A simplified functorial construction of the veblen hierarchy

We give a simple and elementary proof of the following result of Girard and Vauzeilles which is proved in [5]: “The binary Veblen function ψ: On × On — On is a dilator.” Our proof indicates the intimate connection between the traditional theory of ordinal notation systems and Girard's theory of denotation systems. MSC: 03F15.     

A proof of strongly uniform termination for Gödel's T by methods from local predicativity

We estimate the derivation lengths of functionals in G?del's system of primitive recursive functionals of finite type by a purely recursion-theoretic analysis of Schütte's 1977 exposition of Howard's weak normalization proof for . By using collapsing techniques from Pohlers' local predicativity approach to proof theory and based on the Buchholz-Cichon and Weiermann 1994 approach to subrecursive hierarchies we define a collapsing f unction so that for (closed) terms of G?del's we have: If reduces to then By one uniform proof we obtain as corollaries: A derivation lengths classification for functionals in , hence new proof of strongly uniform termination of . A new proof of the Kreisel's classific ation of the number-theoretic functions which can be defined in , hence a classification of the provably total functions of Peano Arithmetic. A new proof of Tait's results on weak normalization for . A new proof of Troelstra's result on strong normalization for . Additionally, a slow growing analysis of G?del's is obtained via Girard's hierarchy comparison theorem. This analyis yields a contribution to two open pro blems posed by Girard in part two of his book on proof theory. For the sake of completeness we also mention the Howard Schütte bound on derivation lengths for the simple typed -calculus. Received August 4, 1995     

Recursion in Partial Type-1 Objects With Well-Behaved Oracles

We refine the definition of II-computability of [12] so that oracles have a “consistent”, but natural, behaviour. We prove a Kleene Normal Form Theorem and closure of semi-recursive relations under ?¹. We also show that in this more inclusive computation theory Post's theorem in the arithmetical hierarchy still holds. Mathematics Subject Classification: 03D65, 03D75.     

A proof of the normal form theorem for the closed terms of Girard's system F by means of computability

In this paper a proof of the normal form theorem for the closed terms of Girard's system F is given by using a computability method à la Tait. It is worth noting that most of the standard consequences of the normal form theorem can be obtained using this version of the theorem as well. From the proof-theoretical point of view the interest of the proof is that the definition of computable derivation here used does not seem to be well founded. MSC: 03F05, 03B15.     

A formalization of Sambins's normalization for GL

Sambin [6] proved the normalization theorem (Hauptsatz) for GL, the modal logic of provability, in a sequent calculus version called by him GLS. His proof does not take into account the concept of reduction, commonly used in normalization proofs. Bellini [1], on the other hand, gave a normalization proof for GL using reductions. Indeed, Sambin's proof is a decision procedure which builds cut-free proofs. In this work we formalize this procedure as a recursive function and prove its recursiveness in an arithmetically formalizable way, concluding that the normalization of GL can be formalized in PA. MSC: 03F05, 03B35, 03B45.     

ON DEFINABILITY OF NORMAL SUBGROUPS OF A SUPERSTABLE GROUP

In this note we treat maximal and minimal normal subgroups of a superstable group and prove that these groups are definable under certain conditions. Main tool is a superstable version of Zil'ber's indecomposability theorem. MSC: 03C60.     

Typical ambiguity and elementary equivalence

A sentence of the usual language of set theory is said to be stratified if it is obtained by “erasing” type indices in a sentence of the language of Russell's Simple Theory of Types. In this paper we give an alternative presentation of a proof the ambiguity theorem stating that any provable stratified sentence has a stratified proof. To this end, we introduce a new set of ambiguity axioms, inspired by Fraïssé's characterization of elementary equivalence; these axioms can be naturally used to give different proofs of the ambiguity theorem (semantic or syntactic, classical or intuitionistic). MSC: 03B15, 03F50, 03F55.     

Intrinsically Hyperarithmetical Sets

The main result proved in the paper is that on every recursive structure the intrinsically hyperarithmetical sets coincide with the relatively intrinsically hyperarithmetical sets. As a side effect of the proof an effective version of the Kueker's theorem on definability by means of infinitary formulas is obtained. Mathematics Subject Classification: 03D70, 03D75.     

LEVELS OF IMPLICATION AND TYPE FREE THEORIES OF CLASSIFICATIONS WITH APPROXIMATION OPERATOR

We investigate a theory of Frege structures extended by the Myhill-Flagg hierarchy of implications. We study its relation to a property theory with an approximation operator and we give a proof theoretical analysis of the basic system involved. MSC: 03F35, 03D60.     

Symmetric functions,codes of partitions and the KP hierarchy

We consider an operator of Bernstein for symmetric functions and give an explicit formula for its action on an arbitrary Schur function. This formula is given in a remarkably simple form when written in terms of some notation based on the code of a partition. As an application, we give a new and very simple proof of a classical result for the KP hierarchy, which involves the Plücker relations for Schur function coefficients in a τ-function for the hierarchy. This proof is especially compact because we are able to restate the Plücker relations in a form that is symmetrical in terms of partition code notation.     

A Nadel vanishing theorem for metrics with minimal singularities on big line bundles

The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki's proof of Kollár's injectivity theorem. Moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics.     

Recursive Structures and Ershov's Hierarchy

Ash and Nerode [2] gave natural definability conditions under which a relation is intrinsically r. e. Here we generalize this to arbitrary levels in Ershov's hierarchy of Δ sets, giving conditions under which a relation is intrinsically α-r. e. Mathematics Subject Classification: 03C57, 03D55.     

Other Proofs of Old Results

We transform the proof of the second incompleteness theorem given in [3] to a proof-theoretic version, avoiding the use of the arithmetized completeness theorem. We give also new proofs of old results: The Arithmetical Hierarchy Theorem and Tarski's Theorem on undefinability of truth; the proofs in which the construction of a sentence by means of diagonalization lemma is not needed.     

Euler’s Pentagonal Number Theorem and the Rogers-Fine Identity

Euler discovered the pentagonal number theorem in 1740 but was not able to prove it until 1750. He sent the proof to Goldbach and published it in a paper that finally appeared in 1760. Moreover, Euler formulated another proof of the pentagonal number theorem in his notebooks around 1750. Euler did not publish this proof or communicate it to his correspondents, probably because of the difficulty of clearly presenting it with the notation at the time. In this paper we show that the method of Euler??s unpublished proof can be used to give a new proof of the celebrated Rogers-Fine identity.     

Cofinal Indiscernibles and some Applications to New Foundations

We prove a theorem about models with indiscernibles that are cofinal in a given linear order. We apply this theorem to obtain new independence results for Quine's set theory New Foundations, thus solving two open problems in this field. Mathematics Subject Classification : 03C55, 03B15, 03B60, 03E70.     

Decomposability of low 2-computably enumerable degrees and turing jumps in the Ershov hierarchy

In this paper we prove the following theorem: For every notation of a constructive ordinal there exists a low 2-computably enumerable degree that is not splittable into two lower 2-computably enumerable degrees whose jumps belong to the corresponding Δ-level of the Ershov hierarchy.     

Remarks on Levy's reflection axiom

Adding higher types to set theory differs from adding inaccessible cardinals, in that higher type arguments apply to all sets rather than just ordinary ones. Levy's reflection axiom is justified, by considering the principle that we can pretend that the universe is a set, together with methods of Gaifman [8]. We reprove some results of Gaifman, and some facts about Levy's reflection axiom, including the fact that adding higher types yields no new theorems about sets. Some remarks on standard models are made. An obvious strengthening of Levy's axiom to higher types is considered, which implies the existence of indescribable cardinals. Other remarks about larger cardinals are made; some questions of Gloede [9] are settled. Finally we argue that the evidence for V = L is strong, and that CH is certainly true. MSC: 03E30, 03E55.     

Jump Theorems for REA Operators

In [2], Jockusch and Shore have introduced a new hierarchy of sets and operators called the REA hierarchy. In this note we prove analogues of the Friedberg Jump Theorem and the Sacks Jump Theorem for many REA operators. MSC: 03D25, 03D55.     

A finitization of Littlewood's Tauberian theorem and an application in Tauberian remainder theory

In this paper we study Littlewood's Tauberian theorem from a proof theoretic perspective. We first use the Dialectica interpretation to produce an equivalent, finitary formulation of the theorem, and then carry out an analysis of Wielandt's proof to extract concrete witnessing terms. We argue that our finitization can be viewed as a generalized Tauberian remainder theorem, and we instantiate it to produce two concrete remainder theorems as a corollary, in terms of rates of convergence and rates metastability, respectively. We rederive the standard remainder estimate for Littlewood's theorem as a special case of the former.     

齐次函数的一个仿射球定理

本文研究了相关齐次函数的仿射球定理.利用Hopf极大值原理,对任意给定的带凹性条件的初等对称曲率问题,获得了此类仿射球定理.特别地,这也给出了Deicke齐次函数定理的一个新证明.