable extra structure) and that a semantic property P of theories is syntactically characterisable if the class of theories with P is an “elementary” class of “algebras.”

When one codes countable theories as real numbers, a syntactically characterisable property will be arithmetical. The converse does not seem reasonable, especially as it is often fairly easy to prove a property arithmetical (using extra predicates, usually), when we may not be able to find a syntactic characterisation.
The characterization of monadic logic
Authors: Tharp Leslie H.
Published: May 2006

The first section of this paper is concerned with the intrinsic properties of elementary monadic logic (EM), and characterizations in the spirit of Lindström [2] are given. His proofs do not apply to monadic logic since relations are used, and intrinsic properties of EM turn out to differ in certain ways from those of the elementary logic of relations (i.e., the predicate calculus), which we shall call EL. In the second section we investigate connections between higher-order monadic and polyadic logics.

EM is the subsystem of EL which results by the restriction to one-place predicate letters. We omit constants (for simplicity) but take EM to contain identity. Let atypebe any finite sequence (possibly empty) of one-place predicate letters. A modelMof typehas a nonempty universe ∣M∣ and assigns to each predicate letterPofa subsetPMof ∣M∣.

Let us take a monadic logicLto be any collection of classes of models, calledL-classes, satisfying the following:

1. All models in a givenL-class are of the same type (called the type of the class).

2. Isomorphic models lie in the sameL-classes.

3. IfandareL-classes of the same type, thenandareL-classes.
Real-valued models with metric equality and uniformly continuous predicates
Authors: Katz Michael
Published: May 2006

Two real-valued deduction schemes are introduced, which agree on ⊢ ⊿ but not on Γ ⊢ ⊿, where Γ and ⊢ are finite sets of formulae. Using the first scheme we axiomatize real-valued equality so that it induces metrics on the domains of appropriate structures. We use the second scheme to reduce substitutivity of equals to uniform continuity, with respect to the metric equality, of interpretations of predicates in structures. This continuity extends from predicates to arbitrary formulae and the appropriate models have completions resembling analytic completions of metric spaces. We provide inference rules for the two deductions and discuss definability of each of them by means of the other.
A complete, decidable theory with two decidable models
Authors: Millar Terrence S.
Published: May 2006

A well-known result of Vaught's is that no complete theory has exactly two nonisomorphic countable models. The main result of this paper is that there is a complete decidable theory with exactly two nonisomorphic decidable models.

A model is decidable if it has a decidable satisfaction predicate. To be more precise, let T be a decidable theory, let {θn∣n < ω} be an effective enumeration of all formulas in L(T), and let be a countable model of T. For any indexing E = {ai∣ i < ω} of ∣∣, and any formula ϕ ∈ L(T), let ‘ϕE’ denote the result of substituting ‘ai’ for every free occurrence of ‘xi’ in ϕ, i < ω. Then is decidable just in case, for some indexing E of ∣∣, {n ∣ ⊨ θnE} is a recursive set of integers. It is easy to show that the decidability of a model does not depend on the choice of the effective enumeration of the formulas in L(T); we omit details. By a simple ‘effectivization’ of Henkin's proof of the completeness theorem (see Chang [1]) we have

Fact 1. Every decidable consistent theory has a decidable model.

Assume next that T is a complete decidable theory and {θn ∣ n < ω} is an effective enumeration of all formulas of L(T).
Logic in the twenties: the nature of the quantifier
Authors: Goldfarb Warren D.
Published: May 2006

We are often told, correctly, that modern logic originated with Frege. For Frege clearly depicted polyadic predication, negation, the conditional, and the quantifier as the bases of logic; moreover, he introduced the idea of a formal system, and argued that mathematical demonstrations, to be fully precise, must be carried out within a formal language by means of explicitly formulated syntactic rules.

Consequently Frege has often been read as providingallthe central notions that constitute our current understanding of quantification. For example, in his recent book on Frege [1973], Michael Dummett speaks of ”the semantics which [Frege] introduced for formulas of the language of predicate logic.” That is, “An interpretation of such a formula … is obtained by assigning entities of suitable kinds to the primitive nonlogical constants occurring in the formula … [T]his procedure is exactly the same as the modern semantic treatment of predicate logic” (pp. 89–90). Indeed, “Frege would therefore have had within his grasp the concepts necessary to frame the notion of the completeness of a formalization of logic as well as its soundness … but he did not do so” (p. 82).

This common appraisal of Frege's work is, I think, quite misleading. Even given Frege's tremendous achievements, the road to an understanding of quantification theory was an arduous one. Obtaining such understanding and formulating those notions which are now common coin in the discussion of logical systems were the tasks of much of the work in logic during the nineteen-twenties.
An effective selection theorem
Authors: Maitra Ashok
Published: May 2006

A recent result of J.P. Burgess [1] states:

Theorem 0. Let F be a multifunction from an analytic subset T of a Polish space to a Polish space X. If F is Borel measurable, Graph(F) is coanalytic in T × X and F(t) is nonmeager in its closure for each t Є T, then F admits a Borel measurable selector.

The above result unifies and significantly extends earlier results of H. Sarbadhikari [8], S.M. Srivastava [9] and G. Debs (unpublished). The reader is referred to [1] for details.

The aim of this article is to give an effective version of Theorem 0. We do this by proving a basis theorem for Π11 sets which are nonmeager in their closure and satisfy a local version of the measurability condition in Theorem 0. Our basis theorem generalizes a well-known result of P.G. Hinman [4] and S.K. Thomason [10] (see also [5] and [7, 4F.20]). Our methods are similar to those used by A. Louveau to prove that a , σ-compact set is contained in a , σ-compact set (see [7, 4F.18]).

The paper is organized as follows. §2 is devoted to preliminaries. In §3, we prove the basis theorem and deduce as a consequence an effective version of Theorem 0. We show in §4 how our methods can be used to give alternative proofs of some known results.

Discussions with R. Barua, B.V. Rao and V.V. Srivatsa are gratefully acknowledged. I am indebted to J.P. Burgess for drawing my attention to an error in an earlier draft of this paper.
Recursion theory on orderings. II
Authors: Remmel J. B.
Published: May 2006

In [6], G. Metakides and the author introduced a general model theoretic setting in which to study the lattice of r.e. substructures of a large class of recursively presented models . Examples included , the natural numbers with equality, 〈 Q, ≤ 〉, the rationals under the usual ordering, and a large class of n-dimensional partial orderings. In this setting, we were able to generalize many of the constructions of classical recursion theory so that the constructions yield the classical results when we specialize to the case of and new results when we specialize to other models. Constructions to generalize Myhill's Theorem on creative sets [8], Friedberg's Theorem on the existence of maximal sets [3], Dekker's Theorem on the degrees of hypersimple sets [2], and Martin's Theorem on the degrees of maximal sets [5] were produced in [6]. In this paper, we give constructions to generalize the Morley-Soare Splitting Theorem [7] and Lachlan's characterization of hyperhypersimple sets [4] in §2, constructions to generalize Lachlan's theorems on the existence of major subsets and r-maximal sets contained in maximal sets [4] in §3, and constructions to generalize Robinson's construction of r-maximal sets that are not contained in any maximal sets [11] and second-order maximal sets [12] in §4.

In §1 of this paper, we give the precise definitions of our model theoretic setting and deal with other preliminaries. Also in §1, we define the notions of “uniformly nonrecursive”, “uniformly maximal”, etc. which are the key notions involved in the generalizations of the various theorems that occur in §§2, 3 and 4.
Sequential method in quantum logic
Authors: Nishimura Hirokazu
Published: May 2006

Since Birkhoff and von Neumann [2] a new area of logical investigation has grown up under the name of quantum logic. However it seems to me that many authors have been inclined to discuss algebraic semantics as such (mainly lattices of a certain kind) almost directly without presenting any axiomatic system, far from developing any proof theory of quantum logic. See, e.g., Gunson [9], Jauch [10], Varadarajan [15], Zeirler [16], etc. In this sense many works presented under the name of quantum logic are algebraic in essence rather than genuinely logical, though it is absurd to doubt the close relationship between algebraic and logical study on quantum mechanics.

The main purpose of this paper is to alter this situation by presenting an axiomatization of quantum logic as natural and as elegant as possible, which further proof-theoretical study is to be based on.

It is true that several axiomatizations of quantum logic are present now. Several authors have investigated the so-called material implication α → β ( = ¬α∨(α ∧ β)) very closely with due regard to its importance. See, e.g., Finch [5], Piziak [11], etc. Indeed material implication plays a predominant role in any axiomatization of a logic in Hilbert-style. Clark [4] has presented an axiomatization of quantum logic with negation ¬ and material implication → as primitive connectives. In this paper we do not follow this approach. First of all, this approach is greatly complicated because orthomodular lattices are only locally distributive.
Determinateness and the separation property
Authors: Steel John R.
Published: May 2006

A pointclass is a class of subsets of the Baire space (ωω) closed under inverse images by continuous functions. The dual of a pointclass Γ, denoted is {~A∣ A ∈ Γ}. (Complements are relative to ωω.) If Γ is nonselfdual, i.e. , then let . We say a nonselfdual pointclass Γ has the first separation property, and write Sep(Γ), iff (∀A, B ∈ Γ)(A ⋂ B = ∅ ⇒ (∃C ∈ Δ)(A ⊆ C ∧ B ⋂ = ∅)). The set C is said to separate A and B.

Descriptive set theory abounds in nonselfdual pointclasses Γ, and for the more natural examples of such Γ one can always show by assuming enough determinateness that exactly one of Sep(Γ) and Sep() holds. Van Wesep [2] provides a partial explanation of this fact by showing that, assuming the full axiom of determinateness, one of Sep(Γ) and Sep() must fail for all nonselfdual pointclasses Γ. We shall complete the explanation by showing that one of Sep(Γ) and Sep() must hold.

The axiom of determinateness has other interesting consequences in the general theory of pointclasses. See e.g. [3].
Locally countable models of Σ1-separation
Authors: Abramson Fred G.
Published: May 2006

Let α be any countable admissible ordinal greater than ω. There is a transitive set A such that A is admissible, locally countable, OnA = α, and A satisfies Σ1-separation. In fact, if B is any nonstandard model of KP + ∀x ⊆ ω (the hyperjump of x exists), the ordinal standard part of B is greater than ω, and every standard ordinal in B is countable in B, then HCB ∩ (standard part of B) satisfies Σ-separation.
On the ultrafilter of closed, unbounded sets
Authors: Martin D. A. , Mitchell W.
Published: May 2006

Solovay proved in 1967 that the axiom of determinateness implies that the filter C generated by closed and unbounded subsets of ω1 is an ultrafilter. It has long been conjectured that a significant part of the theory of the axiom of determinateness should be provable from the hypothesis that C is an ultrafilter, but even the first step of finding inner models with several measurable cardinals has proved elusive. In this paper we show that such models exist. Much of our proof is a modification of Kunen's proof in [3] of the same conclusion from the existence of a measurable cardinal κ such that 2κ > κ+.

Since no proof of Solovay's result seems to have been published, we insert a proof here. We want to show that for any set x ⊂ ω1 there is a closed, unbounded set either contained in or disjoint from x. By the lemma of [4] there is a Turing degree d such that either ω1e Є x for all degrees e ≥T d or ω1e ∉ x for all degrees e ≥T d. By a theorem of Sacks [1], [5] every d-admissible is ω1e for some e ≥T d, so it is enough to show that there is a closed, unbounded set of d-admissibles. Let a ⊂ ω have degree d; then is such a set.
R-maximal Boolean algebras
Authors: Remmel J. B.
Published: May 2006

Metakides and Nerode in [2] suggested the study of what they termed the lattice of recursively enumerable substructures of a recursively presented model. For example, Metakides and Nerode in [3] introduced the lattice of of recusively enumerable subspaces, , of a recursively presented vector space V∞. The similarities and differences between and ℰ, the lattice of recursively enumerable subsets of the natural numbers N as defined in [9], have been studied by Metakides and Nerode, Kalantari, Remmel, Retzlaff, and Shore. In [6], we studied some similarities and differences between ℰ and the lattice of recursively enumerable sub-algebras of a weakly recursively presented Boolean algebra and this paper continues that study. A weakly recursively presented Boolean algebra (W.R.P.B.A.), , consists of a recursive subset of N, ∣∣, called the field of , and operations (meet), (join), and (complement) which are partial recursive and under which becomes a Boolean algebra. We shall write and for the zero and unit of . If S is a subset of , we let (S)* denote the subalgebra generated by S. Given sub-algebras B and C of , we let B + C denote (B ⋃ C)*. A subalgebra B of is recursively enumerable (recursive) if {x ∈ ∣∣ x ∈ B} is a recursively enumerable (recursive) subset of ∣∣. The set of all recursively enumerable subalgebras of , , forms a lattice under the operations of intersection and sum (+).
The optimality of induction as an axiomatization of arithmetic
Authors: Leivant Daniel
Published: May 2006

By induction for a formula φ we mean the schema

(where the terms in brackets are implicitly substituted for some fixed variable, with the usual restrictions). Let be the schema IAφ for φ in Πn (i.e. ); similarly for . Each instance of is Δn+2, and each instance of is Σn+1 Thus the universal closure of an instance α is Πn+2 in either case. Charles Parsons [72] proved that and are equivalent over Z0, where Z0 is essentially Primitive Recursive Arithmetic augmented by classical First Order Logic [Parsons 70].

Theorem. For each n > 0 there is a Πn formula π for whichis not derivable in Z0from

(i) true Πn+1sentences; nor even

(ii) Πn+1sentences consistent withZ0.
Definability in models of set theory
Authors: Guaspari David
Published: May 2006

Call a set A of ordinals “definable” over a theory T if T is some brand of set theory and whenever A appears in the standard part of a (not necessarily standard) model of T, A is “definable”. Two kinds of “definability” are considered, for each of which is provided a complete (or almost complete) characterization of the hereditarily countable sets of ordinals “definable” over true finitely axiomatizable set theories: (1) there is a single formula ϕ such that in any model of T containing A, A is the unique solution to ϕ; (2) the defining formula is allowed to vary from model to model. (Note. The restrictions “finitely axiomatizable”, and “true” are largely for the sake of convenience: such theories provably have lots of models.)

There are few allusions to what a model theorist would regard as his subject—the methods coming from recursion theory and set theory; but the treatment is intended to be intelligible to nonspecialists. The referee's criticisms have greatly improved the exposition.

I would like to thank Leo Harrington for several discussions, both helpful and hapless, and especially for a clever and timely proof which rescued this project from a moribund state. (Further thanks are due to the Movshon family, as a result of whose New Year's Eve party it became clear that the only really magic formulas are Σ1 formulas.)
Degree problems for modular machines
Authors: Cohen Daniel E.
Published: May 2006

Modular machines were introduced in [1] and [2], where they were used to give simple proofs of various unsolvability results in group theory. Here we discuss the degrees of the halting, word, and confluence problems for modular machines, both for their own sake and in the hope that the results may be useful in group theory (see [4] for an application of a related result to group theory).

In the course of the analysis, I found it convenient to compare degrees of these problems for a Turing machine T and for a Turing machine T1 obtained from T by enlarging the alphabet but retaining the same quintuples (or quadruples). The results were surprising. The degree for a problem of T1 depends not just on the corresponding degree for T, but also on the degrees of the corresponding problems when T is restricted to a semi-infinite tape (both semi-infinite to the right and semi-infinite to the left). For the halting and confluence problems, the Turing degrees of the problems for these three machines can be any r.e. degrees. In particular the halting problem of T can be solvable, while that of T1 has any r.e. degree.

A machine M (in the general sense) consists of a countable set of configurations (together with a numbering, which we usually take for granted), a recursive subset of configurations called the terminal configurations, and a recursive function, written C ⇒ C′, on the set of configurations. If, for some n ≥ 0, we have C = C0 ⇒ C1 ⇒ … ⇒ Cn = C′, we write C → C′. We say M halts from C if C → C′ for some terminal C′.
Forcing closed unbounded sets
Authors: Abraham Uri , Shelah Saharon
Published: May 2006

We discuss the problem of finding forcing posets which introduce closed unbounded subsets to a given stationary set.
Incompatible extensions of combinatorial functions
Authors: Ellentuck Erik
Published: May 2006

We find necessary and sufficient conditions for compatibility of the Myhill and the Nerode extensions of a combinatorial function to the isols.
The inverse of a regressive object
Authors: Gross W.F.
Published: May 2006

If C1, …, Ck are members of a certain class of suitable categories (which contains those arising from models with dimension), C = C1 × … × Ck, C′ is a suitable category, F: C → C′ is a partial recursive combinatorial functor satisfying a certain property (which, if C = C1, is that F is nonconstant) and ∈ C, then (1) if F is regressive so is as is each i, and (2) if F is Dedekind then each i is Dedekind.
Type two partial degrees
Authors: Lih Ko-Wei
Published: May 2006

Roughly speaking partial degrees are equivalence classes of partial objects under a certain notion of relative recursiveness. To make this notion precise we have to state explicitly (1) what these partial objects are; (2) how to define a suitable reduction procedure. For example, when the type of these objects is restricted to one, we may include all possible partial functions from natural numbers to natural numbers as basic objects and the reduction procedure could be enumeration, weak Turing, or Turing reducibility as expounded in Sasso [4]. As we climb up the ladder of types, we see that the usual definitions of relative recursiveness, equivalent in the context of type-1 total objects and functions, may be extended to partial objects and functions in quite different ways. First such generalization was initiated by Kleene [2]. He considers partial functions with total objects as arguments. However his theory suffers the lack of transitivity, i.e. we may not obtain a recursive function when we substitute a recursive function into a recursive function. Although Kleene's theory provides a nice background for the study of total higher type objects, it would be unsatisfactory when partial higher type objects are being investigated. In this paper we choose the hierarchy of hereditarily consistent objects over ω as our universe of discourse so that Sasso's objects are exactly those at the type-1 level. Following Kleene's fashion we define relative recursiveness via schemes and indices. Yet in our theory, substitution will preserve recursiveness, which makes a degree theory of partial higher type objects possible. The final result will be a natural extension of Sasso's Turing reducibility. Due to the abstract nature of these objects we do not know much about their behaviour except at the very low types. Here we pay our attention mainly to type-2 objects. In §2 we formulate basic notions and give an outline of our recursion theory of partial higher type objects. In §3 we introduce the definitions of singular degrees and ω-consistent degrees which are two important classes of type-2 objects that we are most interested in.
Some results on measure independent Gödel speed-ups
Authors: Solomon Martin K.
Published: May 2006

We study the measure independent character of Gödel speed-up theorems, in particular, we strengthen Arbib's necessary condition for the occurrence of a Gödel speed-up [2, p. 13] to an equivalence result and generalize Di Paola's speed-up theorem [4]. We also characterize undecidable theories as precisely those theories which possess consistent measure independent Gödel speed-ups and show that a theory τ2 is a measure independent Gödel speed-up of a theory τ1 if and only if the set of undecidable sentences of τ1 which are provable in τ2 is not recursively enumerable.
On the nonexistence of certain normal forms in the logic of provability
Authors: Boolos George
Published: May 2006

G is the system of propositional modal logic whose axioms are all tautologies and all sentences □(A → B) → (□A → □B) and □(□A → A) → □A and whose rules are modus ponens and necessitation. For the connections between G and provability in formal systems, see [3] and [2].

A letterless sentence is a modal sentence that contains no sentence letters at all. In [1] the author showed (in effect) the existence of simple normal forms in G for letterless sentences: every letterless sentence is equivalent in G to a truth-functional compound of sentences ⋄r ⊤ (⋄0B = B; ⋄r+1B = ⋄ ⋄r+1 B. ⊤ is the 0-ary connective that is always evaluated as true; we assume that the language contains ⊤ as a primitive.)

A family of sets {Hn}n∈ω of modal sentences that contain no sentence letters other than some fixed one (say p) was introduced by Solovay in [3]: H0 = the set of sentences equivalent in G to one of p, −p, ⊤, and ⊥; Hn+x = the set of sentences equivalent in G to a truth-functional combination of sentences ⋄rB, where r ∈ ω and B ∈ Hn. (This definition of the Hn's differs inessentially from the one given in [3].) Since r may = 0, Hn ⊆ Hn+1 and thus if m ≤ n, Hm ⊆ Hn; every modal sentence containing no letter other than p is in some Hn. It follows from the normal form theorem for letterless sentences that every letterless sentence is in H1.

In [3] Solovay stated without proof a result about the sets Hn: for every n, Hn ⊊ Hn+1; equivalently, for every n, there is a sentence containing no letter other than p not in Hn. Thus the existence of a normal form for letterless sentences is a very special feature of these sentences, for Solovay's result shows that normal forms like those obtainable for letterless sentences are not to be found in general. He takes this result to be one reason for regarding the Lindenbaum algebra in G of sentences containing no letter other than p as much more complicated than that of letterless sentences.
A lift of a theorem of Friedberg: A Banach-Mazur functional that coincides with no α-recursive functional on the class of α-recursive functions
Authors: di Paola Robert A.
Published: May 2006

R. M. Friedberg demonstrated the existence of a recursive functional that agrees with no Banach-Mazur functional on the class of recursive functions. In this paper Friedberg's result is generalized to both α-recursive functionals and weak α-recursive functionals for all admissible ordinals α such that λ < α*, where α* is the Σ1-projectum of α and λ is the Σ2-cofinality of α. The theorem is also established for the metarecursive case, α = ω1, where α* = λ = ω.
Theories with recursive models
Authors: Lerman Manuel , Schmerl James H.
Published: May 2006

A structure is recursive if the set of quantifier-free sentences in the complete diagram ⊿() of is recursive. It has been known for some time that every decidable theory has a recursive model. In fact, every decidable theory has a decidable model (that is a model such that ⊿() is recursive). In this paper we find other conditions which imply that a theory have a recursive model.

In §1 we study the relation between an ℵ0-categorical theory T having a recursive model and the complexity of the quantificational hierarchy of that theory. We let ∃0 denote the set of quantifier-free sentences, and let ∃n÷1 denote the set of sentences beginning with an existential quantifier and having n alternations of quantifiers. (∀n is defined analogously.) Then we show that if T is an arithmetical ℵ0-categorical theory such that T ⋂ ∃n÷2 is Σn÷10 for each n < ω, then T has a recursive model. We show that this is a best possible result by giving an example of a ⊿n÷20 ℵ0-categorical theory T such that T ⋂ ∃n÷1 is recursive yet T has no recursive model.

In §2 we consider the theory of trees. Ershov [1] had proved that every Σ10 theory of trees has a recursive model. We show this to be best possible by giving an example of a ⊿20 theory of trees which has no recursive model.
Some model theory for game logics
Authors: Green Judy
Published: May 2006

Consistency properties and their model existence theorems have provided an important method of constructing models for fragments of L∞ω. In [E] Ellentuck extended this construction to Suslin logic. One of his extensions, the Borel consistency property, has its extra rule based not on the semantic interpretation of the extra symbols but rather on a theorem of Sierpinski about the classical operation . In this paper we extend that consistency property to the game logic LG and use it to show how one can extend results about and its countable fragments to LG and certain of its countable fragments. The particular formation of LG which we use will allow in the game quantifier infinite alternation of countable conjunctions and disjunctions as well as infinite alternation of quantifiers. In this way LG can be viewed as an extension of Suslin logic.
Sheaves and Boolean valued model theory
Authors: Loullis George
Published: May 2006

In recent years model theorists have been studying various sheaf-theoretic notions as they apply to model theory. For quite a while however, a sheaf of structures was considered to be just a local homeomorphism between topological spaces such that each stalk Sx = p−1(x) is a model-theoretic structure and such that certain maps are continuous. Some of the model-theoretic work done with this notion of a sheaf of structures are the papers by Carson [2] and Macintyre [7]. Soon came the idea of considering a sheaf of structures not just as a collection of structures glued together in some continuous way, but rather as some sort of generalized structure. A significant model-theoretic study of sheaves in this new sense became possible only after the development of the theory of topoi. As F.W. Lawvere pointed out in [6], this represents the advance of mathematics (in our case the advance of model theory) from metaphysics to dialectics.

A topos is the rather ingenious evolution of the notion of a Grothendieck topos [13]. It provides us with the idea that an object of a topos (e.g. the topos of sheaves over a topological space) may be thought of as a generalized set. Furthermore, all first-order logical operations have an interpretation in a topos, hence we may talk about generalized structures. Angus Macintyre suggested that some of his model-theoretic results about sheaves of structures may be understood better and perhaps simplified by doing model theory inside a topos of sheaves.
Isomorphism of structures in S-toposes
Authors: Bell J. L.
Published: May 2006

It is a well-known fact that two structures are ∞ω-equivalent if and only if they are isomorphic in some Boolean extension of the universe of sets (cf. [4]; an early allusion to this result appears in [8]). My principal object here is to show that arbitrary toposes defined over the category of sets may be used instead. Thus ∞ω-equivalence means isomorphism in the extremely general context of some universe of "variable" sets in which not only is much of the usual set-theoretic machinery unavailable but the underlying logic is not even classical. This provides further support for the view that ∞ω-equivalence is a relation between structures of fundamental importance.
Adjoining dominating functions
Authors: Baumgartner James E. , Dordal Peter
Published: May 2006

If dominating functions in ωω are adjoined repeatedly over a model of GCH via a finite-support c.c.c. iteration, then in the resulting generic extension there are no long towers, every well-ordered unbounded family of increasing functions is a scale, and the splitting number (and hence the distributivity number ) remains at ω1.
Definability in number fields
Authors: Tung S. P.
Published: May 2006

We study the sets definable in an algebraic number field by first order formulas of various simple types, showing in particular that N and Z do not have very simple definitions.
Some new natural α-re-degrees
Authors: Bailey Colin G.
Published: May 2006

If α is a singular cardinal (either real or fake) in L, I exhibit many natural α-re subsets, defined uniformly from the ⊿1 subsets of a. If α. is a true cardinal this provides an uppersemilattice (usl) embedding from the lattice of ⊿1 subsets of α into the usl of α-re-degrees. It will also be shown that this embedding cannot be extended to the Σ1 subsets of α.
Sequential discreteness and clopen-I-Boolean classes
Authors: Dougherty Randall
Published: May 2006

Kantorovich and Livenson [6] initiated the study of infinitary Boolean operations applied to the subsets of the Baire space and related spaces. It turns out that a number of interesting collections of subsets of the Baire space, such as the collection of Borel sets of a given type (e.g. the Fσ sets) or the collection of analytic sets, can be expressed as the range of an ω-ary Boolean operation applied to all possible ω-sequences of clopen sets. (Such collections are called clopen-ω-Boolean.) More recently, the ranges of I-ary Boolean operations for uncountable I have been considered; specific questions include whether the collection of Borel sets, or the collection of sets at finite levels in the Borel hierarchy, is clopen-I-Boolean.

The main purpose of this paper is to give a characterization of those collections of subsets of the Baire space (or similar spaces) that are clopen-I-Boolean for some I. The Baire space version can be stated as follows: a collection of subsets of the Baire space is clopen-I-Boolean for some I iff it is nonempty and closed downward and σ-directed upward under Wadge reducibility, and in this case we may take I = ω2. The basic method of proof is to use discrete subsets of spaces of the form K2 to put a number of smaller clopen-I-Boolean classes together to form a large one. The final section of the paper gives converse results indicating that, at least in some cases, ω2 cannot be replaced by a smaller index set.
Transfer principles for pseudo real closed e-fold ordered fields
Authors: Basarab Şerban A.
Published: May 2006

In his famous paper [1] on the elementary theory of finite fields Ax considered fields K with the property that every absolutely irreducible variety defined over K has K-rational points. These fields have been called pseudo algebraically closed (pac) and also regularly closed, and extensively studied by Jarden, Éršov, Fried, Wheeler and others, culminating with the basic works [8] and [11].

The above algebraic-geometric definition of pac fields can be put into the following equivalent model-theoretic version: K is existentially complete (ec) relative to the first order language of fields into each regular field extension of K. It has been this characterization of pac fields which the author extended in [2] to ordered fields. An ordered field (K, <) is called in [2] pseudo real closed (prc) if (K, <) is ec in every ordered field extension (L, <) with L regular over K. The concept of pre ordered field has also been introduced by McKenna in his thesis [15] by analogy with the original algebraic-geometric definition of pac fields.

Given a positive integer e, a system K = (K; P1, …, Pe), where K is a field and P1, …, Pe are orders of K (identified with the corresponding positive cones), is called an e-fold ordered field (e-field). In his thesis [9] van den Dries developed a model theory for e-fields. The main result proved in [9, Chapter II] states that the theory e-OF of e-fields is model con. panionable, and the models of the model companion e-OF are explicitly described.
Stationary reflections for uncountable cofinality
Authors: Komjáth Péter
Published: May 2006

It was J. E. Baumgartner who in [1] proved that when a weakly compact cardinal is Lévy-collapsed to ω2 the new ω2 inherits some of the large cardinal properties; e.g. if S is a stationary set of ω-limits in ω2 then for some α < ω2, S ∩ α is stationary in α. Later S. Shelah extended this to the following theorem: if a supercompact cardinal κ is Lévy-collapsed to ω2, then in the resulting model the following holds: if S ⊆ λ is a stationary set of ω-limits and cf(λ) ≥ ω2 then there is an α. < λ such that S ∩ α is stationary in α, i.e. stationary reflection holds for countable cofinality (see [1] and [3]). These theorems are important prototypes of small cardinal compactness theorems; many applications and generalizations can be found in the literature. One might think that these results are true for sets with an uncountable cofinality μ as well, i.e. when an appropriate large cardinal is collapsed to μ++. Though this is true for Baumgartner's theorem, there remains a problem with Shelah's result. The point is that the lemma stating that a stationary set of ω-limits remains stationary after forcing with an ω2-closed partial order may be false in the case of μ-limits in a cardinal of the form λ+ with cf(λ) < μ, as was shown in [8] by Shelah. The problem has recently been solved by Baumgartner, who observed that if a universal box-sequence on the class of those ordinals with cofinality ≤ μ exists, the lemma still holds, and a universal box-sequence of the above type can be added without destroying supercompact cardinals beyond μ.
Pragmatic truth and approximation to truth
Authors: Mikenberg Irene , da Costa Newton C. A. , Chuaqui Rolando
Published: May 2006

There are several conceptions of truth, such as the classical correspondence conception, the coherence conception and the pragmatic conception. The classical correspondence conception, or Aristotelian conception, received a mathematical treatment in the hands of Tarski (cf. Tarski [1935] and [1944]), which was the starting point of a great progress in logic and in mathematics. In effect, Tarski's semantic ideas, especially his semantic characterization of truth, have exerted a major influence on various disciplines, besides logic and mathematics; for instance, linguistics, the philosophy of science, and the theory of knowledge.

The importance of the Tarskian investigations derives, among other things, from the fact that they constitute a mathematical, formal mark to serve as a reference for the philosophical (informal) conceptions of truth. Today the philosopher knows that the classical conception can be developed and that it is free from paradoxes and other difficulties, if certain precautions are taken.

We believe that is not an exaggeration if we assert that Tarski's theory should be considered as one of the greatest accomplishments of logic and mathematics of our time, an accomplishment which is also of extraordinary relevance to philosophy, as we have already remarked.

In this paper we show that the pragmatic conception of truth, at least in one of its possible interpretations, has also a mathematical formulation, similar in spirit to that given by Tarski to the classical correspondence conception.
Answer to a problem raised by J. Robinson: The arithmetic of positive or negative integers is definable from successor and divisibility
Authors: Richard Denis
Published: May 2006

In this paper we give a positive answer to Julia Robinson's question whether the definability of + and · from S and ∣ that she proved in the case of positive integers is extendible to arbitrary integers (cf. [JR, p. 102]).
On the proof theory of the intermediate logic MH
Authors: Seldin Jonathan P.
Published: May 2006

A natural deduction formulation is given for the intermediate logic called MH by Gabbay in [4]. Proof-theoretic methods are used to show that every deduction can be normalized, that MH is the weakest intermediate logic for which the Glivenko theorem holds, and that the Craig-Lyndon interpolation theorem holds for it.
Supercompact cardinals, trees of normal ultrafilters, and the partition property
Authors: Barbanel Julius B.
Published: May 2006

Suppose κ is a supercompact cardinal. It is known that for every λ ≥ κ, many normal ultrafilters on Pκ(λ) have the partition property. It is also known that certain large cardinal assumptions imply the existence of normal ultrafilters without the partition property. In [1], we introduced the tree T of normal ultrafilters associated with κ. We investigate the distribution throughout T of normal ultrafilters with and normal ultrafilters without the partition property.
Small decidable sheaves
Authors: Blass Andreas , Scedrov Andre
Published: May 2006

Fred Richman conjectured that the following principle is not constructive:

(*) If A is a decidable subset of the set N of natural numbers and if, for every decidable subset B of N, either A ⊆ B or A ⊆ N − B, then, for some n ∈ N, A ⊆ {n}.

A set A of natural numbers is called decidable if ∀n(n ∈ A ∨ ⌉ (n ∈ A)) holds. In recursive models, this agrees with the recursion-theoretic meaning of decidability. In other contexts, “complemented” and “detachable” are often used.

Richman's conjecture was motivated by the problem of uniqueness of divisible hulls of abelian groups in constructive algebra. Richman showed that a countable discrete abelian p-group G has a unique (up to isomorphism over G) divisible hull if the subgroup pG is decidable. He also showed that the converse implies.

We confirm the nonconstructive nature of by showing (in §1) that it is not provable in intuitionistic set theory, IZF. Thus, in the models we construct, there are countable discrete abelian p-groups G whose divisible hulls are unique but whose subgroups pG are not decidable.

Our models do not satisfy further conditions imposed by Richman, namely Church's Thesis and Markov's Principle, so the full conjecture remains an open problem. We do, however, show (in §2) how to embellish our first model so that the fan theorem (i.e., compactness of 2N) fails. (Church's Thesis implies the stronger statement that the negation of the fan theorem holds.)

Our models will be constructed by the method of sheaf semantics [1], [3]. That is, we shall construct Grothendieck topoi in whose internal logic fails.
A simple solution to Friedman's fourth problem
Authors: Caicedo Xavier
Published: May 2006

It is shown that Friedman's problem, whether there exists a proper extension of first order logic satisfying the compactness and interpolation theorems, has extremely simple positive solutions if one considers extensions by generalized (finitary) propositional connectives. This does not solve, however, the problem of whether such extensions exist which are also closed under relativization of formulas.
On the orbits of hyperhypersimple sets
Authors: Maass Wolfgang
Published: May 2006

This paper contributes to the question of under which conditions recursively enumerable sets with isomorphic lattices of recursively enumerable-supersets are automorphic in the lattice of all recursively enumerable sets. We show that hyperhypersimple sets (i.e. sets where the recursively enumerable supersets form a Boolean algebra) are automorphic if there is a -definable isomorphism between their lattices of supersets. Lerman, Shore and Soare have shown that this is not true if one replaces by .
On the compactness of some Boolean algebras
Authors: Cichoń Jacek
Published: May 2006

We say that the Boolean algebra B is λ-compact, where λ is a cardinal number, if for every family Z ⊆ B∖{0} of power at most λ, if inf Z = 0 then for some finite subfamily Z0 ⊆ Z we have inf Z0 = 0.

On the set of all finite subsets of a cardinal number κ, which is denoted [κ]<ω, the sets of the form for any p Є [κ]<ω generate the filter Tκ.

This filter is a standard example of a κ-regular filter (see [2]). Because of the importance of κ-regular filters in studying the saturatedness of ultraproducts and reduced products by model-theoretic methods, the question of compactness of the algebra Bκ = P([κ]<ω/Tκ was natural. This question in the most optimistical way was formulated by M. Benda [1, Problem 5c]: is the algebra Bκω-compact for every uncountable κ?

In this paper we show that for most of the cardinal numbers which are greater or equal to 2ω the algebra Bκ is not ω-compact. Hence, in view of obtained results, the following question appears: does there exist an uncountable κ such that the algebra Bκ is κ-compact?

We use standard set-theoretical notations. CH denotes the Continuum Hypothesis, GCH denotes the General Continuum Hypothesis and MA denotes Martin's Axiom.
Generic graph construction
Authors: Baumgartner James E.
Published: May 2006

It is shown that if ZF is consistent, then so is ZFC + GCH + “There is a graph with cardinality ℵ2 and chromatic number ℵ2 such that every subgraph of cardinality ≤ ℵ1, has chromatic number ≤ ℵ0”. This partially answers a question of Erdös and Hajnal.
Regularity in models of arithmetic
Authors: Mills George , Paris Jeff
Published: May 2006

This paper investigates the quantifier “there exist unboundedly many” in the context of first-order arithmetic. An alternative axiomatization is found for Peano arithmetic based on an axiom schema of regularity: The union of boundedly many bounded sets is bounded. We also obtain combinatorial equivalents of certain second-order theories associated with cuts in nonstandard models of arithmetic.
Functors and ordinal notations. I: A functorial construction of the veblen hierarchy
Authors: Girard Jean-Yves , Vauzeilles Jacqueline
Published: May 2006

This paper is the first of several works: the matter is the systematic rewriting of usual results of ordinal notations, using Π½-logic. The purpose of these articles is already realized in Π½-logic, in a relatively abstract form; also, they are examples for Π½-logic. The first and second articles in this series are complete versions of unpublished results that one of authors had announced in 1976.

Veblen introduced his hierarchy in 1908. The hierarchy has since been widely used in proof theory; in fact, what is actually needed is the more natural hierarchy: , with (not necessarily a Cantor normal form) and K the Veblen hierarchy.

We shall show:

i) φ is in fact a two-variables functor (denoted by V in the text) commuting with direct limits and with pull-backs.

ii) The property of commutation with pull-backs permits one to give a representation of each ordinal x < φα(0) by means of a finite sequence of ordinals < α. This representation is independent of α it is ⊆ x, and can be used in the building up of notations systems. This representation is the “blowing-up” of the Cantor normal form, and it corresponds exactly to the denotation of ordinals < φα(0) by means of the functions (φβ)β<α.
Functors and ordinal notations. IV: The Howard ordinal and the functor ∧.
Authors: Vauzeilles Jacqueline
Published: May 2006

In this paper, we prove a result that J. Y. Girard has conjectured for the last few years. That is,

This result is part of a study of ordinal notations. For the proof, we use some concepts introduced in Parts I, II and III of this article (see [GV1], [GV2] and [V]); in particular, in Part III (see [V]) we defined the category GAR of gardens (of type Ω), and we have shown that there is an isomorphism between GAR and the category DIL Ω of dilators which send Ω into Ω. To describe this isomorphism, we defined a functor SYN from GAR to DIL Ω and a functor DEC from DIL Ω to GAR which are inverse to each other. The functor SYN is constructed by induction on the height x of the garden Jx, iterating, at each step of cofinality Ω, the operation UN (unification of variables; see [G1, Chapter 3]). Here, we define the functor S⊿ from GAR to DIL Ω: this construction is very close to that of SYN, but we iterate, at each step of cofinality Ω, the operation ⊿ (identification of variables; see [G, Chapter 3]). Technically, the definition of S⊿ is easier and more natural than that of SYN (as ⊿ is easier than UN) but is not inversible (as ⊿); for example, if we consider the (regular) garden GεΩ+1 introduced in [GV2],S⊿(GεΩ+1) is exactly the dilator N = ⋃0<n<ωNn, with N1 = 2 + Id, and , but we are incapable of expressing SYN(GεΩ+1).