bluuebunny/crossref_metadata_2025 huggingface dataset:
The formal language of recursion
Authors: Moschovakis Yiannis N.
Published: May 2006

This is the first of a sequence of papers in which we will develop a foundation for the theory of computation based on a precise, mathematical notion of abstract algorithm. To understand the aim of this program, one should keep in mind clearly the distinction between an algorithm and the object (typically a function) computed by that algorithm. The theory of computable functions (on the integers and on abstract structures) is obviously relevant to this work, but we will focus on making rigorous and identifying the mathematical properties of the finer (intensional) notion of algorithm.

It is characteristic of this approach that we take recursion to be a fundamental (primitive) process for constructing algorithms, not a derived notion which must be reduced to others—e.g. iteration or application and abstraction, as in the classical λ-calculus. We will model algorithms by recursors, the set-theoretic objects one would naturally choose to represent (syntactically described) recursive definitions. Explicit and iterative algorithms are modelled by (appropriately degenerate) recursors.

The main technical tool we will use is the formal language of recursion, FLR, a language of terms with two kinds of semantics: on each suitable structure, the denotation of a term t of FLR is a function, while the intension of t is a recursor (i.e. an algorithm) which computes the denotation of t. FLR is meant to be intensionally complete, in the sense that every (intuitively understood) “algorithm” should “be” (faithfully modelled, in all its essential properties by) the intension of some term of FLR on a suitably chosen structure.
A proof of morley's conjecture
Authors: Hart Bradd
Published: May 2006

In the 1960's, it was conjectured that a complete first order theory in a countable language would have a nondecreasing spectrum on uncountable cardinals. This conjecture became known as Morley's conjecture. Shelah has proved this in [10]. The intent of this paper is to give a different proof which resembles a more naive way of approaching this theorem.

Let I(T, λ) = the number of nonisomorphic models of T in cardinality λ. We prove:

Theorem 0.1. If T is a complete countable first order theory then for ℵ0 < κ < λ, I(T,K) ≤ I(T, λ).

In some sense, one can view Shelah's work on the classification of first order theories as an attack on Morley's conjecture. Over the years, he has shown that certain assumptions on a first order theory would lead to its having maximal spectrum in powers larger than the cardinality of its language (see §6 for precise references). At some point it must have seemed that Morley's conjecture would be a corollary to an exact calculation of all possible spectrums. In the end, this did not occur and, in fact, the exact spectrum functions are still not known (see [10]). Let us consider a naive approach to the proof.

If we have two nonisomorphic models of the same cardinality and their cardinality is “large enough” then there should be some reason, irrespective of their cardinalities, which causes this nonisomorphism. If we could isolate this property and extend these models to a larger cardinality preserving this property, then the larger models would also be nonisomorphic. The notion of extendibility introduced in §2 is such a property which allows a version of this naive proof to work. Let us preview the sections.
The classification of excellent classes
Authors: Grossberg R. , Hart B.
Published: May 2006

In [9] and [12], Shelah defined a certain type of Scott sentence which he called excellent. He proved, among other things, that if a Scott sentence is excellent and categorical in some uncountable power then it is categorical in all uncountable powers: the analog of the Morley categoricity theorem. Proving such an analog is often the starting point in the classification of a family of classes. Before beginning this classification in the case of excellent Scott sentences, let us say a few words about what this paper is and what it is not.

It is not the beginning of a classification theory for complete sentences in where is countable. Although excellence arises in the study of the model theory of Scott sentences, it is not a dividing line in a classification of them. In particular, the assumption of nonexcellence does not yield much information. In fact, in [3] there is an example of a nonexcellent Scott sentence, categorical in ℵ1 which is. not fully categorical. It seems to the second author that a classification of sentences analogous to the classification of first order theories is a long way off and may not be accomplishable in ZFC.

This is not to say that the study of excellent Scott sentences (or the class of models of such which we will call excellent classes) is unproductive. Besides its extreme usefulness in [12], Mekler and Shelah have shown that excellence plays a decisive role in the study of almost free algebras (see [7]). Moreover, as the class of ω-saturated models of an ω-stable theory is an example of an excellent class, the study of excellent classes is at least as difficult as the study of first order ω-stable theories.
Algorithmic information theory
Authors: van Lambalgen Michiel
Published: May 2006

We present a critical discussion of the claim (most forcefully propounded by Chaitin) that algorithmic information theory sheds new light on Gödel's first incompleteness theorem.
The consistency problem for positive comprehension principles
Authors: Forti M. , Hinnion R.
Published: May 2006

Since Gilmore showed that some theory with a positive comprehension scheme is consistent when the axiom of extensionality is dropped and inconsistent with it (see [1] and [2]), the problem of the consistency of various positive comprehension schemes has been investigated. We give here a short classification, which shows clearly the importance of the axiom of extensionality and of the abstraction operator in these consistency problems. The most difficult problem was to show the consistency of the comprehension scheme for positive formulas, with extensionality but without abstraction operator. In his unpublished thesis, Set theory in which the axiom of foundation fails [3], Malitz solved partially this problem but he needed to assume the existence of some unusual kind of large cardinal; as his original construction is very interesting and his thesis is unpublished, we give a short summary of it. M. Forti solved the problem completely by working in ZF with a free-construction principle (sometimes called an anti-foundation axiom), instead of ZF with the axiom of foundation, as Malitz did.

This permits one to obtain the consistency of this positive theory, relative to ZF. In his general investigations about “topological set theories” (to be published), E. Weydert has independently proved the same result. The authors are grateful to the Mathematisches Forshungsinstitut Oberwolfach for giving them the opportunity of discussing these subjects and meeting E. Weydert during the meeting “New Foundations”, March 1–7, 1987.
Interpolation in fragments of intuitionistic propositional logic
Authors: de Lavalette Gerard R. Renardel
Published: May 2006

We show in this paper that all fragments of intuitionistic propositional logic based on a subset of the connectives ∧, ∨, →, ¬ satisfy interpolation. Fragments containing ↔ or ¬¬ are briefly considered.
Reflecting on incompleteness
Authors: Feferman Solomon
Published: May 2006

To what extent can mathematical thought be analyzed in formal terms? Gödel's theorems show the inadequacy of single formal systems for this purpose, except in relatively restricted parts of mathematics. However at the same time they point to the possibility of systematically generating larger and larger systems whose acceptability is implicit in acceptance of the starting theory. The engines for that purpose are what have come to be called reflection principles. These may be iterated into the constructive transfinite, leading to what are called recursive progressions of theories. A number of informative technical results have been obtained about such progressions (cf. Feferman [1962], [1964], [1968] and Kreisel [1958], [1970]). However, for some years I had hoped to give a more realistic and perspicuous finite generation procedure. This was first done in a rather special way in Feferman [1979] for the characterization of predicativity, which may be regarded as that part of mathematical thought implicit in our acceptance of elementary number theory. What is presented here is a new and simple notion of the reflective closure of a schematic theory which can be applied quite generally.

Two examples of schematic theories in the sense used here are versions of Peano arithmetic and Zermelo set theory.
Normalization theorems for full first order classical natural deduction
Authors: Stålmarck Gunnar
Published: May 2006

In this paper we prove the strong normalization theorem for full first order classical N.D. (natural deduction)—full in the sense that all logical constants are taken as primitive. We also give a syntactic proof of the normal form theorem and (weak) normalization for the same system.

The theorem has been stated several times, and some proofs appear in the literature. The first proof occurs in Statman [1], where full first order classical N.D. (with the elimination rules for ∨ and ∃ restricted to atomic conclusions) is embedded in a system for second order (propositional) intuitionistic N.D., for which a strong normalization theorem is proved using strongly impredicative methods.

A proof of the normal form theorem and (weak) normalization theorem occurs in Seldin [1] as an extension of a proof of the same theorem for an N.D.-system for the intermediate logic called MH.

The proof of the strong normalization theorem presented in this paper is obtained by proving that a certain kind of validity applies to all derivations in the system considered.

The notion “validity” is adopted from Prawitz [2], where it is used to prove the strong normalization theorem for a restricted version of first order classical N.D., and is extended to cover the full system. Notions similar to “validity” have been used earlier by Tait (convertability), Girard (réducibilité) and Martin-Löf (computability).

In Prawitz [2] the N.D. system is restricted in the sense that ∨ and ∃ are not treated as primitive logical constants, and hence the deductions can only be seen to be “natural” with respect to the other logical constants. To spell it out, the strong normalization theorem for the restricted version of first order classical N.D. together with the well-known results on the definability of the rules for ∨ and ∃ in the restricted system does not imply the normalization theorem for the full system.
Where do sets come from?
Authors: Hodes Harold T.
Published: May 2006

Many philosophers take set-theoretic discourse to be about objects of a special sort, namely sets; correlatively, they regard truth in such discourse as quite like truth in discourse about nonmathematical objects. There is a thin “disquotational” way of construing this construal; but that may candy-coat a philosophically substantive semantic theory: the Mathematical-Object theory of the basis for the distribution of truth and falsehood to sentences containing set-theoretic expressions. This theory asserts that truth and falsity for sentences containing set-theoretic expressions are grounded in semantic facts (about the relation between language and the world) of the sort modelled by the usual model-theoretic semantics for an uninterpreted formal first-order language. For example, it would maintain that ‘{ } ∈ {{ }}” is true in virtue of the set-theoretic fact that the empty set is a member of its singleton, and the semantic facts that ‘{ }’ designates the empty set,‘{{ }}’ designates its singleton, and ‘∈’ applies to an ordered pair of objects iff that pair's first component is a member of its second component.

Now this theory may come so naturally as to seem trivial. My purpose here is to loosen its grip by “modelling” an alternative account of the alethic underpinnings of set-theoretic discourse. According to the Alternative theory, the point of having set-theoretic expressions (‘set’ and ‘∈’ will do) in a language is not to permit its speakers to talk about some special objects under a special relation; rather it is to clothe a higher-order language in lower-order garments.
There are reasonably nice logics
Authors: Hodges Wilfrid , Shelah Saharon
Published: May 2006

A well-known question of Feferman asks whether there is a logic which extends the logic , is ℵ0-compact and satisfies the interpolation theorem. (Cf. Makowsky [M] for background and terminology.)

The same question was open when ℵ1 in is replaced by any other uncountable cardinal κ. We shall show that when κ is an uncountable strongly compact cardinal and there is a strongly compact cardinal > κ, then there is such a logic. It is impossible to prove the existence of uncountable strongly compact cardinals in ZFC. However, the logic that we describe has a simple and natural definition, together with several other pleasant properties. For example it satisfies Robinson's lemma, PPP (pair preservation property, viz. the theory of the sum of two models is the sum of their theories), versions of the elementary chain lemma for chains of length < λ, and isomorphism of (suitable) ultralimits.

This logic is described in §2 below; we call it 1. It is not a new logic—it was introduced in [Sh, Part II, §3] as an example of a logic which has the amalgamation and joint embedding properties. See the transparent presentation in [M]. But we shall repeat all the definitions. In [HS] we presented a logic with some of the same properties as 1, also based on a strongly compact cardinal λ; but unlike 1, it was not a sublogic of λ,λ.
Strong negative partition above the continuum
Authors: Shelah Saharon
Published: May 2006

For e.g. λ = μ+, μ regular, λ larger than the continuum, we prove a strong nonpartition result (stronger than λ → [λ; λ]2). As a consequence, the product of two topological spaces of cellularity <λ may have cellularity λ, or, in equivalent formulation, the product of two λ-c.c. Boolean algebras may lack the λ-c.c. Also λ-S-spaces and λ-L-spaces exist. In fact we deal not with successors of regular λ but with regular λ above the continuum which has a nonreflecting stationary subset of ordinals with uncountable cofinalities; sometimes we require λ to be not strong limit.

The paper is self-contained. On the nonpartition results see the closely related papers of Todorčević [T1], Shelah [Sh276] and [Sh261], and Shelah and Steprans [ShSt1].

On the cellularity of products see Todorčević [T2] and [T3], where such results were obtained for (e.g.) cf and ; the class of cardinals he gets is quite disjoint from ours. In [Sh282] such results were obtained for more successors of singulars (mainly λ+, λ > 2cf λ). Also, concerning S and L spaces, Todorčević gets existence.

Todorčević's work on cardinals like relies on [Sh68] (see more in [ShA2, Chapter XIII]) (the scales appearing in the proof of ). The problem was stressed in a preliminary version of the surveys of Juhasz and Monk. We give a detailed proof for one strong nonpartition theorem (1.1) and then give various strengthenings. We then use 1.10 to get the consequences (in 1.11 and 1.12).
Every recursive linear ordering has a copy in DTIME-SPACE(n,log(n))
Authors: Grigorieff Serge
Published: May 2006

This paper is a contribution to the following natural problem in complexity theory:

(*) Is there a complexity theory for isomorphism types of recursive countable relational structures? I.e. given a recursive relational structure ℛ over the set N of nonnegative integers, is there a nontrivial lower bound for the time-space complexity of recursive structures isomorphic (resp. recursively isomorphic) to ℛ?

For unary recursive relations R, the answer is trivially negative: either R is finite or coinfinite or 〈N, R〉 is recursively isomorphic to 〈N, {x ϵ N: x is even}〉.

The general problem for relations with arity 2 (or greater) is open.

Related to this problem, a classical result (going back to S. C. Kleene [4], 1955) states that every recursive ordinal is in fact primitive recursive.

In [3] Patrick Dehornoy, using methods relevant to computer science, improves this result, showing that every recursive ordinal can be represented by a recursive total ordering over N which has linear deterministic time complexity relative to the binary representation of integers. As he notices, his proof applies to every recursive total order type α such that the isomorphism type of α is not changed if points are replaced by arbitrary finite nonempty subsets of consecutive points.

In this paper we extend Dehornoy's result to all recursive total orderings over N and get minimal complexity for both time and space simultaneously.
Consistency results about filters and the number of inequivalent growth types
Authors: Blass Andreas , Laflamme Claude
Published: May 2006

We use models of set theory described in [2] and [3] to prove the consistency of several combinatorial principles, for example:

If ℱ is any filter on N containing all the cofinite sets, then there is a finite-to-one function f: N → N such that f(ℱ) is either the filter of cofinite sets or an ultrafilter.

As a consequence of our combinatorial principles, we also obtain the consistency of:

The partial ordering P of slenderness classes of abelian groups, denned and studied in [4], is a four-element chain.

In the remainder of this Introduction, we shall define our terminology and state the combinatorial principles to be considered. In §2, we shall establish some implications between these principles. In §3, we shall prove our consistency results by showing that the strongest of our principles holds in models of set theory constructed in [2] and [3].

A filter on N will always mean a proper filter containing all cofinite sets; in particular, an ultrafilter will necessarily be nonprincipal. We write N ↗ N for the set of nondecreasing functions from the set N of positive integers into itself. A subset ℐ of N ↗ N is called an ideal if it is closed downward (if f(n) ≤ g(n) for all n and if g ∈ ℐ, then f ∈ ℐ) and closed under binary maximum (if f(n) = max(g(n), h(n)) for all n and if g, h ∈ ℐ then f ∈ ℐ).
Models with the ω-property
Authors: Kossak Roman
Published: May 2006

In [KP] we have studied the problem of determining when a subset of a (countable) model M of PA can be coded in an elementary end extension of M. Sets with this property are called elementary extensional. In particular we can ask whether there are elementary extensional subsets of a model which have order type ω. It turns out that having elementary extensional subsets of order type ω is an interesting property connected with other structural properties of models of PA. We will call this property the ω-property. In [KP] the problem of characterizing models with the ω-property was left open. It is still open, and the aim of this paper is to present a collection of results pertaining to it. It should be mentioned that the same notion was studied by Kaufmann and Schmerl in [KS2] in connection with some weak notions of saturation which they discuss there. Our notion of a model with the ω-property corresponds to the notion of an upward monotonically ω-lofty cut.

It is fairly easy to see that countable recursively saturated models (or in fact all recursively saturated models with cofinality ω) and all short recursively saturated models have the ω-property (Proposition 1.2 below). On the other hand, if we had asked the question about the existence of models with the ω-property before 1975 (when recursively saturated models were introduced) the answer would probably not have been that easy and we would have to come to notions close to recursive saturation.
Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken
Authors: Luckhardt H.
Published: May 2006

A previously unexplored method, combining logical and mathematical elements, is shown to yield substantial numerical improvements in the area of Diophantine approximations. Kreisel illustrated the method abstractly by noting that effective bounds on the number of elements are ensured if Herbrand terms from ineffective proofs ofΣ2-finiteness theorems satisfy certain simple growth conditions. Here several efficient growth conditions for the same purpose are presented that are actually satisfied in practice, in particular, by the proofs of Roth's theorem due to Roth himself and to Esnault and Viehweg. The analysis of the former yields an exponential bound of order exp(70ε−2d2) in place of exp(285ε−2d2) given by Davenport and Roth in 1955, whereαis (real) algebraic of degreed≥ 2 and ∣α−pq−1∣ <q−2−ε. (Thus the new bound is less than the fourth root of the old one.) The new bounds extracted from the other proof arepolynomial of low degree(inε−1and logd). Corollaries: Apart from a new bound for the number of solutions of the corresponding Diophantine equations and inequalities (among them Thue's inequality), log logqν, <Cα, εν5/6+ε, whereqνare the denominators of the convergents to the continued fraction ofα.
Sentences of type theory: the only sentences preserved under isomorphisms
Authors: Marshall M. Victoria , Chuaqui Rolando
Published: May 2006

Sometimes it is said that mathematics talks only about truths that are invariant under isomorphisms. Thus, a syntactical characterization of sentences preserved under isomorphisms seems to be important to understand what mathematics is.

The importance of such sentences is underlined, for instance, by Bourbaki's requirement in [1, Chapter IV] that all axioms for mathematical structures satisfy the condition of being what they call transportable, that is, preserved under isomorphisms. In [2, §1], a formalization of Bourbaki's notion of mathematical structure and of the notion of transportable sentence was given and discussed. We adopt here, as in [2], Bourbaki's terminology, and call a sentence φ transportable if φ is true in a structure E if and only if φ is true in any structure F isomorphic to E.

An alternative formulation for Bourbaki's notion of structure is offered in [2, §3], which is also intended as a formalization of Suppes' notion of set-theoretical predicate, as formulated, for instance, in [3]. It is this alternative formulation that we adopt here.

In [2, p. 106], it was conjectured that a set-theoretical sentence is transportable if and only if it is equivalent to a sentence of type theory. One direction of the conjecture is obvious, namely, that any sentence of type theory is transportable. It is also clear that there are sentences of set theory which are not transportable. The main purpose of the present paper is to prove the rest of the conjecture. That is, we prove here that any transportable set-theoretical sentence is equivalent to a sentence of type theory.
Some remarks on modular regular types
Authors: Pillay Anand
Published: May 2006

Here we consider some problems concerning regular types. In the first place we consider a strongly minimal set D. One can ask what is the strength of the assumption that D has (full) elimination of imaginaries (namely, every definable set X over D has as canonical parameter some tuple from D). We show that D cannot be locally modular. Nontriviality of D is immediate. However, to exclude the locally modular nontrivial case one has to understand structures of the form G/E, where G is a modular strongly minimal group and E is a definable equivalence relation on G with finite classes. We show that the quotient structure G/E can be obtained in two steps. First quotient by a finite subgroup K of G to obtain a strongly minimal group H. Now let Γ be a finite subgroup of the group Aff(H) of definable affine automorphisms of H (namely maps of the form x → αx + a, where α is a definable automorphism of H and a ∈ H), and quotient H by Γ (namely form the orbit space of H under Γ). It can clearly be arranged that Γ contains no nontrivial subgroup of translations.

In the second place we look at a nontrivial modular regular type p whose pregeometry is actually a geometry. The geometry is then known to be (infinite-dimensional) projective geometry over a division ring F. We ask whether F is definable (internally to p). If F is finite, this is clear. In fact in this case p must have U-rank 1. So we assume F to be infinite. We are only able to show definability of F in the case where F is a field, using some results on 2-transitive subgroups of PGL [V]. Moreover in the superstable case we also observe that p is isolated.
Making the hyperreal line both saturated and complete
Authors: Keisler H. Jerome , Schmerl James H.
Published: May 2006

In a nonstandard universe, the κ-saturation property states that any family of fewer than κ internal sets with the finite intersection property has a nonempty intersection. An ordered field F is said to have the λ-Bolzano-Weierstrass property iff F has cofinality λ and every bounded λ-sequence in F has a convergent λ-subsequence. We show that if κ < λ are uncountable regular cardinals and βα < λ whenever α < κ and β < λ then there is a κ-saturated nonstandard universe in which the hyperreal numbers have the λ-Bolzano-Weierstrass property. The result also applies to certain fragments of set theory and second order arithmetic.
Diagonals and semihyperhypersimple sets
Authors: Kummer Martin
Published: May 2006

The most basic construction of an r.e. nonrecursive set—e.g. of the halting problem—proceeds by taking the diagonal of a recursive enumeration of all r.e. sets. We will answer the question of which r.e. sets can be constructed in this manner.

If ψ is a computable numbering of some class of partial recursive functions, we define the diagonal of ψ to be the set Kψ ≔ {i ∈ ω ∣ ψi(i)↓}- It is well known that Kφ is creative if φ is a Gödelnumbering, and that for each creative set K there exists a Gödelnumbering φ such that K = Kφ. That is to say, the class of diagonals of Gödelnumberings is characterized as the class of creative sets. This class was shown to be elementary lattice theoretic (e.l.t.) by Harrington (see [So87, XV. 1.1]).

We give a characterization of diagonals of arbitrary computable numberings of the class P1 of all partial recursive functions. To this end we introduce the notion of a semihyperhypersimple (shhs) set, which generalizes the notion of hyperhypersimplicity to nonsimple sets. It is shown that the diagonals of numberings of P1 are exactly the non-shhs sets. Then, properties of shhs sets are discussed. For example, for each nonrecursive r.e. set A there exists a nonrecursive shhs set B ≤TA, but not every r.e. T-degree contains a shhs set. These results build upon previous work by Downey and Stob [DSta].

The question whether the property “shhs” is (elementary) lattice theoretic remains open. A positive answer would give both an analog of Harrington's result mentioned above, and a generalization of the well-known fact, due to Lachlan [La68], that hyperhypersimplicity is e.l.t. Therefore, we suspect that shhs sets turn out to be useful in the study of the lattice of r.e. sets.

Previously, for several constructions from recursion theory the role of the underlying numbering of P1 was investigated; see Martin ([Ma66a] or [So87, V.4.1]) and Lachlan ([La75] or [Od89, III.9.2]) for Post's simple set, and Jockusch and Soare ([JS73]; cf. also [So87, XII.3.6, 3.7]) for Post's hypersimple set. However, only Gödelnumberings were considered. An explanation for the greater variety which arises when arbitrary numberings of P1 are admitted is provided by the fact that the index set of Gödelnumberings is less complex than the index set of all numberings of P1. The former is Σ1-complete; the latter is Π4-complete.
Quadratic forms in normal open induction
Authors: Otero Margarita
Published: May 2006

Models of normal open induction (NOI) are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the language of ordered semirings.

Here we study the problem of representability of an element a of a model M of NOI (in some extension of M) by a quadratic form of the type X2 + b Y2 where b is a nonzero integer. Using either a trigonometric or a hyperbolic parametrization we prove that except in some trivial cases, M[x, y] with x2 + by2 = a can be embedded in a model of NOI.

We also study quadratic extensions of a model M of NOI; we first prove some properties of the ring of Gaussian integers of M. Then we study the group of solutions of a Pell equation in NOI; we construct a model in which the quotient group by the squares has size continuum.
ℵ0-categorical tree-decomposable structures
Authors: Lachlan A. H.
Published: May 2006

Our purpose in this note is to study countable ℵ0-categorical structures whose theories are tree-decomposable in the sense of Baldwin and Shelah. The permutation group corresponding to such a structure can be decomposed in a canonical manner into simpler permutation groups in the same class. As an application of the analysis we show that these structures are finitely homogeneous.
Standard foundations for nonstandard analysis
Authors: Ballard David , Hrbacek Karel
Published: May 2006

In the thirty years since its invention by Abraham Robinson, nonstandard analysis has become a useful tool for research in many areas of mathematics. It seems fair to say, however, that the search for practically satisfactory foundations for the subject is not yet completed. New proposals, intended to remedy various shortcomings of older approaches, continue to be put forward. The objective of this paper is to show that nonstandard concepts have a natural place in the usual (more or less “standard”) set theory, and to argue that this approach improves upon various aspects of hitherto considered systems, while retaining most of their attractive features. We do this by working in Zermelo-Fraenkel set theory with non-well-founded sets. It has always been clear that the axiom of regularity may fail for external sets. The previous approaches either avoid non-well-foundedness by considering only that fragment of nonstandard set theory that is well-founded (over individuals; enlargements of Robinson and Zakon [17]) or reluctantly live with it (various axiomatic nonstandard set theories). Ballard and Davidon [2] were the first to propose constructive use for non-well-foundedness in the foundations of nonstandard analysis. In the present paper we adopt a very strong anti-foundation axiom. In the resulting more or less “usual” set theory, the (to the “standard” mathematician) unfamiliar concepts of standard, external and internal sets can be defined and their requisite properties proved (rather than postulated, as is the case in axiomatic nonstandard set theories).
On the relationship between the partition property and the weak partition property for normal ultrafilters on Pκλ
Authors: Barbanel Julius B.
Published: May 2006

Suppose κ is a supercompact cardinal and λ > κ. We study the relationship between the partition properly and the weak partition properly for normal ultrafilters on Pκλ. On the one hand, we show that the following statement is consistent, given an appropriate large cardinal assumption: The partition property and the weak partition properly are equivalent, there are many normal ultrafilters that satisfy these properties, and there are many normal ultrafilters that do not satisfy these properties. On the other hand, we consider the assumption that, for some λ > κ, there exists a normal ultrafilter U on Pκλ such that U satisfies the weak partition property but does not satisfy the partition property. We show that this assumption is implied by the assertion that there exists a cardinal γ > κ such that γ is γ+-supercompact, and, assuming the GCH, it implies the assertion that there exists a cardinal γ > κ such that γ is a measurable cardinal with a normal ultrafilter concentrating on measurable cardinals.
Quantifying over propositions in relevance logic: nonaxiomatisability of primary interpretations of ∀pand ∃p
Authors: Kremer Philip
Published: May 2006

A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take amodelto be one of these structures, α, together with some function or relation which associates with every formulaAa subset ofα. (This is the approach of, among others, Urquhart, Routley and Meyer and Fine.) In some cases there are restrictions on the class of subsets of α with which a formula can be associated: for example, in the semantics of Routley and Meyer [1973], a formula can only be associated with subsets which are closed upwards. It is natural to take apropositionof α to be such a subset of α, and, further, to take the propositional quantifiers to range over these propositions. (Routley and Meyer [1973] explicitly consider this interpretation.) Given such an algebraic semantics, we call (following Routley and Meyer [1973], who follow Henkin [1950]) the above-described interpretation of the quantifiers theprimaryinterpretation associated with the semantics.
Extensions of Hilbert's tenth problem
Authors: Pheidas Thanases
Published: May 2006

The present article is an attempt to bridge the gap between the researchers that work in the areas adjacent to Hilbert's Tenth Problem (for short, HTP), mainly, number theory and mathematical logic. It presents the main results that have been obtained and asks some of the open questions in the area, leading to the main unanswered question (at least in the opinion of the author)—the analogue of HTP for the rational numbers. In this respect, the article is a successor of [Da] where the reader can find more information on the solution of the initial problem. Most of the results that are presented are not new, but some of the proofs are. As the purpose of a survey article is the “unification” of the methods involved, we present a proof of the analogue of HTP in the case of polynomial rings which is more geometric in nature than the initial proof of Denef (although it uses essentially the same tools). This allows, in our opinion, an easy introduction to the utilisation of the concepts used in the more recent approaches, like that of elliptic curves. We must say that the approach to the subject that we suggest through the questions we ask (mostly consisting of generalisations of HTP to various algebraic and analytic domains) is not the only one.
Extender based forcings
Authors: Gitik Moti , Magidor Menachem
Published: May 2006

The paper is a continuation of [The SCH revisited], In § 1 we define a forcing with countably many nice systems. It is used, for example, to construct a model “GCH below κ, c f κ = ℵ0, and 2κ > κ+ω” from 0(κ) = κ+ω. In §2 we define a triangle iteration and use it to construct a model satisfying “{μ ≤ λ∣c f μ = ℵ0 and pp(μ) > λ} is countable for some λ”. The question of whether this is possible was asked by S. Shelah. In §3 a forcing for blowing the power of a singular cardinal without collapsing cardinals or adding new bounded subsets is presented. Answering a question of H. Woodin, we show that it is consistent to have “c f κ = ℵ0. GCH below κ, 2κ > κ+, and ”. In §4 a variation of the forcing of [The SCH revisited, §1] is defined. It behaves nicely in iteration processes. As an application, we sketch a construction of a model satisfying:

“κ is a measurable and 2κ ≥ κ+α for some α, κ < c f α < α” starting with 0(κ) = κ+α. This answers the question from Gitik's On measurable cardinals violating the continuum hypothesis.
Machine learning of higher-order programs
Authors: Baliga Ganesh , Case John , Jain Sanjay , Suraj Mandayam
Published: May 2006

A generator program for a computable function (by definition) generates an infinite sequence of programs all but finitely many of which compute that function. Machine learning of generator programs for computable functions is studied. To motivate these studies partially, it is shown that, in some cases, interesting global properties for computable functions can be proved from suitable generator programs which cannot be proved from any ordinary programs for them. The power (for variants of various learning criteria from the literature) of learning generator programs is compared with the power of learning ordinary programs. The learning power in these cases is also compared to that of learning limiting programs, i.e., programs allowed finitely many mind changes about their correct outputs.
Countable structures of given age
Authors: Macpherson H. D. , Pouzet M. , Woodrow R. E.
Published: May 2006

Let L be a finite relational language. The age of a structure over L is the set of isomorphism types of finite substructures of . We classify those ages for which there are less than 2ω countably infinite pairwise nonisomorphic L-structures of age .
An undecidable problem in correspondence theory
Authors: Chagrova L. A.
Published: May 2006

In this paper we prove undecidability of first-order definability of propositional formulas. The main result is proved for intuitionistic formulas, but it remains valid for other kinds of propositional formulas by analogous arguments or with the help of various translations.

For general background on correspondence theory the reader is referred to van Benthem [1], [2] (see [3] for a survey of recent results).

The method for the proofs of undecidability in this paper will be to simulate calculations of a Minsky machine by intuitionistic formulas. §3 concerns this simulation. Effective procedures for the construction of simulating modal formulas can be found in the literature (cf. [4]).

The principal results of the paper are in §4. §5 gives some further undecidability results, that will be proved in another paper by modification of the method of this paper.

I am indebted to the referee for drawing my attention to some errors in an earlier version of this paper.
Variations of the Martin-Solovay tree
Authors: Hjorth Greg
Published: May 2006

Assuming determinacy, the model L[T2] does not depend on the choice of T2.
The proofs of α → α in P – W
Authors: Hirokawa Sachio
Published: May 2006

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

Errol Martin (1982) proved that no formula with form α → α is provable in P − W without using the axiom I. We show that a β-normal form λ-term M in the class is η reducible to λx.x if the translated BB′ I-combinator M+ contains I. Using this theorem and Martin's result, we prove that a λ-term in the class is βη-reducible to λx.x if the λ-term has a type α → α. Hence the structure of proofs of α → α in P − W is determined.
Subformula property in many-valued modal logics
Authors: Takano Mitio
Published: May 2006

Fitting, in [1] and [2], investigated two families of many-valued modal logics. The first, which is somewhat familiar in the literature, is that of the logics characterized using a many-valued version of the Kripke model (binary modal modelin his terminology) with a two-valued accessibility relation. On the other hand, those logics which are characterized using another many-valued version of the Kripke model (implicational modal model), with a many-valued accessibility relation, form the second family. Although he gave a sequent calculus for each of these logics, it is far from having the cut-elimination property (CEP) or the subformula property. So we will give a substitute for his system enjoying the subformula property, though it is not of ordinary sequent calculus but of the many-valued version of sequent calculus initiated by Takahashi [7] and Rousseau [3].

The author, unaware of the deduction systems with CEP, had given in [8] and [9], after Rousseau [4], the deduction systems for the intuitionistic many-valued logics which enjoy CEP only for a certain restricted class of proofs. Then in [10], he gave for three-valued modal logics the ones with CEP, but these systems have a rule of inference which is unnecessary if the Cut rule is present. Why are we particular about CEP? The author's answer is that a cut-free proof is easy to examine since it is composed solely of subformulas of the formulas which form its conclusion. In this direction, the author has given, for modal logics with the Brouwerian axiom [11], the ones without CEP which nevertheless enjoy the subformula property. This paper is a sequel to the study in [11].
Step by step – Building representations in algebraic logic
Authors: Hirsch Robin , Hodkinson Ian
Published: May 2006

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finte relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is ω-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable.

An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras.

Other instances of this approach are looked at, and include the step by step method.
Degree theoretic definitions of the low2 recursively enumerable sets
Authors: Downey Rod , Shore Richard A.
Published: May 2006

The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤TB, where arbitrary (Turing) machines, φe, can be used; access to information about (the oracle) B is unlimited and the lengths of computations are potentially unbounded. Many other interesting reducibilities result from restricitng one or more of these facets of the procedure. Thus, for example, the strongest notion considered is one-one reducibility on sets: A ≤1B iff there is a one-one recursive (= effective) function f such that x Є A ⇔ f(x) Є B. Many-one (≤m) reducibility simply allows f to be many-one. Other intermediate reducibilities include truth-table (≤tt) and weak truth-table (≤wtt). The latter imposes a recursive bound f(x) on the information about B that can be used to compute A(x). The former also bounds the length of computations by requiring that the computation of A(x) from B halt in at most f(x) many steps.

Each such reducibility r defines a notion of degree, degr(A) = {B : A ≤rB ∧ B ≤rA}, and a corresponding structure of the r-degrees ordered by r-reducibility. (We typically denote the degree of A by a.) A major theme in recursion theory has been the investigation of the relation between a set's place in these orderings (the algebraic properties of its degree) and other algorithmic, set-theoretic or definability type notions of complexity. Important examples of such other notions include rates of growth of functions, the types of approximation procedures which converge to the given function or set and the (syntactic) complexity of defining the set (or function) in arithmetic or analysis.
The finite model property in tense logic
Authors: Wolter Frank
Published: May 2006

Tense logics in the bimodal propositional language are investigated with respect to the Finite Model Property. In order to prove positive results techniques from investigations of modal logics above K4 are extended to tense logic. General negative results show the limits of the transfer.
Separably closed fields with higher derivations I
Authors: Messmer Margit , Wood Carol
Published: May 2006

We define a complete theory SHFe of separably closed fields of finite invariant e (=degree of imperfection) which carry an infinite stack of Hasse-derivations. We show that SHFe has quantifier elimination and eliminates imaginaries.
On ℵ1 many minimal models
Authors: Hjorth Greg
Published: May 2006

The existence of a countable complete theory with exactly ℵ1 many minimal models is independent of ZFC + ¬CH.
A few special ordinal ultrafilters
Authors: Laflamme Claude
Published: May 2006

We prove various results on the notion of ordinal ultrafiters introduced by J. Baumgartner. In particular, we show that this notion of ultrafilter complexity is independent of the more familiar Rudin-Keisler ordering.
Notes on polynomially bounded arithmetic
Authors: Zambella Domenico
Published: May 2006

We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general model-theoretical investigations on fragments of bounded arithmetic.
Some characterization theorems for infinitary universal Horn logic without equality
Authors: Dellunde Pilar , Jansana Ramon
Published: May 2006

In this paper we mainly study preservation theorems for two fragments of the infinitary languagesLκκ, withκregular, without the equality symbol: the universal Horn fragment and the universal strict Horn fragment. In particular, whenκisω, we obtain the corresponding theorems for the first-order case.

The universal Horn fragment of first-order logic (with equality) has been extensively studied; for references see [10], [7] and [8]. But the universal Horn fragment without equality, used frequently in logic programming, has received much less attention from the model theoretic point of view. At least to our knowledge, the problem of obtaining preservation results for it has not been studied before by model theorists. In spite of this, in the field of abstract algebraic logic we find a theorem which, properly translated, is a preservation result for the strict universal Horn fragment of infinitary languages without equality which, apart from function symbols, have only a unary relation symbol. This theorem is due to J. Czelakowski; see [5], Theorem 6.1, and [6], Theorem 5.1. A. Torrens [12] also has an unpublished result dealing with matrices of sequent calculi which, properly translated, is a preservation result for the strict universal Horn fragment of a first-order language. And in [2] of W. J. Blok and D. Pigozzi we find Corollary 6.3 which properly translated corresponds to our Corollary 19, but for the case of a first-order language that apart from its function symbols has only oneκ-ary relation symbol, and for strict universal Horn sentences. The study of these results is the basis for the present work. In the last part of the paper, Section 4, we will make these connections clear and obtain some of these results from our theorems. In this way we hope to make clear two things: (1) The field of abstract algebraic logic can be seen, in part, as a disguised study of universal Horn logic without equality and so has an added interest. (2) A general study of universal Horn logic without equality from a model theoretic point of view can be of help in the field of abstract algebraic logic.