Reflection using the derivability conditions

Reflection principles are a way to build non-conservative true extensions of a theory. However the application of a reflection principle needs a proof predicate, and the effort needed to provide this is so great as to be not really practical. We look at a possible way to avoid this effort by using, instead of a proof predicate, a predicate defined using only necessary `modal' properties. Surprisingly, we can produce powerful non-conservative extensions this way. But a reflection principle based on such a predicate is essentially weaker, and we also consider its limitations

Metalevel and reflexive extension in mechanical theorem proving

In spite of many years of research into mechanical assistance for mathematics it is still much more difficult to construct a proof on a machine than on paper. Of course this is partly because, unlike a proof on paper, a machine checked proof must be formal in the strictest sense of that word, but it is also because usually the ways of going about building proofs on a machine are limited compared to what a mathematician is used to. This thesis looks at some possible extensions to the range of tools available on a machine that might lend a user more flexibility in proving theorems, complementing whatever is already available.In particular, it examines what is possible in a framework theorem prover. Such a system, if it is configured to prove theorems in a particular logic T, must have a formal description of the proof theory of T written in the framework theory F of the system. So it should be possible to use whatever facilities are available in F not only to prove theorems of T, but also theorems about T that can then be used in their turn to aid the user in building theorems of T.The thesis is divided into three parts. The first describes the theory FS₀, which has been suggested by Feferman as a candidate for a framework theory suitable for doing meta-theory. The second describes some experiments with FS₀, proving meta-theorems. The third describes an experiment in extending the theory PRA, declared in FS₀, with a reflection facility.More precisely, in the second section three theories are formalised: propositional logic, sorted predicate logic, and the lambda calculus (with a deBruijn style binding). For the first two the deduction theorem and the prenex normal form theorem are respectively proven. For the third, a relational definition of beta-reduction is replaced with an explicit function.In the third section, a method is proposed for avoiding the work involved in building a full Godel style proof predicate for a theory. It is suggested that the language be extended with quotation and substitution facilities directly, instead of providing them as definitional extensions. With this, it is possible to exploit an observation of Solovay's that the Lob derivability conditions are sufficient to capture the schematic behaviour of a proof predicate. Combining this with a reflection schema is enough to produce a non-conservative extension of PRA, and this is demonstrated by some experiments

A Framework for Program Development Based on Schematic Proof

Often, calculi for manipulating and reasoning about programs can be recast as calculi for synthesizing programs. The difference involves often only a slight shift of perspective: admitting metavariables into proofs. We propose that such calculi should be implemented in logical frameworks that support this kind of proof construction and that such an implementation can unify program verification and synthesis. Our proposal is illustrated with a worked example developed in Paulson's Isabelle system. We also give examples of existent calculi that are closely related to the methodology we are proposing and others that can be profitably recast using our approach

T. S. Eliot, D. H. Lawrence, and the Structure of Feeling of Modernism

Eliot and Lawrence have long been considered the opposed critical, social and intellectual poles of Modernism. F. R. Leavis's indignant catalogue of Eliot's attacks on Lawrence established the orthodox position: Eliot was 'the essential opposition in person'. However, more recent research-drawing on the many newly accessible materials by and about Eliot-demonstrates that Eliot's relationship with Lawrence was far more complex, volatile and intriguing than this orthodoxy allows. Not only is the extent, intensity and acuity of Eliot's readings of Lawrence overlooked, so also is the pattern of intertextual echoes and references which mark his work, and the uncanny overlap of the two men's social circles. Bertrand Russell, John Middleton Murry, Richard Aldington and Aldous Huxley were close friends of both men, and they also shared a wide circle of acquaintances which included Ottoline Morrell, Katherine Mansfield, Brigid Patmore, Ezra Pound and others. This essay explores the ways in which Lawrence's example, his life and writing, perplexed and provoked Eliot, revealing new aspects of the emergent modernist structure of feeling

Testate amoebae as palaeohydrological proxies in Sürmene Aǧaçbaşi Yaylasi Peatland (Northeast Turkey)

Testate amoebae as palaeohydrological proxies in sürmene ağaçbaşi yaylasi peatland (northeast Turkey) Testate amoebae are unicellular micro-organisms whose hydrological sensitivity and good preservation in peats make them valuable proxies for past peatland surface wetness, and therefore climate. Previous testate amoebae transfer functions have been spatially restricted with no studies from Asia. To derive a transfer function, a sequence of samples was extracted from an ombrotrophic peatland in Turkey and amoebae counted. The internal structure of the data was explored using principal components analysis and relationships with the environmental data tested by redundancy analyses. Transfer function models were developed using a variety of techniques. As in other regions, depth to water table is the most important control on amoebae community composition. Transfer function performance was initially poor, primarily due to the inclusion of samples from areas of the site that had been heavily affected by peat cutting and had distinctly different amoebae communities. Model performance is improved by selective sample exclusion, reducing jack-knifed root mean square error of prediction to 7.1 cm. The model was tested using an initial palaeoecological data-set. Overlap with the training set was limited, although a hydrological reconstruction using this model produces similar results to a transfer function derived from northern European peatlands. This study provides the first testate amoebae transfer function from Asia and demonstrates that hydrological preferences of many of the key taxa are consistent across a large area of the Northern Hemisphere. The transfer function will allow detailed palaeoclimate reconstruction from this peatland, adding to our knowledge of Holocene climatic change in southwest Asia

ALMA Observations of the Largest Proto-Planetary Disk in the Orion Nebula, 114-426: A CO Silhouette

We present ALMA observations of the largest protoplanetary disk in the Orion Nebula, 114-426. Detectable 345 GHz (856 micron) dust continuum is produced only in the 350 AU central region of the ~1000 AU diameter silhouette seen against the bright H-alpha background in HST images. Assuming optically thin dust emission at 345 GHz, a gas-to-dust ratio of 100, and a grain temperature of 20 K, the disk gas-mass is estimated to be 3.1 +/- 0.6 Jupiter masses. If most solids and ices have have been incorporated into large grains, however, this value is a lower limit. The disk is not detected in dense-gas tracers such as HCO+ J=4-3, HCN J=4-3, or CS =7-6. These results may indicate that the 114-426 disk is evolved and depleted in some light organic compounds found in molecular clouds. The CO J=3-2 line is seen in absorption against the bright 50 to 80 K background of the Orion A molecular cloud over the full spatial extent and a little beyond the dust continuum emission. The CO absorption reaches a depth of 27 K below the background CO emission at VLSR ~6.7 km/s about 0.52 arcseconds (210 AU) northeast and 12 K below the background CO emission at VLSR ~ 9.7 km/s about 0.34 arcseconds (140 AU) southwest of the suspected location of the central star, implying that the embedded star has a mass less than 1 Solar mass .Comment: 20 pages, 4 figure

ALMA Observations of the Orion Proplyds

We present ALMA observations of protoplanetary disks ("proplyds") in the Orion Nebula Cluster. We imaged 5 individual fields at 856um containing 22 HST-identified proplyds and detected 21 of them. Eight of those disks were detected for the first time at submillimeter wavelengths, including the most prominent, well-known proplyd in the entire Orion Nebula, 114-426. Thermal dust emission in excess of any free-free component was measured in all but one of the detected disks, and ranged between 1-163 mJy, with resulting disk masses of 0.3-79 Mjup. An additional 26 stars with no prior evidence of associated disks in HST observations were also imaged within the 5 fields, but only 2 were detected. The disk mass upper limits for the undetected targets, which include OB stars, theta1Ori C and theta1Ori F, range from 0.1-0.6 Mjup. Combining these ALMA data with previous SMA observations, we find a lack of massive (>3 Mjup) disks in the extreme-UV dominated region of Orion, within 0.03 pc of O-star theta1Ori C. At larger separations from theta1Ori C, in the far-UV dominated region, there is a wide range of disk masses, similar to what is found in low-mass star forming regions. Taken together, these results suggest that a rapid dissipation of disk masses likely inhibits potential planet formation in the extreme-UV dominated regions of OB associations, but leaves disks in the far-UV dominated regions relatively unaffected.Comment: ApJ, in pres

First evidence of cryptotephra in palaeoenvironmental records associated with Norse occupation sites in Greenland

The Norse/Viking occupation of Greenland is part of a dispersal of communities across the North Atlantic coincident with the supposed Medieval Warm Period of the late 1st millennium AD. The abandonment of the Greenland settlements has been linked to climatic deterioration in the Little Ice Age as well as other possible explanations. There are significant dating uncertainties over the time of European abandonment of Greenland and the potential influence of climatic deterioration. Dating issues largely revolve around radiocarbon chronologies for Norse settlements and associated mire sequences close to settlement sites. Here we show the potential for moving this situation forward by a combination of palynological, radiocarbon and cryptotephra analyses of environmental records close to three ‘iconic’ Norse sites in the former Eastern Settlement of Greenland – Herjolfsnes, Hvalsey and Garðar (the modern Igaliku). While much work remains to be undertaken, our results show that palynological evidence can provide a useful marker for both the onset and end of Norse occupation in the region, while the radiocarbon chronologies for these sequences remain difficult. Significantly, we here demonstrate the potential for cryptotephra to become a useful tool in resolving the chronology of Norse occupation, when coupled with palynology. For the first time, we show that cryptotephra are present within palaeoenvironmental sequences located within or close to Norse settlement ruin-groups, with tephra horizons detected at all three sites. While shard concentrations were small at Herjolfsnes, concentrations sufficient for geochemical analyses were detected at Igaliku and Hvalsey. WDS-EPMA analyses of these tephra indicate that, unlike the predominantly Icelandic tephra sources reported in the Greenland ice core records, the tephra associated with the Norse sites correlate more closely with volcanic centres in the Aleutians and Cascades. Recent investigations of cryptotephra dispersal from North American centres, along with our new findings, point to the potential for cryptotephra to facilitate hypothesis testing, providing a key chronological tool for refining the timing of Norse activities in Greenland (e.g. abandonment) and of environmental contexts and drivers (e.g. climate forcing)

Genetic disorders of vitamin B12 metabolism: eight complementation groups – eight genes

Vitamin B12 (cobalamin, Cbl) is an essential nutrient in human metabolism. Genetic diseases of vitamin B12 utilisation constitute an important fraction of inherited newborn disease. Functionally, B12 is the cofactor for methionine synthase and methylmalonyl CoA mutase. To function as a cofactor, B12 must be metabolised through a complex pathway that modifies its structure and takes it through subcellular compartments of the cell. Through the study of inherited disorders of vitamin B12 utilisation, the genes for eight complementation groups have been identified, leading to the determination of the general structure of vitamin B12 processing and providing methods for carrier testing, prenatal diagnosis and approaches to treatment