bluuebunny/crossref_metadata_2025_split huggingface dataset:
The continuum hypothesis in intuitionism
Authors: Gielen W. , de Swart H. , Veldman W. 
Published: May 2006

Although Brouwer became famous for his vehement attacks upon classical logic and set theory, his work did not develop in a vacuum and strongly depended on that of Cantor.His mind bent on shifting aside nonconstructive arguments, he tried to rebuild Cantor's edifice along new, intuitionistic lines. The continuum hypothesis, lying at the core of set theory, also confronted Brouwer, and he had to face the farthest conclusion Cantor had been able to reach in trying to solve it: every nondenumerable closed subset of the real line has the power of the continuum.Brouwer's thinking about it seems to have been subject to some development. In 1914 we hear him saying: “Wir sahen oben dass das Cantorsche Haupttheorem für den Intuitionisten keines Beweises bedarf” (“As we saw above, for us, being intuitionists, Cantor's Main Theorem does not need a proof”) [3]. Nevertheless, five years later, he publishes an essay: Theorie der Punktmengen, which might be described as an attempt to reconstruct Cantor's reasonings in detail [4].This attempt was not entirely successful, as Brouwer comes to admit in 1952, probably having lost, now, some of his youthful rashness [10]. So the question of what the constructive content of Cantor's Main Theorem is, still awaits an answer.We do not think the answer we will give can be considered a conclusive one, but, in any case, it is a beginning.
Sous-groupes définissables d'un groupe stable
Authors: Poizat Bruno 
Published: May 2006

On sait, depuis les travaux de Zil'ber et de Cherlin, que le degré de Morley de la théorie T d'un groupe G totalement transcendant est l'indice du plus petit sous-groupe définissable d'indice fini de G. Il est clair qu'il lui est supérieur, et l'inégalité inverse peut s'obtenir de la manière suivante: on fait agir G sur les types de S1(G) de rang de Morley maximum en associant à p, type de x au-dessus de G, le type ap de ax au-dessus de G; on montre alors que cette action est définissable, que le fait que ap = q équivaut au fait que a satisfasse une certaine formule à paramètres dans G, ce qui est bien facile si on n'oublie pas que dans une théorie stable tous les types sont définissables; on montre ensuite que cette action est transitive, que si p et q sont de rang de Morley maximum il existe a dans G tel que ap = q, et la méthode la plus rapide, mais qui est aussi la plus sophistiquée, est d'utiliser l'argument de symétrie de la déviation employé dans la preuve de la Proposition 1 de présent article; on conclut alors puisque le degré de Morley, qui est par définition le nombre de types de rang de Morley maximum, ést egal à l'indice du stabilisateur de p, qui est définissable.Ce comportement des types de rang de Morely maximum se retrouve sans peine, si G est seulement superstable, dans celui des types de “plus petit rang continu” (encore appellé “degré de Shelah”) maximum. Pour trouver ce qui leur correspond dans le cas où G est seulement stable, il faut être un peu plus soigneux, et considérer les types p de S1(G), où G aura éventuellement été remplacé par une extension élémentaire suffisament saturée, tels que pour tout a de G ap ne dévie pas sur ∅: on montre qu'ils existent, et qu'ils sont tous conjugués par action de G; le fait que ap = q s'exprimera cette fois par une infinité de formules et non plus par une seule.
Weakly compact cardinals: A combinatorial proof
Authors: Shelah S. 
Published: May 2006

We give here a direct purely combinatorial proof that weak compactness is equivalent to a combinatorial property (2). This property (2) is apparently stronger, and from it, all other usual equivalent definitions and usual properties of weakly compact cardinals can be deduced. So this proof may be useful for books which want to present weakly compact cardinals, but not logic.A direct simple proof of a weaker implication (e.g., weakly compact μ is not the first inaccessible, and every stationary set has an initial segment which is stationary) was given by Kunen [K] and independently by the author [Sh]. Baumgartner [B] had another proof.We were motivated by the manuscript of Erdös, Hajnal, Mate and Rado's book on partition calculus, and by conversations with A. Levi who was writing a book on naive set theory.Notation. Let i, j, α, β, γ, η, σ be ordinals, μ be a cardinal, f, g be functions.Let cf α be the cofinality of α.A partially ordered set T is a tree if for any a ∈ T, {b: b < a, b ∈ T} is well ordered; its order type (an ordinal) is called the level of a, and Tα is the set of a ∈ T of level a.
Logic and time
Authors: Burgess John P. 
Published: May 2006

‘In Time rigorous abstraction, in Time the highest, in Time divine knowledge, is comprehended’Atharva VedaFollowing McTaggart [17], we may distinguish two aspects of time: The A-series, running through the past to the present and on into the future, and the B-series, running from earlier to later. In Indo-European languages at least, verbs are tensed, so we cannot help but place whatever we speak of in one of the three divisions of the A-series. But these divisions are not permanent: what is present was future and will be past. Hence a typical statement, e.g. ‘Socrates is sitting’, may well be true at one time and false at another. The instability of truth-value over time was a commonplace among pre-Renaissance logicians, but most modern writers have ‘abstracted from’, i.e. ignored, this feature of ordinary language.Early logicians were quite interested in time: Aristotle questioned the applicability of the excluded middle to predictions of future contingencies in the famous ‘sea-fight’ passage of On interpretation. Later Greek logicians debated whether that which neither is nor will be can legitimately be called possible, and whether, in order for the conditional ‘if p, then q’ to be true, it is required that ‘not both p and not ∼q’ be true (not just now but) always. Mediaeval logicians in Western Europe struggled with logical difficulties created by the Dogmas of the Church, Omniscience and Free Will. Their counterparts in the Islamic world puzzled over the semantics of the temporal adverbs, ‘always’, ‘usually’, ‘often’, ‘sometimes’, ‘seldom’, ‘never’.
Chains and antichains in
Authors: Baumgartner James E. 
Published: May 2006

Consider the following propositions:(A) Every uncountable subset of contains an uncountable chain or antichain (with respect to ⊆).(B) Every uncountable Boolean algebra contains an uncountable antichain (i.e., an uncountable set of pairwise incomparable elements).Until quite recently, relatively little was known about these propositions. The oldest result, due to Kunen [4] and the author independently, asserts that if the Continuum Hypothesis (CH) holds, then (A) is false. In fact there is a counter-example 〈Aα: α < ω1〉 such that α < β implies Aβ −Aα is finite. Kunen also observed that Martin's Axiom (MA) + ¬CH implies that no such counterexample 〈Aα: α < ω1〉 exists.Much later, Komjáth and the author [2] showed that ◊ implies the existence of several kinds of uncountable Boolean algebras with no uncountable chains or antichains. Similar results (but motivated quite differently) were obtained independently by Rubin [5]. Berney [3] showed that CH implies that (B) is false, but his algebra has uncountable chains. Finally, Shelah showed very recently that CH implies the existence of an uncountable Boolean algebra with no uncountable chains or antichains.Except for Kunen's result cited above, the only result in the other direction was the theorem, due also to Kunen, that MA + ¬CH implies that any uncountable subset of with no uncountable antichains must have both ascending and decending infinite sequences under ⊆.
On the consistency of an impredicative subsystem of Quine's NF
Authors: Crabbé Marcel 
Published: May 2006

NFP is thepredicativefragment of NF. In this system we do not allow a set to exist if it cannot be defined without using quantifiers ranging over its type or parameters of a higher type. NFI is a less restrictive fragment located between NFP and NF.We show that NFP is really weaker than NFI; similarly, NFI is weaker than NF. This result will be obtained in the following manner: on the one hand, we will show that NFP can be proved consistent in elementary arithmetic and that second order arithmetic is interpretable in NFI; on the other hand, we will prove the consistency of NFI in third order arithmetic, which is contained in NF.The paper is divided in four sections. In §1, we define the concepts needed and collect a few results together in such a way that they will be ready for later use. In §2, we will present a model-theoretic (quick) proof of the consistency of NFI (and thus of NFP). The proof will be chosen (it is not the quickest!) so as to motivate in a natural manner the details of the proof-theoretical version of it that will be presented in §3. §4 will be devoted to the axiom of infinity in NFP and NFI.
A result of relative consistency about the predicate WO(δ, x)
Authors: David René 
Published: May 2006

In [K] Keisler introduces a set theoretical relation WO(δ, x), where δ is an ordinal. This relation is characterized in ZF set theory by the following properties:(1) WO(0, x) if and only if there is a wellordering on x.(2) For δ > 0, WO(δ, x) if and only if there is a function ƒ with domain an ordinal λ such that:We denote WO∞ the collection denned by: WO∞(x) = ∃δ WO(δ, x).In [K] Keisler shows that countable transitive models of ZF + ∃x ¬ WO∞(x) have transitive uncountable elementary extensions with the same ordinals. Moreover for transitive models, satisfying ∃x ¬ WO∞(x) is also a necessary condition for the existence of an elementary extension with the same ordinals. (See [K bis] and also [K-M] where the connection with forcing is analysed.)
The 1-section of a countable functional
Authors: Normann Dag , Wainer Stan S. 
Published: May 2006

The continuous or countable functionals were independently denned by Kleene [9] and Kreisel [10]. They were intended as a suitable basis for constructive mathematics, and thus it is interesting to investigate various notions of recursion on the countable functionals.There have been two main streams in this investigation, the study of countable recursion and the study of computability or Kleene-recursion.Countable recursion is the theory of recursion on the associates. Gandy and Hyland [3] and Hyland [7] are good sources for the recent development of countable recursion.This paper will mostly be concerned with Kleene-recursion on the countable functionals as denned in Kleene [8] and [9]. We assume some familiarity with the countable functionals and associates, as presented in Kleene [9], Bergstra [1] or any other paper on the subject.Pioneering work with recursion in nonnormal objects was done by Grilliot [4], who proved that a functional F of type 2 is normal if and only if its 1-section (that is the set of functions recursive in F) is closed under ordinary jump, and if and only if F is continuous on 1-section (F).Hinman [6] constructed a countable functional that is not recursively equivalent to a function, and thereby showed that recursion in nonnormal functionals is an extension of ordinary recursion in functions.In [6], Hinman asked if there are functionals with topless 1-sections, i.e. with no maximal elements in the semi-lattice of degrees. This was answered in the affirmative by Bergstra [1], using a spoiling construction. Thus the class of 1-sections of functionals extends the class of 1-sections of functions.
Forcing in intuitionistic systems without power-set
Authors: Grayson R.J. 
Published: May 2006

AbstractIt is shown how to define forcing semantics within metatheories not containing the power-set construction, in particular, how to construct exponents assuming only (a slightly strengthened form of) exponents in the metatheory. Some straightforward applications (consistency and independence results, and derived rules) are obtained for such systems.
Expansions of models and turing degrees
Authors: Knight Julia , Nadel Mark 
Published: May 2006

If is a countable recursively saturated structure and T is a recursively axiomatizable theory that is consistent with Th(), then it is well known that can be expanded to a recursively saturated model of T [7, p. 186]. This is what has made recursively saturated models useful in model theory. Recursive saturation is the weakest notion of saturation for which this expandability result holds. In fact, if is a countable model of Pr = Th(ω, +), then can be expanded to a model of first order Peano arithmetic P just in case is recursively saturated (see [3]).In this paper we investigate two natural sets of Turing degrees that tell a good deal about the expandability of a given structure. If is a recursively saturated structure, I() consists of the degrees of sets that are recursive in complete types realized in . The second set of degrees, D(), consists of the degrees of sets S such that is recursive in S-saturated. In general, I() ⊆ D(). Moreover, I() is obviously an “ideal” of degrees. For countable structures , D() is “closed” in the following sense: For any class C ⊆ 2ω, if C is co-r.e. in S for some set S such that , then there is some σ ∈ C such that . For uncountable structures , we do not know whether D() must be closed.
Deux ou trois choses que je sais de Ln
Authors: Poizat Bruno 
Published: May 2006

Si vous ne gardez du calcul des prédicats classique (que j'appellerai Lω) que les formules qui ne font intervenir, tant en occurences libres que liées, que les n premières variables, vous obtenez le langage Ln. Avant de poursuivre, et de façon un peu cavalière, je vous invite à résoudre l'exercice suivant:Exercice 1. Dans le langage comprenant seulement un symbole de relation binaire, caractériser à isomorphisme près la chaîne à n éléments au moyen d'un seul énoncé de L2.Vous avez vu que Ln a plus de possibilités d'expression que vous ne l'aviez d'abord pensé, qu'il est ennemi de la forme prénexe, que sa force au contraire réside dans l'utilisation d'une même variable en différentes places d'une formule.Ces langages ont été étudiés par Léon Henkin dans [1]; il y remarque que L1 est très pauvre: c'est le “calcul des classes”, on y peut seulement exprimer que des combinaisons booléennes de formules atomiques sont vides ou non, et nous n'en parlerons plus; il montre ensuite que tout théorème de L2est prouvable par une suite de déductions (d'un système classique) qui ne fait intervenir que des énoncés de L3. Dans [2], Donald Monk construit pour tout n un théorème de L3 dont toute preuve utilise des énoncés avec au moins n variables. Dans [3], Dana Scott prouve que L2 est décidable; ce résultat est précisé par Michael Mortimer dans [4], qui montre que tout énoncé consistant f de L2 a un modèle fini, de cardinal borné effectivement en fonction de f. Pour n supérieur ou égal à 3, Ln n'est plus décidable car il a même pouvoir d'expression que Lω, comme il est montré dans l'exercice suivant.
An axiomatic version of positive semilattice relevance logic
Authors: Charlwood G. 
Published: May 2006

In [8] and [9] Urquhart presented the semilattice semantics for a relevance logic with conjunction and disjunction. These semantics combined the simplest known semantics for relevant implication, with minimal modification of the truth conditions for conjunction and disjunction. A slightly different approach leads to the distinct positive relevance logic of [7] and [5]. Sound and complete axiomatic logics for the semilattice semantics were readily available, so long as disjunction was left out.The purpose of this paper is to report the remedy of this lack. In [1] the author developed work of Fine ([3], announced in [4]) and Prawitz [6] to show soundness and completeness of an axiomatic logic and two natural deduction logics, with respect to Urquhart's semantics. Thus Prawitz' positive relevance logic is equivalent to the semilattice logic, rather than the logic of [5]. We present here the axiomatic logic and show its equivalence to the semilattice semantics.In §1 the languages and logic are given, along with a few proof-theoretic remarks. In §2 the semantics are defined, and the soundness proof is sketched. In §3, the completeness proof is given.
Nonstandard natural number systems and nonstandard models
Authors: Kamo Shizuo 
Published: May 2006

AbstractIt is known (see [1, 3.1.5]) that the order type of the nonstandard natural number system *N has the form ω + (ω* + ω) θ, where θ is a dense order type without first or last element and ω is the order type of N. Concerning this, Zakon [2] examined *N more closely and investigated the nonstandard real number system *R, as an ordered set, as an additive group and as a uniform space. He raised five questions which remained unsolved. These questions are concerned with the cofinality and coinitiality of θ (which depend on the underlying nonstandard universe *U). In this paper, we shall treat nonstandard models where the cofinality and coinitiality of θ coincide with some appropriated cardinals. Using these nonstandard models, we shall give answers to three of these questions and partial answers to the other to questions in [2].
Blunt and topless end extensions of models of set theory
Authors: Kaufmann Matt 
Published: May 2006

AbstractLet be a well-founded model of ZFC whose class of ordinals has uncountable cofinality, such that has a Σn end extension for each n ∈ ω. It is shown in Theorem 1.1 that there is such a model which has no elementary end extension. In the process some interesting facts about topless end extensions (those with no least new ordinal) are uncovered, for example Theorem 2.1: If is a well-founded model of ZFC, such that has uncountable cofinality and has a topless Σ3 end extension, then has a topless elementary end extension and also a well-founded elementary end extension, and contains ordinals which are (in ) highly hyperinaccessible. In §3 related results are proved for κ-like models (κ any regular cardinal) which need not be well founded. As an application a soft proof is given of a theorem of Schmerl on the model-theoretic relation κ → λ. (The author has been informed that Silver had earlier, independently, found a similar unpublished proof of that theorem.) Also, a simpler proof is given of (a generalization of) a characterization by Keisler and Silver of the class of well-founded models which have a Σn end extension for each n ∈ ω. The case κ = ω1 is investigated more deeply in §4, where the problem solved by Theorem 1.1 is considered for non-well-founded models. In Theorems 4.1 and 4.4, ω1-like models of ZFC are constructed which have a Σn end extension for all n ∈ ω but have no elementary end extension. ω1-like models of ZFC which have no Σ3 end extension are produced in Theorem 4.2. The proof uses a notion of satisfaction class, which is also applied in the proof of Theorem 4.6: No model of ZFC has a definable end extension which satisfies ZFC. Finally, Theorem 5.1 generalizes results of Keisler and Morley, and Hutchinson, by asserting that every model of ZFC of countable cofinality has a topless elementary end extension. This contrasts with the rest of the paper, which shows that for well-founded models of uncountable cofinality and for κ-like models with κ regular, topless end extensions are much rarer than blunt end extensions.
On the semantics of the Henkin quantifier
Authors: Krynicki Michał , Lachlan Alistair H. 
Published: May 2006

In [5] Henkin defined a quantifier, which we shall denote by QH: linking four variables in one formula. This quantifier is related to the notion of formulas in which the usual universal and existential quantifiers occur but are not linearly ordered. The original definition of QH wasHere (QHx1x2y1y2)φ is true if for every x1 there exists y1 such that for every x2 there exists y2, whose choice depends only on x2 not on x1 and y1 such that φ(x14, x2, y1, y2). Another way of writing this isIn [5] it was observed that the logic L(QH) obtained by adjoining QH defined as in (1) is more powerful than first-order logic. In particular, it turned out that the quantifier “there exist infinitely many” denoted Q0 was definable from QH because
Recursive Boolean algebras with recursive atoms
Authors: Remmel Jeffrey B. 
Published: May 2006

A Boolean algebra (henceforth abbreviated B.A.) is said to be recursive if B is a recursive subset of the natural numbers N and the operations ∧ (meet), ∨ (join), and ¬ (complement) are partial recursive. Let denote the set of atoms of and denote the ideal generated by the atoms of . Given recursive B.A.s and , we write ≈ if is isomorphic to and ≈r if is recursively isomorphic to , i.e., if there is a partial recursive isomorphism from onto .Recursive B.A.s have been studied by several authors including Ershov [2], Fiener [3], [4], Goncharov [5], [6], [7], LaRoche [8], Nurtazin [7], and the author [10], [11]. This paper continues a study of the recursion theoretic relationships among , , and the recursive isomorphism type of a recursive B.A. we started in [11]. We refer the reader to [11] for any unexplained notation and definitions. In [11], we were mainly concerned with the possible recursion theoretic properties of the set of atoms in recursive B.A.s. We found that even if we insist that be recursive, there is considerable freedom for the properties of . For example, we showed that if is a recursive B.A. such that is recursive and is infinite, then (i) there exists a recursive B.A. such that and both and are recursive and (ii) for any nonzero r.e. degree δ, there exist recursive B.A.s , , … such that for each i, is of degree δ, is recursive, is immune if i is even and is not immune if i is odd, and no two B.A.s in the sequence are recursively isomorphic.
A result on implications of Σ1-sentences and its application to normal form theorems
Authors: Girard Jean-Yves , Päppinghaus Peter 
Published: May 2006

In this paper we are concerned with the formal provability of certain true implications of Σ1-sentences. Old completeness and incompleteness results already give some information about this. For example by Σ1-completeness of PRA (primitive recursive arithmetic) every true implication of the form D → E, where D is a Σ0-sentence and E a Σ1-sentence, or D a Σ1-sentence and E a true Σ1-sentence, is provable in PRA. On the other hand, by Gödel's incompleteness theorems one can define for every suitable theory S a false Σ1-sentence Ds such that for every false Σ0-sentence E the true implication Ds → E is not provable in S, but is provable in PRA + Cons. So one sees that for a suitable fixed conclusion the provability of true implications of Σ1-sentences depends on the content of the premise.Now we ask, if and how for a suitable fixed premise the provability of true implications of Σ1-sentences depends on the conclusion. As remarked above, by Σ1-completeness of PRA this question is settled, if the premise is true. For a false premise it is answered in § 1 as follows: Let D be a false Σ1-sentence, S an extension of PRA, and S+ ≔ PRA + IAΣ1 + RFNΣ1S). Depending on D and S one can define a Σ1-sentence Es such that S+ ⊢ D → Es, but S ⊬ D → Es provided that S+ is not strong enough to refute D. (IAΣ1 denotes the scheme of the induction axiom for Σ1-formulas, and RFNΣ1(S) the uniform Σ1-reflection principle for S.)
Vopěnka's principle and compact logics
Authors: Makowsky J. A. 
Published: May 2006

AbstractWe study the effects of Vopěnka's principle on properties of model theoretic logics. We show that Vopěnka's principle is equivalent to the assumption that every finitely generated logic has a compact cardinal. We show also that it is equivalent to the assumption that every such logic has a global Hanf number.