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

Rethinking the off-grid city

This is the final version. Available on open access from Routledge via the DOI in this recordThere has been a resurgence in interest in the off-grid city, with a focus on off-grid urban spaces in the Global South, and on how the off-grid functions as a collection of places, lived spaces, and dynamic infrastructural configurations. As scholars and practitioners working in the off-grid urban context in South Africa, we contend it is necessary to question the assumptions of the “off-grid” concept in urban geography in terms of its implications for conducting research. We thereby identify four areas for further conceptual and empirical elaboration. The first area concerns the importance of continuing to redefine academic and practical understandings of the “grid”, ultimately moving to redefine its meaning in the city. The second is a need to decolonise and decentre the relationship between global and technocratic urban development “standards”, practices and discourses, and the granular off-grid context. The third area is the imperative of critically engaging with narratives of inadequacy and imperfection as often applied to off-grid, informal urban spaces. The fourth is the priority of moving towards a needs-based approach to off-grid development, with a focus on co-production of urban knowledge with local communities to ensure their needs and interests are met.British Academ

Beyond the Grid: The Micropolitics of Off-Grid Energy in Qandu-Qandu, South Africa

This is the final version. Available on open access from Wiley via the DOI in this recordIn this paper, we argue using smart technologies beyond the grid disrupts the access and use of existing energy sources, with profound impacts on everyday social life. We show how off-grid smart energy solutions constitute their own politics when considering existing conceptualisations of urban infrastructures in geography and the social sciences. To expose its politics, or “micropolitics”, we consider how tensions occur at the interface between infrastructures, where there are additions and modifications. We draw on an empirical example of Qandu-Qandu, an informal settlement in South Africa, to highlight how the placement, technical capabilities, and flexible financing options associated with off-grid solar energy create micropolitics with profound implications for everyday life. To conclude, we reflect on the value of using disruptions for understanding and enhancing equity in off-grid settings, contributing to the broader sustainability transitions narrative, and its “liveliness”

Smart and disruptive infrastructures: Re-building knowledge on the informal city

This is the final version. Available on open access from SAGE Publications via the DOI in this recordSmart urbanism is an established research area in geography and the social sciences. We draw on ‘worlding-provincialising’ strategies identified in an Urban Studies Special Issue from February 2021 to explore how smart infrastructures, a form of smart urbanism, disrupt representations of informality and urban development in new and productive ways. Focussing on the disruptive or troublesome implications of smart infrastructures reveals site-level considerations for developing policy and practice, where acknowledging the nuanced context for its use can present opportunities for not only understanding energy transitions in the Global South, but also creates opportunities for cross-learning. Drawing on our collective insights on a solar mini-grid project in Qandu-Qandu, Cape Town, we sketch out three ways the disruptive aspects of solar energy can be helpful for re-building knowledge on the informal city by: (i) re-positioning notions of ‘formal’ and ‘informal’ infrastructure(s) in urban planning and policymaking; (ii) highlighting new avenues for citizen autonomy; and (iii) recasting the informal city as a site for continuous innovation and learning.British AcademyUK Newton Fun

Debating the urban dimension of territorial cohesion

The Territorial Cohesion goal was only included in the EU Treaty by 2009, with a view to promote a more balanced and harmonious European territory. One year earlier (2008), the European Commission (EC) published the ‘Green Paper on Territorial Cohesion—Turning territorial diversity into strength’. Neither one, nor the other, clearly defines the meaning of the Territorial Cohesion concept. The later, however, proposes three main policy responses towards more balanced and harmonious development: (i) Concentration: overcoming differences in density; (ii) Connecting territories: overcoming distance; and (iii) Cooperation: overcoming division. Although not explicitly, this document identifies several ‘urban questions’ to be dealt when promoting territorial cohesive policies: avoiding diseconomies of very large agglomerations and urban sprawl processes, combating urban decay and social exclusion, avoiding excessive concentrations of growth, promoting access to integrated transport systems and creating metropolitan bodies. In this light, this chapter proposes to debate the importance of the urban dimension to achieve the goal of territorial cohesion at several territorial levels.info:eu-repo/semantics/acceptedVersio

Yield Function of the DOSimetry TELescope Count and Dose Rates Aboard the International Space Station

The Earth is constantly hit by energetic particles originating from galactic sources. The flux of these particles is altered by the magnetized solar wind in the heliosphere and the Earth's magnetic field. For this reason, the ability of a particle to approach a spacecraft in low Earth orbit depends on its energy and the position of the spacecraft within the Earth's magnetosphere. Moreover, there are some areas (radiation belts) where the particles are trapped for a long time and, therefore, the flux of energetic particles is particularly high. Occasionally, solar energetic particles contribute to the energetic particle flux too. DOSimetry TELescope (DOSTEL) is one of the instruments aboard the International Space Station (ISS) that monitors the radiation field within the European module Columbus. Because being installed inside the ISS, particles produced by the interaction between the “primary” radiation and the ISS materials are also measured. To describe the observations in such a complex radiation field, we follow the method by Caballero‐Lopez and Moraal (2012, Journal of Geophysical Research, 117[A12], A12103. doi:10.1029/2012JA017794) to compute the so‐called yield function using precise measurements of the proton and helium energy spectra obtained by alpha magnet spectrometer and the systematic variation of the DOSTEL measurements within the Earth's magnetosphere.Key Points: Energetic particles Influence of the Earth magnetosphere on the count‐ and dose rate aboard International Space Station (ISS) Yield functionBundesministerium Wirtschaf