Completeness Of First-order Logic Research Articles

Visions of Henkin

Leon Henkin (1921–2006) was not only an extraordinary logician, but also an excellent teacher, a dedicated professor and an exceptional person. The first two sections of this paper are biographical, discussing both his personal and academic life. In the last section we present three aspects of Henkin’s work. First we comment part of his work fruit of his emphasis on teaching. In a personal communication he affirms that On mathematical induction, published in 1969, was the favourite among his articles with a somewhat panoramic nature and not meant exclusively to specialists. This subject is covered in the first subsection. Needless to say that we also analyse Henkin’s better known contribution: his completeness method. His renowned results on completeness for both type theory and first order logic were part of his thesis, The Completeness of Formal Systems, presented at Princeton in 1947 under the advise of Alonzo Church. It is interesting to note that he obtained the proof of completeness for first order logic readapting the argument for the theory of types. The last subsection is devoted to philosophy. The work most directly related to philosophy is an article entitled: Some Notes on Nominalism which appeared in the Journal of Symbolic Logic in 1953. Unfortunately, we are not covering his contribution to the field of cylindric algebras. As a matter of fact, Henkin spent many years investigating algebraic structures with Alfred Tarski and Donald Monk, among others.

Fictional Propositions and the Unprovability of Consistency

We introduce an epistemic version of validity and completeness of first order logic, based on the notions of and . We then show how the perspective here considered may help to solve an epistemic puzzle arising from Gödel's second incompleteness theorem.

Foundations of Mathematics in the Twentieth Century

1. INTRODUCTION AND EARLY DEVELOPMENTS. Logic and foundations are a domain of mathematics concerned with basic mathematical structures (in terms of which one can define all other mathematical structures), with the correctness and significance of mathematical reasoning, and with the effectiveness of mathematical computations. In the twentieth century, these areas have crystallized into three large chapters of mathematics: set theory, mathematical logic (including model theory), and computability theory, which are intertwined in significant ways. In this paper we describe the evolution and present state of each of them. In modern times the study of logic and foundations has attracted eminent mathematicians and philosophers such as Cantor, Cohen, Frege, Godel, Hilbert, Kleene, Martin, Russell, Solovay, Shelah, Skolem, Tarski, Turing, Zermelo, and others, and has given rise to a large body of knowledge. Although our paper is only a brief sketch of this development, we discuss essential results such as G6del's theorem on the completeness of first-order logic and his theorems on the incompleteness of most mathematical theories, some independence theorems in set theory, the role of axioms of existence of large cardinal numbers, Turing's work on computability, and some recent developments. There are still many interesting unsolved problems in logic and foundations. For example, logic does not explain what mathematics is. We know that mathematics has a very precise structure: axioms, definitions, theorems, proofs. Thus we know what is correct mathematics but not why the works of certain mathematicians delight us while others strike us as downright boring. Nor do foundations tell us how mathematicians construct proofs of their conjectures. Since we have no good theoretical model of the process for constructing proofs, we are far from having truly effective procedures for automatically proving theorems, although spectacular successes have been achieved in this area. We mention other unsolved problems at the end of this paper. We now give a brief sketch of the history of logic and foundations prior to 1900. The ancient Greeks asked: what are correct arguments? and: what are the real numbers? As partial answers they created the theory of syllogisms and a theory of commensurable and incommensurable magnitudes. These questions resurfaced in the 18th century, due to the development of analysis and to the lack of sufficiently clear concepts of sets, functions, continuity, convergence, etc. In a series of papers (1878-1897) Georg Cantor created set theory. In 1879 Gottlob Frege described a formal system of logic that explained precisely the logical structure of all mathematical proofs. In 1858 Richard Dedekind gave a definitive answer to the question: what are the real numbers? by defining them in terms of sets of rational numbers. He proved the axiom of continuity of the real line, an axiom accepted hitherto (beginning with the Greeks) without proof.

Tablas semánticas y metalógica (El caso de la lógica de segundo orden)

Tablas semánticas y metalógica (El caso de la lógica de segundo orden)

Minimal models of Heyting arithmetic

In this paper, we give a constructive nonstandard model of intuitionistic arithmetic (Heyting arithmetic). We present two axiomatisations of the model: one finitary and one infinitary variant. Using the model these axiomatisations are proven to be conservative over ordinary intuitionistic arithmetic. The definition of the model along with the proofs of its properties may be carried out within a constructive and predicative metatheory (such as Martin-Löf's type theory). This paper gives an illustration of the use of sheaf semantics to obtain effective proof-theoretic results.The axiomatisations of nonstandard intuitionistic arithmetic (to be calledHAIandHAIωrespectively) as well as their model are based on the construction in [5] of a sheaf model for arithmetic using a site of filters. In this paper we present a “minimal” version of this model, built instead on a suitable site of provable filter bases. The construction of this site can be viewed as an extension of the well-known construction of the classifying topos for a geometric theory which uses “syntactic sites”. (Such sites can in fact be used to prove semantical completeness of first order logic in a strictly constructive framework, see [6].)We should mention that for classical nonstandard arithmetics there are several nonconstructive methods of proving conservativity over arithmetic, e.g. the compactness theorem, Mac Dowell–Specker's theorem [3].

Strong conceptual completeness for first-order logic

Etude de la completude conceptuelle forte pour la logique du premier ordre. Resultats de reconstruction