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

An Investigation into the Effect of Hydrodynamic Cavitation on Diesel using Optical Extinction

A conventional diesel and paraffinic-rich model diesel fuel were subjected to sustained cavitation in a custom-built high-pressure recirculation flow rig. Changes to the spectral extinction coefficient at 405 nm were measured using a simple optical arrangement. The spectral extinction coefficient at 405 nm for the conventional diesel sample was observed to increase to a maximum value and then asymptotically decrease to a steady-state value, while that for the paraffinic-rich model diesel was observed to progressively decrease. It is suggested that this is caused by the sonochemical pyrolysis of mono-aromatics to form primary soot-like carbonaceous particles, which then coagulate to form larger particles, which are then trapped by the filter, leading to a steady-state spectral absorbance

A Survey on Retrieval of Mathematical Knowledge

We present a short survey of the literature on indexing and retrieval of mathematical knowledge, with pointers to 72 papers and tentative taxonomies of both retrieval problems and recurring techniques.Comment: CICM 2015, 20 page

On the Role of OpenMath in Interactive Mathematical Documents

AbstractThe standard OpenMath is an enabling technology for creating an integrated computer environment in which software packages for computer algebra and for proof checking can be combined. Here we demonstrate how OpenMath can be employed for generating interactive mathematical documents containing primality proofs. Our case study takes place within a browser; once a prime number is specified, a document appears summarizing the proof in a number of assertions. By clicking an assertion regarding the truth of an arithmetic equality, a computer algebra calculation is invoked verifying the equality. By clicking an assertion regarding a specific mathematical lemma called Pocklington’s Criterion, a verification of the corresponding formal proof is carried out by a proof checker. Moreover, the whole document is structured in such a way that it can be easily translated to a formal proof object. OpenMath supports the interaction between the document as it appears in the browser and the mathematical software packages. This paper begins with an introduction to OpenMath and a brief comparison with MathML

[Rheumatoid arthritis: echographic study of lesions of the periskeletal soft tissues].

The role of US was investigated in the study of rheumatoid arthritis, since the method depicts the changes in the periskeletal soft tissues--i.e., where the disorder preferably locates in both its early and late phases. A hundred and fifty-eight patients affected with rheumatoid arthritis according to American Rheumatism Association criteria were examined: the hand (wrist, carpus, metacarpus and fingers), the knee and the foot (metatarsus and toes) were studied in all patients. The study population was divided into two groups according to the time of onset of the disease: in 82 of them (52\%) the onset of symptoms dated back to less than a year, while 76 of them (48\%) had been suffering for over a year. US appears as the most accurate method to study the early phases of rheumatoid arthritis, for it makes early diagnosis possible, thus allowing the correct treatment to be chosen and preventing the disease from causing the irreversible lesions which progressively disable the patient. In the early phases of rheumatoid arthritis, US detects the exudative effects of synovial inflammation in periskeletal soft tissues. Joint effusions and synovial pannus are also depicted by US, as well as the thickening of tendon sheaths and tendon ruptures and rheumatoid nodules. In the late phases of rheumatoid arthritis, US supports conventional radiology, the latter remaining the irreplaceable method of choice to demonstrate skeletal lesions. Nonetheless, in such phases US yields further information on periarticular soft tissue involvement which no other method would make available--e.g., the presence of effusions, bulgings, synovial pannus, joint cartilage erosions, damaged tendons and sheaths, hypoplasia of the muscles ending on the involved joint and finally periarticular changes. Finally, US proves of great value in the early demonstration of reactivating phases, with unquestionable prognostic advantages

Integrating Computational and Deduction Systems Using OpenMath

AbstractThe standard OpenMath is a crucial ingredient for creating an integrated environment combining systems for computer algebra with proof checkers. OpenMath consists of a formal grammar of OpenMath objects, their encodings, Content Dictionaries, Phrasebooks and other tools. The OpenMath standard allows integration of computational systems of different kind. Here we demonstrate how OpenMath works by setting up an environment in which Maple expressions are type-checked by the proof checkers Lego and Coq