Are There Enough Injective Sets?

The axiom of choice ensures precisely that, in ZFC, every set is projective: that is, a projective object in the category of sets. In constructive ZF (CZF) the existence of enough projective sets has been discussed as an additional axiom taken from the interpretation of CZF in Martin-Löf’s intuitionistic type theory. On the other hand, every non-empty set is injective in classical ZF, which argument fails to work in CZF. The aim of this paper is to shed some light on the problem whether there are (enough) injective sets in CZF. We show that no two element set is injective unless the law of excluded middle is admitted for negated formulas, and that the axiom of power set is required for proving that “there are strongly enough injective sets”. The latter notion is abstracted from the singleton embedding into the power set, which ensures enough injectives both in every topos and in IZF. We further show that it is consistent with CZF to assume that the only injective sets are the singletons. In particular, assuming the consistency of CZF one cannot prove in CZF that there are enough injective sets. As a complement we revisit the duality between injective and projective sets from the point of view of intuitionistic type theory.     

Proleptic Creation1

Proleptic theology endeavours for a convergence of theology and modern science by proclaiming that the world we discover is a world where creation is still going on. Ted Peters, the creator of the idea, suggests that God creates from the future by drawing contingent beings into a harmonious whole. The future kingdom reveals itself through prolepsis, especially in the Christ event. Peters strives to combine theology and natural science in terms of hypothetical consonance, i.e. the idea that they both study the same reality and are supposed to be in consonance, if not now, later on in the future. In spite of certain unclear points and internal tensions, Peters’ idea about proleptic creation is one of the most up-to-date versions of the doctrine of creation.     

Mathematical symbols as epistemic actions

Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used to express mathematical concepts??they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition.     

Busting a Myth about Leśniewski and Definitions

A theory of definitions which places the eliminability and conservativeness requirements on definitions is usually called the standard theory. We examine a persistent myth which credits this theory to Le?niewski, a Polish logician. After a brief survey of its origins, we show that the myth is highly dubious. First, no place in Le?niewski's published or unpublished work is known where the standard conditions are discussed. Second, Le?niewski's own logical theories allow for creative definitions. Third, Le?niewski's celebrated ‘rules of definition’ lay merely syntactical restrictions on the form of definitions: they do not provide definitions with such meta-theoretical requirements as eliminability or conservativeness. On the positive side, we point out that among the Polish logicians, in the 1920s and 1930s, a study of these meta-theoretical conditions is more readily found in the works of ?ukasiewicz and Ajdukiewicz.     

Are fear-avoidance beliefs related to the inception of an episode of back pain? A prospective study

Abstract

Fear-avoidance beliefs and catastrophizing have been implicated in chronic pain and theoretical models have been developed that feature these factor in the transition from acute to chronic pain. However, little has been done to determine whether these factors occur in the general population or whether they arc associated with the inception of an episode of neck or back pain. The aim of this study was to evaluate prospectively the effects of fear-avoidance beliefs and catastrophizing on the development of an episode of self-reported pain and associated physical functioning. To achieve this, we selected a sample of 415 people from the general population who reported no spinal pain during the past year. At the pretest a battery of questionnaires was administered to assess beliefs about pain and activity and it featured the Pain Catastrophizing Scale and a modified version of the Fear-Avoidance Beliefs Questionnaire. One year later outcome was evaluated by self-reports of the occurrence of a pain episode as well as a self-administered physical function test. The results showed that scores on both fear-avoidme and cabstrophizing were quite low. During the one year follow-up, 19% of the sample suffered an episode of back pain. Those with scores above the median on fear-avoidance beliefs at the pretest had twice the risk of suffering an episode of back pain and a 1.7 times higher risk of lowered physical function at the follow-up. Catastrophizing was somewhat less salient, increasing the risk of pain or lowered function by 1.5. but with confidence intervals falling below unity. These data indicate that fear-avoidance beliefs may be involved at a very early pint in the development of pain and associated activity problems in people with back pain. Theoretically. our results support the idea that fear-avoidance beliefs may develop in an interaction with the experience of pain. Clinically, the results suggest that catastrophizing and particularly fear-avoidance beliefs are important in the development of a pain problem and might be of use in screening procedures.     

Response of the pituitary-adrenal axis in terms of type a behaviour,hostility and vital exhaustion in healthy middle-aged men

Abstract

Pituitary-adrenal axis was studied in terms of Type A behaviour, hostility and vital exhaustion among 69 healthy middle-aged men. The results showed that psychological factors could explain a significant proportion of the biologically manipulated responses of HPA axis, but they worked in different ways. Type A behaviour was related to a high level of mean basal ACTH and a low level of cortisol response to ACTH stimulation after dexamethasone suppression; hostility was related to a high level of mean basal cortisol and a high cortisol in cortisol/ACTH ratio, while vital exhaustion was characterized by a low level of mean basal ACTH and a decreased ACTH in relation to cortisol. The adrenocortical patterns, i.e. a high ACTH-low cortisol; a high cortisol; and a low ACTH-low mean basal cortisol, as related to Type A behaviour, hostility and exhaustion, respectively, are in line with the traditional physiological stress model and suggest that different adrenocortical responses might be able to identify different mental stress processes. Sense of control has been suggested to be a key concept for psychological understanding of this finding.     

Demographic,health-related and psychosocial predictors of changes in depressive symptoms and anxiety in late middle-aged and older persons with fall-related injuries

Research on anxiety and depression after fall-related injuries to the extremities and its pre-fall determinants is scarce. The present study sought to examine change between pre-fall and post-fall anxiety and depression and its (pre-fall) predictors. Data were obtained from 181 Dutch persons, aged 57 and older, who sustained fall-related injuries between 1993 and 1997. Demographic (e.g. socio-economic status), health-related (e.g. number of chronic conditions), and psychosocial (e.g. social support, neuroticism) factors were determined in 1993 (prior to the fall). Depressive symptoms and anxiety were determined prior to the fall, and two, five and twelve months after the fall. Taking into account pre-fall anxiety and depression, pre-fall neuroticism was the most consistent, independent predictor of adverse changes in anxiety and depression. Other factors, including severity of the fall and age were not related to changes in anxiety and depression. Our prospective findings may be interpreted as support for the importance of personality for mental health outcomes in persons who have sustained fall-related injuries.     

A constitutive model for quasicrystal plasticity

High-pressure X-ray diffraction studies on gadolinium sesquioxide (Gd₂O₃) have been carried out up to a pressure of ～25 GPa in a diamond-anvil cell at room temperature. Gadolinium oxide, which has a cubic or bixbyite structure under ambient conditions, undergoes an irreversible structural phase at around 12 GPa. The high-pressure phase has been identified as a hexagonal La₂O₃-type structure. The bulk modulus and its pressure derivative of this phase have been calculated.