17 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
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
Type inference in mathematics
In the theory of programming languages, type inference is the process of
inferring the type of an expression automatically, often making use of
information from the context in which the expression appears. Such mechanisms
turn out to be extremely useful in the practice of interactive theorem proving,
whereby users interact with a computational proof assistant to construct formal
axiomatic derivations of mathematical theorems. This article explains some of
the mechanisms for type inference used by the Mathematical Components project,
which is working towards a verification of the Feit-Thompson theorem
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
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
Quantitative Continuity and Computable Analysis in Coq
We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the sense of computable analysis. The development is done in the proof assistant Coq and heavily relies on the Incone library for information theoretic continuity. This library is developed by one of the authors and the results of this paper extend the library. While full executability in a formal development of mathematical statements about real numbers and the like is not a feature that is unique to the Incone library, its original contribution is to adhere to the conventions of computable analysis to provide a general purpose interface for algorithmic reasoning on continuous structures. The paper includes a brief description of the most important concepts of Incone and its sub libraries mf and Metric.
The results that provide complete computational content include that the algebraic operations and the efficient limit operator on the reals are computable, that the countably infinite product of a space with itself is isomorphic to a space of functions, compatibility of the enumeration representation of subsets of natural numbers with the abstract definition of the space of open subsets of the natural numbers, and that continuous realizability implies sequential continuity. We also describe many non-computational results that support the correctness of definitions from the library. These include that the information theoretic notion of continuity used in the library is equivalent to the metric notion of continuity on Baire space, a complete comparison of the different concepts of continuity that arise from metric and represented space structures and the discontinuity of the unrestricted limit operator on the real numbers and the task of selecting an element of a closed subset of the natural numbers
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
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
A Mechanised Proof of Gödelâs Incompleteness Theorems Using Nominal Isabelle
An Isabelle/HOL formalisation of G\"odel's two incompleteness theorems is
presented. The work follows \'Swierczkowski's detailed proof of the theorems
using hereditarily finite (HF) set theory. Avoiding the usual arithmetical
encodings of syntax eliminates the necessity to formalise elementary number
theory within an embedded logical calculus. The Isabelle formalisation uses two
separate treatments of variable binding: the nominal package is shown to scale
to a development of this complexity, while de Bruijn indices turn out to be
ideal for coding syntax. Critical details of the Isabelle proof are described,
in particular gaps and errors found in the literature.Jesse Alama drew my attention to Swierczkowski, the source material for this ÂŽ
project. Christian Urban assisted with nominal aspects of some of the proofs, even
writing code. Brian Huffman provided the core formalisation of type hf. Dana Scott
offered advice and drew my attention to Kirby. Matt Kaufmann and the referees
made many insightful comments.This is the author accepted manuscript. The final version is available from Springer at http://link.springer.com/article/10.1007%2Fs10817-015-9322-