The Prospects for Mathematical Logic in the Twenty-First Century

Abstract

Abstract. The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.We can only see a short distance ahead, but we can see plenty there that needs to be done.A. Turing, 1950. §1. Introduction. The annual meeting of the Association for Symbolic Logic held in Urbana-Champaign, June 2000, included a panel discussion on "The Prospects for Mathematical Logic in the Twenty-First Century". The panel discussions included independent presentations by the four panel members, followed by approximately one hour of lively discussion with members of the audience.The main themes of the discussions concerned the directions mathematical logic should or could pursue in the future. Some members of the audience strongly felt that logic needs to find more applications to mathematics; however, there was disagreement as to what kinds of applications were most likely to be possible and important. Many people also felt that applications to computer science will be of great importance. On the other hand, quite a few people, while acknowledging the importance of applications of logic, felt that the most important progress in logic comes from internal developments.It seems safe to presume that the future of mathematical logic will include a multitude of directions and a blend of these various elements. Indeed, it speaks well for the strength of the field that there are multiple compelling directions for future progress. It is to be hoped that logic will be driven both by internal developments and by external applications, Supported in part by NSF grant DMS-9803515.

Syntax Completeness and Its Properties on Formal Systems

“…In various formal systems, the classical first order logic is most important, because other first order logic systems are consistent extensions of the classical first order logic system [3]. By Theorem 2 and Theorem 4, we have the following corollary 1.…”

“…A k is not a valid formula. Thus, A k is not a theorem of L because the first order logical system has soundness [3], [4], [5]. Now, we add A k as a new axiom (or axiom scheme) to obtain L * from L.…”

Differentiation and Processing of Contradictory Knowledge and Opposite Knowledge

“…Therefore Because of the medium predicate logic MF has abundant results for the formal relation and formal inference, and its the completeness, soundness and consistence had been proven [9], [10], [11], so, the all formal relation and the laws of formal inference on contradictory predicates and opposite predicates had been reflected in MF. We omit them in this paper because limit length, the detailed results see References [8].…”

“…We can transform the contradictory concepts and opposite concepts into data using truth-valued of concept. For the three-valued model [9], [10] and the infinite-valued mode Ф of the medium predicate logic MF [11], we consider that the infinitevalued model Ф is optimum on description of contradictory concepts and opposite concepts.…”

Specifying confidentiality

“…An article derived from the workshop appeared in the Bulletin of Symbolic Logic [Halpern et al 2001], and it is an exceedingly good read. If you get a copy of that issue of the Bulletin, make sure to also have a look at the article by Buss et al [2001], which discusses the current state of mathematical logic.…”