11,361 research outputs found

    Essential Incompleteness of Arithmetic Verified by Coq

    Get PDF
    A constructive proof of the Goedel-Rosser incompleteness theorem has been completed using the Coq proof assistant. Some theory of classical first-order logic over an arbitrary language is formalized. A development of primitive recursive functions is given, and all primitive recursive functions are proved to be representable in a weak axiom system. Formulas and proofs are encoded as natural numbers, and functions operating on these codes are proved to be primitive recursive. The weak axiom system is proved to be essentially incomplete. In particular, Peano arithmetic is proved to be consistent in Coq's type theory and therefore is incomplete.Comment: This paper is part of the proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2005). For the associated Coq source files see the TeX sources, or see <http://r6.ca/Goedel20050512.tar.gz

    Hilbert's Program Then and Now

    Get PDF
    Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.Comment: 43 page

    Perspectives for proof unwinding by programming languages techniques

    Get PDF
    In this chapter, we propose some future directions of work, potentially beneficial to Mathematics and its foundations, based on the recent import of methodology from the theory of programming languages into proof theory. This scientific essay, written for the audience of proof theorists as well as the working mathematician, is not a survey of the field, but rather a personal view of the author who hopes that it may inspire future and fellow researchers

    Consequences of a Goedel's misjudgment

    Full text link
    The fundamental aim of the paper is to correct an harmful way to interpret a Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the Goedel's fault is rather venial, its misreading has produced and continues to produce dangerous fruits, as to apply the incompleteness Theorems to the full second-order Arithmetic and to deduce the semantic incompleteness of its language by these same Theorems. The first three paragraphs are introductory and serve to define the languages inherently semantic and its properties, to discuss the consequences of the expression order used in a language and some question about the semantic completeness: in particular is highlighted the fact that a non-formal theory may be semantically complete despite using a language semantically incomplete. Finally, an alternative interpretation of the Goedel's unfortunate comment is proposed. KEYWORDS: semantic completeness, syntactic incompleteness, categoricity, arithmetic, second-order languages, paradoxesComment: English version, 19 pages. Fixed and improved terminolog
    • …
    corecore