Gödel mathematics versus Hilbert mathematics. I. The Gödel incompleteness (1931) statement: axiom or theorem?

The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering “theorem”, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for “Hilbert mathematics” accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel’s papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part

Relations between logic and mathematics in the work of Benjamin and Charles S. Peirce.

Charles Peirce (1839-1914) was one of the most important logicians of the nineteenth century. This thesis traces the development of his algebraic logic from his early papers, with especial attention paid to the mathematical aspects. There are three main sources to consider. 1) Benjamin Peirce (1809-1880), Charles's father and also a leading American mathematician of his day, was an inspiration. His memoir Linear Associative Algebra (1870) is summarised and for the first time the algebraic structures behind its 169 algebras are analysed in depth. 2) Peirce's early papers on algebraic logic from the late 1860s were largely an attempt to expand and adapt George Boole's calculus, using a part/whole theory of classes and algebraic analogies concerning symbols, operations and equations to produce a method of deducing consequences from premises. 3) One of Peirce's main achievements was his work on the theory of relations, following in the pioneering footsteps of Augustus De Morgan. By linking the theory of relations to his post-Boolean algebraic logic, he solved many of the limitations that beset Boole's calculus. Peirce's seminal paper `Description of a Notation for the Logic of Relatives' (1870) is analysed in detail, with a new interpretation suggested for his mysterious process of logical differentiation. Charles Peirce's later work up to the mid 1880s is then surveyed, both for its extended algebraic character and for its novel theory of quantification. The contributions of two of his students at the Johns Hopkins University, Oscar Mitchell and Christine Ladd-Franklin are traced, specifically with an analysis of their problem solving methods. The work of Peirce's successor Ernst Schröder is also reviewed, contrasting the differences and similarities between their logics. During the 1890s and later, Charles Peirce turned to a diagrammatic representation and extension of his algebraic logic. The basic concepts of this topological twist are introduced. Although Peirce's work in logic has been studied by previous scholars, this thesis stresses to a new extent the mathematical aspects of his logic - in particular the algebraic background and methods, not only of Peirce but also of several of his contemporaries

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..

Az adverbiumok mondattani és jelentéstani kérdései = The syntax and syntax-semantics interface of adverbial modification

A határozószók és a határozók alaktani, mondattani és funkcionális kérdéseit vizsgáltuk a generatív nyelvelmélet keretében, főként magyar anyag alapján. Olyan leírásra törekedtünk, melyből a különféle határozófajták mondattani viselkedése, hatóköre, valamint hangsúlyozása egyaránt következik. A különféle határozótípusok PP-ként való elemzésének lehetőségét bizonyítottuk. A határozók mondatbeli elhelyezése tekintetében a specifikálói pozíció (Cinque 1999) ellen és az adjunkciós elemzés (Ernst 2002) mellett érveltünk. Megmutattuk, hogy a határozók szórendjének levezetéséhez bal- és jobboldali adjunkció feltételezése egyaránt szükséges. A különféle határozófajták szórendi helyét mondattani, jelentéstani és prozódiai tényezők összjátékával magyaráztuk. A jelentéstani tényezők között pl. a határozók inkorporálhatóságát korlátozó típusmegszorítást, a negatív határozók kötelező fókuszálását előidéző skaláris megszorítást, egyes határozófajták és igefajták komplex eseményszerkezetének inkompatibilitását vizsgáltuk. Az ige mögötti határozók szórendjét befolyásoló prozódiai tényező például a növekvő összetevők törvénye. Megfigyeltük az intonációskifejezés- újraelemzés kiváltódásának feltételeit és jelentéstani következményeit is. A helyhatározói igekötők egy típusát a mozgatási láncok sajátos fonológiai megvalósulásaként (a fonológiailag redukált kópia inkorporációjaként) elemeztük. A tárgykörben mintegy 60 tanulmányt publikáltunk. Adverbs and Adverbial Adjuncts at the Interfaces (489 old.) c. könyvünket kiadja a Mouton de Gruyter (Berlin). | This project has aimed to clarify (on the basis of mainly Hungarian data) basic issues concerning the category "adverb", the function "adverbial", and the grammar of adverbial modification. We have argued for the PP analysis of adverbials, and have claimed that they enter the derivation via left- and right-adjunction. Their merge-in position is determined by the interplay of syntactic, semantic, and prosodic factors. The semantically motivated constraints discussed also include a type restriction affecting adverbials semantically incorporated into the verbal predicate, an obligatory focus position for scalar adverbs representing negative values of bidirectional scales, cooccurrence restrictions between verbs and adverbials involving incompatible subevents, etc. The order and interpretation of adverbials in the postverbal domain is shown to be affected by such phonologically motivated constraints as the Law of Growing Constituents, and by intonation-phrase restructuring. The shape of the light-headed chain arising in the course of locative PP incorporation is determined by morpho-phonological requirements. The types of adverbs and adverbials analyzed include locatives, temporals, comitatives, epistemic adverbs, adverbs of degree, manner, counting, and frequency, quantificational adverbs, and adverbial participles. We have published about 60 studies; our book Adverbs and Adverbial Adjuncts at the Interfaces (pp. 489) is published in the series Interface Explorations of Mouton de Gruyter, Berlin