237 research outputs found
Essential Incompleteness of Arithmetic Verified by Coq
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
Perspectives for proof unwinding by programming languages techniques
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
Synthetic Undecidability and Incompleteness of First-Order Axiom Systems in Coq
We mechanise the undecidability of various frst-order axiom systems in Coq, employing
the synthetic approach to computability underlying the growing Coq Library of Undecidability Proofs. Concretely, we cover both semantic and deductive entailment in fragments
of Peano arithmetic (PA) as well as ZF and related fnitary set theories, with their undecidability established by many-one reductions from solvability of Diophantine equations, i.e.
Hilbertâs tenth problem (H10), and the Post correspondence problem (PCP), respectively.
In the synthetic setting based on the computability of all functions defnable in a constructive foundation, such as Coqâs type theory, it sufces to defne these reductions as metalevel functions with no need for further encoding in a formalised model of computation.
The concrete cases of PA and the considered set theories are supplemented by a general
synthetic theory of undecidable axiomatisations, focusing on well-known connections to
consistency and incompleteness. Specifcally, our reductions rely on the existence of standard models, necessitating additional assumptions in the case of full ZF, and all axiomatic
extensions still justifed by such standard models are shown incomplete. As a by-product of
the undecidability of set theories formulated using only membership and no equality symbol, we obtain the undecidability of frst-order logic with a single binary relation
Mechanised metamathematics : an investigation of first-order logic and set theory in constructive type theory
In this thesis, we investigate several key results in the canon of metamathematics, applying the contemporary perspective of formalisation in constructive type theory and mechanisation in the Coq proof assistant. Concretely, we consider the central completeness, undecidability, and incompleteness theorems of first-order logic as well as properties of the axiom of choice and the continuum hypothesis in axiomatic set theory. Due to their fundamental role in the foundations of mathematics and their technical intricacies, these results have a long tradition in the codification as standard literature and, in more recent investigations, increasingly serve as a benchmark for computer mechanisation. With the present thesis, we continue this tradition by uniformly analysing the aforementioned cornerstones of metamathematics in the formal framework of constructive type theory. This programme offers novel insights into the constructive content of completeness, a synthetic approach to undecidability and incompleteness that largely eliminates the notorious tedium obscuring the essence of their proofs, as well as natural representations of set theory in the form of a second-order axiomatisation and of a fully type-theoretic account. The mechanisation concerning first-order logic is organised as a comprehensive Coq library open to usage and contribution by external users.In dieser Doktorarbeit werden einige SchlĂŒsselergebnisse aus dem Kanon der Metamathematik untersucht, unter Verwendung der zeitgenössischen Perspektive von Formalisierung in konstruktiver Typtheorie und Mechanisierung mit Hilfe des Beweisassistenten Coq. Konkret werden die zentralen VollstĂ€ndigkeits-, Unentscheidbarkeits- und UnvollstĂ€ndigkeitsergebnisse der Logik erster Ordnung sowie Eigenschaften des Auswahlaxioms und der Kontinuumshypothese in axiomatischer Mengenlehre betrachtet. Aufgrund ihrer fundamentalen Rolle in der Fundierung der Mathematik und ihrer technischen Schwierigkeiten, besitzen diese Ergebnisse eine lange Tradition der Kodifizierung als Standardliteratur und, besonders in jĂŒngeren Untersuchungen, eine zunehmende Bedeutung als MaĂstab fĂŒr Mechanisierung mit Computern. Mit der vorliegenden Doktorarbeit wird diese Tradition fortgefĂŒhrt, indem die zuvorgenannten Grundpfeiler der Methamatematik uniform im formalen Rahmen der konstruktiven Typtheorie analysiert werden. Dieses Programm ermöglicht neue Einsichten in den konstruktiven Gehalt von VollstĂ€ndigkeit, einen synthetischen Ansatz fĂŒr Unentscheidbarkeit und UnvollstĂ€ndigkeit, der groĂteils den berĂŒchtigten, die Essenz der Beweise verdeckenden, technischen Aufwand eliminiert, sowie natĂŒrliche ReprĂ€sentationen von Mengentheorie in Form einer Axiomatisierung zweiter Ordnung und einer vollkommen typtheoretischen Darstellung. Die Mechanisierung zur Logik erster Ordnung ist als eine umfassende Coq-Bibliothek organisiert, die offen fĂŒr Nutzung und BeitrĂ€ge externer Anwender ist
Type classes for efficient exact real arithmetic in Coq
Floating point operations are fast, but require continuous effort on the part
of the user in order to ensure that the results are correct. This burden can be
shifted away from the user by providing a library of exact analysis in which
the computer handles the error estimates. Previously, we [Krebbers/Spitters
2011] provided a fast implementation of the exact real numbers in the Coq proof
assistant. Our implementation improved on an earlier implementation by O'Connor
by using type classes to describe an abstract specification of the underlying
dense set from which the real numbers are built. In particular, we used dyadic
rationals built from Coq's machine integers to obtain a 100 times speed up of
the basic operations already. This article is a substantially expanded version
of [Krebbers/Spitters 2011] in which the implementation is extended in the
various ways. First, we implement and verify the sine and cosine function.
Secondly, we create an additional implementation of the dense set based on
Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order
on undecidable structures, while it was limited to decidable structures before.
This hierarchy, based on type classes, allows us to share theory on the
naturals, integers, rationals, dyadics, and reals in a convenient way. Finally,
we obtain another dramatic speed-up by avoiding evaluation of termination
proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275
An Analysis of Tennenbaum's Theorem in Constructive Type Theory
Tennenbaum's theorem states that the only countable model of Peano arithmetic
(PA) with computable arithmetical operations is the standard model of natural
numbers. In this paper, we use constructive type theory as a framework to
revisit, analyze and generalize this result. The chosen framework allows for a
synthetic approach to computability theory, exploiting that, externally, all
functions definable in constructive type theory can be shown computable. We
then build on this viewpoint and furthermore internalize it by assuming a
version of Church's thesis, which expresses that any function on natural
numbers is representable by a formula in PA. This assumption provides for a
conveniently abstract setup to carry out rigorous computability arguments, even
in the theorem's mechanization. Concretely, we constructivize several classical
proofs and present one inherently constructive rendering of Tennenbaum's
theorem, all following arguments from the literature. Concerning the classical
proofs in particular, the constructive setting allows us to highlight
differences in their assumptions and conclusions which are not visible
classically. All versions are accompanied by a unified mechanization in the Coq
proof assistant.Comment: 23 pages, extension of conference paper published at FSCD 202
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A MACHINE-ASSISTED PROOF OF GĂDEL'S INCOMPLETENESS THEOREMS FOR THE THEORY OF HEREDITARILY FINITE SETS
A formalisation of G\"odel's incompleteness theorems using the Isabelle proof
assistant is described. This is apparently the first mechanical verification of
the second incompleteness theorem. The work closely follows {\'S}wierczkowski
(2003), who gave a detailed proof using hereditarily finite set theory. The
adoption of this theory is generally beneficial, but it poses certain technical
issues that do not arise for Peano arithmetic. The formalisation itself should
be useful to logicians, particularly concerning the second incompleteness
theorem, where existing proofs are lacking in detail.This is the author accepted manuscript. The final version is available from Cambridge University Press via https://doi.org/10.1017/S175502031400011
A formally verified abstract account of Gödel's incompleteness theorems
We present an abstract development of Gödelâs incompleteness theorems, performed with the help of the Isabelle/HOL theorem prover. We analyze sufficient conditions for the theoremsâ applicability to a partially specified logic. In addition to the usual benefits of generality, our abstract perspective enables a comparison between alternative approaches from the literature. These include Rosserâs variation of the first theorem, Jeroslowâs variation of the second theorem, and the S ÌwierczkowskiâPaulson semantics-based approach. As part of our frameworkâs validation, we upgrade Paulsonâs Isabelle proof to produce a mech- anization of the second theorem that does not assume soundness in the standard model, and in fact does not rely on any notion of model or semantic interpretation
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