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

[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

On communicating proofs in interactive mathematical documents

There is a wealth of interactive mathematics available on the web. Examples range from animated geometry to computing the nth digit in the expansion of p. However, proofs seem to remain static and at most they provide interaction in the form of links to definitions and other proofs. In this paper, we want to show how interactivity can be included in proofs themselves by making them executable, human-readable, and yet formal. The basic ingredients are formal proof-objects, OpenMathrelated languages, and the latest eXtensible Markup Language (XML) technology. We exhibit, by an example taken from a formal development in number theory, the final product of which we believe to be a truly interactive mathematical document