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Gӧdel’s incompleteness theorem asserts that if formal arithmetic is consistent then there exists an arithmetic statement such that neither the statement nor its negation can be deduced from the axioms of formal arithmetic. Previously [3], it was proved that formal arithmetic remains incomplete if, instead of the set of all arithmetic statements, we consider any set of some class of “almost all” statements (the class of so-called strongly generic subsets). This result is strengthened as follows: formal arithmetic is incomplete for any generic subset of arithmetic statements (i.e., a subset of asymptotic density 1).

A well-known result in Reverse Mathematics is the equivalence of the formalized version of the Gödel completeness theorem [8] – i.e. every countable, consistent set of first-order sentences has a model – and Weak König's Lemma [WKL] – i.e. every infinite tree of 0-1 sequences contains an infinite path– over the base theory RCA0. It is less well known how the Completeness Theorem came to be studied in the setting of second-order arithmetic and computability theory. The first goal of this note will be to recount these developments against the backdrop of the latter phases of the Hilbert program, culminating in the publication of the second volume of Hilbert and Bernays’s [13] Grundlagen der Mathematiks in 1939. This work contains a detailed formalization of the Completeness Theorem in a system similar to first-order Peano arithmetic [PA] – a result which has come to be known as the Arithmetized Completeness Theorem. Its second goal will be to illustrate how reflection on this result informed Bernays’s views about the philosophy of mathematics, in particular in regard to his engagement with the maxim “consistency implies existence”.

Recently, Hasankhani et al. proved that any Felbin-fuzzy inner product space can be imbedded in a complete Felbin-fuzzy inner product space or Felbin-fuzzy Hilbert space. In this paper, it is showed a general result that any classical Hilbert space is a Felbin-fuzzy Hilbert space, so it shows that all results in classical Hilbert spaces are immediate consequences of the corresponding results for Felbin-fuzzy Hilbert spaces. Moreover by an example, it is showed that the spectrum of the category of Felbin-fuzzy Hilbert spaces is broader than the category of classical Hilbert spaces. Finally the authors are able to state a transformation theorem from an ascending family of crisp inner product into Felbin-fuzzy inner product that shows how to formulate the recent results.

Hilbert’s Fourth Problem is one of the most important mathema tical problems, formulated by Hilbert in 1900. Unfortunately, attempts to solve this problem during 20 th century did not lead to the generally recognized solution, and now modern mathematicians believe that the problem has been formulated by Hilbert "very vague" and therefore it can not be solved. Th e main purpose of this article is to develop a new view on authors’ original solution to this problem a nd to interpret this problem as MILLENNIUM PROBLEM in Geometry what has an interdisciplinary importance and affects not only on geometry, but also on all theoretical natural sciences. The source of a new approach to solving this problem is a new branch of mathematics, the Mathematics of Harmony, which goes back in its origins to Euclid’s Elements and has interdisciplinary importance for modern science.

In this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one of the main problems that Hilbert encountered in his foundational studies consisted in securing a link between formalization and intuition. We will also analyze a related problem, that we will call "Frege's Problem", form the time of the foundation of geometry and investigate the role of the Axiom of Completeness in its solution.

In this paper I will consider the axioms for propositional logic which were presented by Hilbert in his conferences during the year 1922, and those which were presented by Hilbert and Bernays in the book Grundlagen der Mathematik, I (1934). I will describe a general procedure in order to translate Hilbert's axioms into rules on sequents and I will show that, following this procedure, Hilbert's axioms become particular cases of (derived or primitive) rules of Gentzen's Sequent Calculus and contain ideas which will be focused and developed in Gentzen's Sequent Calculus and also in more recent logical investigations.

Hilbert first mentioned ideal elements in his 1898-99 lectures on geometry. He described them as important, fruitful, and of frequent occurrence in mathematics, pointing to the examples of negative, irrational, imaginary, ideal and transfinite numbers. In geometry, he had in mind the examples of points, lines, and planes at infinity, whose introduction gives geometry a certain completeness, by making theorems such as those of Pappus and Desargues universally valid. In this article I will discuss how Hilbert transformed our view of the Pappus and Desargues theorems by showing that they express the underlying algebraic structure of projective geometry. I will compare this result with another of Hilbert's great contributions, his calculus of ends. By studying the ideal elements of the hyperbolic plane, Hilbert similarly extracted algebraic structure from the axioms of hyperbolic geometry. Hilbert's treatments of projective and hyperbolic geometry have another important common element: construction of real numbers. To achieve this, Hilbert has to add an axiom of continuity to the geometry axioms, but he evidently wants to show that the real numbers can be put on a geometric foundation.

In this paper I provide a brief reconstruction of Otto Holder’s conception of proof. My reconstruction focuses on Holder’s critical assessment of David Hilbert’s account of axiomatics in general, and of Hilbert’s conception of metamathematics in particular. I argue that Holder’s analysis of Hilbert’s general methodological ideas and, more importantly, Holder’s analysis of the logical structure of the proofs provided by Hilbert in his Grundlagen der Geometrie of 1899 are helpful in reaching a clearer understanding of van der Waerden’s claim linking Holder’s conception of proof to the tradition established by Kurt Godel.

We consider Hilbert’s contributions to the foundations of mathematics, including his conception of the axiomatic method, his notion of mathematical proof, and his approach to infinity through finitary tools. We describe his “Programme”, and we comment its collapse due to Godel’s Incompleteness Theorems. Despite that defeat, Hilbert’s project remains topical today.

The objective of this paper is to discuss existence, uniqueness and regularity issues of minimizers of one dimensional variational problems in Hilbert spaces. We obtain existence of C2 local minimizers and prove that the value function of an optimal control problem solves corresponding Hamilton-Jacobi equation in a viscosity sense.

Hilbert C ⁎ -modules are the analogues of Hilbert spaces where a C ⁎ -algebra plays the role of the scalar field. With the advent of Kasparovʼs celebrated KK -theory they became a standard tool in the theory of operator algebras. While the elementary properties of Hilbert C ⁎ -modules can be derived basically in parallel to Hilbert space theory the lack of an analogue of the Projection Theorem soon leads to serious obstructions and difficulties. In particular the theory of unbounded operators is notoriously more complicated due to the additional axiom of regularity which is not easy to check. In this paper we present a new criterion for regularity in terms of the Hilbert space localizations of an unbounded operator. We discuss several examples which show that the criterion can easily be checked and that it leads to nontrivial regularity results.

The development of the theory of extensions of Hilbert C*-modules was initiated by Bakic and Guljas. An easy observation shows that, if the underlying C*-algebra extension is commutative and the Hilbert C*-modules are projective of finite type, then the algebraic properties of the corresponding Busby invariant enable one to identify extensions with isometric mappings of the corresponding vector bundles. For the case in which the Hilbert C*-modules are free of rank one, we evaluate the set of extensions in topological terms.

In this paper, we will study the class of Hilbert algebras with supremum, i.e., Hilbert algebras where the associated order is a join-semilattice. First, we will give a simplified topological duality for Hilbert algebras using sober topological spaces with a basis of open-compact sets satisfying an additional condition. Next, we will extend this duality to Hilbert algebras with supremum. We shall prove that the ordered set of all ideals of a Hilbert algebra with supremum has a lattice structure. We will also see that in this lattice, it is possible to define an implication, but the resulting structure is neither a Heyting algebra nor an implicative semilattice. Finally, we will give a dual description of the lattice of ideals of a Hilbert algebra with supremum.

Deligne proved that the Hilbert series of all Artin monoids are rational functions. We give an algorithm to compute the Hilbert series of the braid monoids [Formula: see text]. We also show that the Hilbert series of the positive words in [Formula: see text] with a given prefix are rational functions.

The aim of this article is to establish some interesting inequalities involving arithmetic functions.

Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel's second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical instrumentalism are defeated by Gödel's theorem, not all are. By considering inductive reasons in mathematics, we show that some mathematical instrumentalisms survive the theorem. © 2010 Elsevier Ltd.

This essay consists of two parts. In the first part, I focus my attention on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. This gives me the opportunity to comment also a little on H. Hankel, J. Thomae—Frege’s main targets when he comes to criticize “formal theories of arithmetic” in Die Grundlagen der Arithmetik (1884) and the second volume of Grundgesetze der Arithmetik (1903)—G. Cantor, L. E. J. Brouwer and D. Hilbert (1899). Part 2 is mainly devoted to Hilbert’s proof theory of the 1920s (1922–1931). I begin with an account of his early attempt to prove directly, and thus not by reduction or by constructing a model, the consistency of (a fragment of) arithmetic. In subsequent sections, I give a kind of overview of Hilbert’s metamathematics of the 1920s and try to shed light on a number of difficulties to which it gives rise. One serious difficulty that I discuss is the fact, widely ignored in the pertinent literature on Hilbert’s programme, that his language of finitist metamathematics fails to supply the conceptual resources for formulating a consistency statement qua unbounded quantification. Along the way, I shall comment on W. W. Tait’s objection to an interpretation of Hilbert’s finitism by Niebergall and Schirn, on G. Gentzen’s allegedly finitist consistency proof for Peano Arithmetic as well as his ideas on the provability and unprovability of initial cases of transfinite induction in pure number theory. Another topic I deal with is what has come to be known as partial realizations of Hilbert’s programme, chiefly advocated by S. G. Simpson. Towards the end of this essay, I take a critical look at Wittgenstein’s views about (in)consistency and consistency proofs in the period 1929–1933. I argue that both his insouciant attitude towards the emergence of a contradiction in a calculus and his outright repudiation of metamathematical consistency proofs are unwarranted. In particular, I argue that Wittgenstein falls short of making a convincing case against Hilbert’s programme. I conclude with some philosophical remarks on consistency proofs and soundness and raise a question concerning the consistency of analysis.

In this paper we give some new results for g-frames in Hilbert C∗-modules and then we introduce a bounded A-linear operator L, by means of this operator L we character the properties of the g-frames and g-Riesz basis in Hilbert C∗-modules. Finally, we establish some important equalities and inequalities for frames and g-frames in Hilbert C∗-modules.

Inspired by examples of small Hilbert matrices, the author proves a property of symmetric totally positive Cauchy matrices, called AT-property, and consequences for the Hilbert matrix.

It is typically recognized that Godel’s negative results undermined Hilbert’s Program, at least in its strictest form. In this paper we point out the way in which Hilbert himself had created the possibility of Godel doing so. We highlight the crucial role Hilbert played in the evolution and stabilization of the term “completeness”, and analyze the impact of his work on results obtained by researchers working close to him, but also by authors belonging to different traditions of mathematical practice, and whose views (that is, mathematical and philosophical views) on the nature of their field did not coincide with those of Hilbert; the major example here is Kurt Godel. Furthermore, we show that an informal version of the modern notion of completeness is already presented by Hilbert as early as 1899. Our historiographical objective is thus twofold: on the one hand, to suggest a new way of showing what we mean when we say that the completeness theorem took about thirty years to be stated as such, and merely ...

In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic. We explore how this duality is related to the duality given in [6] for finite Hilbert algebras, and with the topological duality developed in [7] for Tarski algebras.

We present a functional interpretation of Peano arithmetic that uses Godel’s computable functionals and which systematically injects uniformities into the statements of finite-type arithmetic. As a consequence, some uniform boundedness principles (not necessarily set-theoretically true) are interpreted while maintaining unmoved the Π 2 0 -sentences of arithmetic. We explain why this interpretation is tailored to yield conservation results.

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollstandigkeitsaxiom).

Bounds for the Castelnuovo-Mumford regularity and Hilbert coefficients are given in terms of the arithmetic degree (if the ring is reduced) or in terms of the defining degrees. From this it follows that there exists only a finite number of Hilbert functions associated to reduced algebras over an algebraically closed field with a given arithmetic degree and dimension. A good bound is also given for the Castelnuovo-Mumford regularity of initial ideals which depends neither on term orders nor on the coordinates, and holds for any field., Comment: To appear in Trans. Amer. Math. Soc

1. INTRODUCTION. In this article we explore a variety of pleasing connections between analysis, number theory, and operator theory, while revisiting a number of beautiful inequalities originating with Hilbert, Hardy and others. We shall first discuss the aforementioned Hilbert inequality [14], [18] and then apply it to various multiple zeta values. In consequence we obtain the norm of the classical Hilbert matrix, in the process illustrating the interplay of numerical and symbolic computation with classical mathematics. 2. HILBERT'S (EASIER) INEQUALITY. The inequality in question is

In November and December 1915, Hilbert presented two communications to the Gottingen Academy of Sciences under the common title ‘The Foundations of Physics’. Versions of each eventually appeared in the Nachrichten of the Academy. Hilbert's first communication has received significant reconsideration in recent years, following the discovery of printer's proofs of this paper, dated 6 December 1915. The focus has been primarily on the ‘priority dispute’ over the Einstein field equations. Our contention, in contrast, is that the discovery of the December proofs makes it possible to see the thematic linkage between the material that Hilbert cut from the published version of the first communication and the content of the second, as published in 1917. The latter has been largely either disregarded or misinterpreted, and our aim is to show that (a) Hilbert's two communications should be regarded as part of a wider research program within the overarching framework of ‘the axiomatic method’ (as Hilbert expressly stated was the case), and (b) the second communication is a fine and coherent piece of work within this framework, whose principal aim is to address an apparent tension between general invariance and causality (in the precise sense of Cauchy determination), pinpointed in Theorem I of the first communication. This is not the same problem as that found in Einstein's ‘hole argument’—something that, we argue, never confused Hilbert.

In the context of Hilbert’s irreducibility theorem, it is an open question whether there exists a bound for the least hilbertian specialization in N that is polynomial in the degree d and the logarithmic height log(H) of the polynomial P (T,Y ) in question. A positive answer would be useful, notably for algorithmic applications. We obtain a polynomial bound in log(H) and dHi(P) where Hi(P) — the Hilbert index of P — is a pure group-theoretical invariant we define and which we show to be absolutely bounded for many classes of polynomials. We also discuss further questions related to effectiveness in Hilbert’s irreducibility theorem. 2000 MSC. Primary 12E25, 14G05 ; Secondary 11C08, 12E05

We give alternative proofs of results due to Paris and Wilkie concerning the existence of end extensions of countable models of $B\Sigma_{1}$ , that is, the theory of $\Sigma_{1}$ collection.

In this section we shall be interested in discussing proofs, generalizations and geometric formulations of the so-called Hilbert Irreducibility Theorem (HIT in the sequel).

There are many beautiful constructions of the Hilbert scheme of closed subschemes of projective schemes. In many cases where Hilbert schemes are involved, it suffices to know they exist. On the other hand, there are many situations when it is crucial to have an explicit description of the Hilbert schemes. In this article we give a simple construction of Hilbert schemes of points, under general circumstances, that provides such a description. In addition to being useful for computations the construction is short, elementary, and explicit. Our methods rely on simple algebraic constructions. It gives explicit expressions of members of an affine covering of the Hilbert schemes, involving few variables satisfying natural equations. In particular, it explains the connections with commuting schemes of matrices. We also give some examples showing how our method can be used.

International audience; We generalize to Hilbert spaces a theorem of Glaeser concerning minimal Lipschitz extensions of m-jets of one variable with range in R. The results contained in this paper can be seen as a small contribution to the general problem of the minimal Lipschitz extensions from m-jets for a Hilbert space to another Hilbert space.

We establish the following model theoretic characterization of the fragment I¢0+Exp+B§1 of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment I¢0 of Peano arithmetic with induction limited to ¢0-formulae).

In this work we shall give a characterization of the Hilbert algebras given by the order and we will prove a duality for finite Hilbert algebras by means of finite ordered sets endowed with a distinguished set of subsets. We will also study the case when the finite Hilbert algebras are join-semilattices or meet-semilattices relative to the natural order defined by the implication. Finally we will prove that Hilbert do not admit a natural duality.

We study the arithmetical schema asserting that every eventually decreasing elementary recursive function has a limit. Some other related principles are also formulated. We establish their relationship with restricted parameter-free induction schemata. We also prove that the same principle, formulated as an inference rule, provides an axiomatization of the Σ 2 -consequences of I Σ 1 . Using these results we show that ILM is the logic of Π 1 -conservativity of any reasonable extension of parameter-free Π 1 -induction schema. This result, however, cannot be much improved: by adapting a theorem of D. Zambella and G. Mints we show that the logic of Π 1 -conservativity of primitive recursive arithmetic properly extends ILM . In the third part of the paper we give an ordinal classification of Σ n 0 -consequences of the standard fragments of Peano arithmetic in terms of reflection principles. This is interesting in view of the general program of ordinal analysis of theories, which in the most standard cases classifies Π -classes of sentences (usually Π 1 1 or Π 2 0 ).

Termination proofs are of critical importance for establishing the correct behavior of both transformational and reactive computing systems. A general setting for establishing termination proofs involves the use of the ordinal numbers, an extension of the natural numbers into the transfinite that were introduced by Cantor in the nineteenth century and are at the core of modern set theory. We present the first comprehensive treatment of ordinal arithmetic on compact ordinal notations and give efficient algorithms for various operations, including addition, subtraction, multiplication, and exponentiation. Using the ACL2 theorem proving system, we implemented our ordinal arithmetic algorithms, mechanically verified their correctness, and developed a library of theorems that can be used to significantly automate reasoning involving the ordinals. To enable users of the ACL2 system to fully utilize our work required that we modify ACL2, e.g., we replaced the underlying representation of the ordinals and added a large library of definitions and theorems. Our modifications are available starting with ACL2 version 2.8.

A finite Hilbert transformation associated with a polynomial is the analogue of a Hilbert transformation associated with an entire function which is a generalization of the classical Hilbert transformation. The weighted Hilbert inequality, which has applications in analytic number theory, is closely related to the finite Hilbert transformation associated with a polynomial. In this note, we study a relation between the finite Hilbert transformation and the weighted Hilbert’s inequality. A question is posed about the finite Hilbert transformation, of which an affirmative answer implies the weighted Hilbert inequality.

By RCA 0 , we denote a subsystem of second order arithmetic based on Δ0 1 comprehension and Δ0 1 induction. We show within this system that the real number system R satisfies all the theorems (possibly with non-standard length) of the theory of real closed fields under an appropriate truth definition. This enables us to develop linear algebra and polynomial ring theory over real and complex numbers, so that we particularly obtain Hilbert’s Nullstellensatz in RCA 0 .

A finite Hilbert transformation associated with a polynomial is the analogue of a Hilbert transformation associated with an entire function, which is a generalization of the classical Hilbert transformation (cf. [4]). Hilbert's inequality, which has applications in analytic number theory (see Vaaler [11]), is closely related to the finite Hilbert transformation associated with a polynomial. In this paper, an explicit construction is given for the finite Hilbert transformation.

The theory that ``consistency implies existence'' was put forward by Hilbert on various occasions around the start of the last century, and it was strongly and explicitly emphasized in his correspondence with Frege. Since (Godel's) completeness theorem, abstractly speaking, forms the basis of this theory, it has become common practice to assume that Hilbert took for granted the semantic completeness of second order logic. In this paper I maintain that this widely held view is untrue to the facts, and that the clue to explain what Hilbert meant by linking together consistency and existence is to be found in the role played by the completeness axiom within both geometrical and arithmetical axiom systems.

This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T21 and S22 can prove the well-foundedness on bounded domains of the ordinal notations below e0 and Γ0. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below e0 and Γ0 without increasing the class PLS.

A generalization of the Hilbert basis theorem in the geometric setting is proposed. It asserts that, for any well-describable (in a certain sense) family of polynomials, there exists a number C such that if P is an everywhere dense (in a certain sense) subfamily of this family, a is an arbitrary point, and the first C polynomials in any sequence from P vanish at the point a, then all polynomials from P vanish at a.

This paper presents an algorithm for computing numerical evidence for a conjecture whose validity is predicted by the requirement that the equivariant Tamagawa number conjectures for Tate motives as formulated by Burns and Flach are compatible with the functional equation of the Artin L-series. The algorithm includes methods for the computation of Fitting ideals and projective lattices over the integral group ring.

The shortest definition of a number by a first order formula with one free variable, where the notion of a formula defining a number extends a notion used by Boolos in a proof of the Incompleteness Theorem, is shown to be non computable. This is followed by an examination of the complexity of sets associated with this function.