In the study of the arithmetic degrees the $\omega \text {-REA}$ sets play a role analogous to the role the r.e. degrees play in the study of the Turing degrees. However, much less is known about the arithmetic degrees and the role of the $\omega \text {-REA}$ sets in that structure than about the Turing degrees. Indeed, even basic questions such as the existence of an $\omega \text {-REA}$ set of minimal arithmetic degree are open. This paper makes progress on this question by demonstrating that some promising approaches inspired by the analogy with the r.e. sets fail to show that no $\omega \text {-REA}$ set is arithmetically minimal. Finally, it constructs a $\prod ^0_{2}$ singleton of minimal arithmetic degree. Not only is this a result of considerable interest in its own right, constructions of $\prod ^0_{2}$ singletons often pave the way for constructions of $\omega \text {-REA}$ sets with similar properties. Along the way, a number of interesting results relating arithmetic reducibility and rates of growth are established.

Whether the decline of birth weight (BW) reported in developed countries in the early 2000s is ongoing remains unknown. Furthermore, despite recent sharp increases in twin births, comparing secular trends of BW between singletons and twins is difficult, as studies have rarely examined secular trends of BW in twins and singletons simultaneously. Therefore, this study aimed to investigate the most recent 20-year trends (2000–2020) of BW in twins and singletons in South Korea. Annual natality files from 2000 to 2020 obtained from the Korean Statistical Information Service were analyzed. A yearly decrease of BW was 3 g among singletons and 5 to 6 g in twins from 2000 to 2020, indicating a widening gap of BW between twins and singletons with increasing years. Gestational age (GA) also decreased in twins and singletons with yearly decreases of 0.28 days in singletons and 0.41 days in twins. Whereas BW decreased in term (GA ≥ 37 weeks), and very preterm groups (28 weeks ≤ GA < 32 weeks) from 2000 to 2020 in twins and singletons, it increased in moderate to late preterm (32 weeks ≤ GA < 37 weeks) groups, indicating a non-linear relationship between BW and GA. The prevalence of macrosomia (BW > 4000 g) in singletons decreased from 2000 to 2020, whereas low birth weight (LBW; BW < 2500 g) increased in twins and singletons. LBW is associated with adverse health outcomes. Effective public health strategies aiming at reduction in the incidence of LBW in the population should be developed.

There are no studies and only limited data that compare the difference in mortality between twins and singletons in the Arab world. We studied the survival of 306,966 children, including 9,280 twins, over the period 1970–2013 in six Arab countries (Algeria, Egypt, Iraq, Mauritania, Sudan and Tunisia) based on the Multiple Indicator Cluster Survey (MICS) database. With the use of relative survival models, we estimated the mortality of twins relative to singletons by including socioeconomic and demographic variables. This study confirms the results of previous studies on the excess risk of death of twins compared to singletons. There is evidence that excess mortality decreases with follow-up; in addition, male twins have a higher risk of death compared to females for all countries except Tunisia. Wealth index and education levels of women are factors that influence the risk of mortality. It is recommended that these findings are considered when performing future health and population strategies in these Arab countries.

Almost all set theorists pay at least lip service to Cantor’s definition of a set as a collection of many things into one whole; but empty and singleton sets do not fit with it. Adapting Dana Scott’s axiomatization of the cumulative theory of types, we present a ‘Cantorian’ system which excludes these anomalous sets. We investigate the consequences of their omission, examining their claim to a place on grounds of convenience, and asking whether their absence is an obstacle to the theory’s ability to represent ordered pairs or to support the arithmetization of analysis or the development of the theory of cardinals and ordinals.

More than 200 million children aged <5 years fail to reach their full cognitive potential, and children born as twins are particularly at risk. In this article, we review studies that examined differences in the neurodevelopmental outcomes of twins compared to singletons. We searched the Medline database for articles on twins, singletons, neuro, and cognitive development. We also inspected bibliographies of relevant publications to identify related articles from 2011 to 2017. Our search criteria yielded 162 studies, 8 of which met the inclusion criteria. Of the eight studies examined, four were prospective follow-up studies, three were cross-sectional studies, and one was a randomized controlled trial. Five of these studies were carried out in developed countries, and they found no statistically significant difference in neurodevelopmental outcomes among twins and singletons. However, two of the three studies carried out in developing countries found a difference with singletons having significantly higher academic ratings than twins. Studies in which neurodevelopmental outcomes were measured early in life (1–5 years) showed no significant twin–singleton differences, while those in which it was measured later in life showed mixed twin–singleton differences. Overall, these studies may have been underpowered and may not have been optimally designed and implemented. There is need for studies with adequate sample sizes, good design, and optimal measurement of all relevant covariates in order to resolve the conflicting reports in the literature.

To date, several studies have investigated the intergenerational effect of preterm and small-for-gestational-age (SGA) births. However, most studies excluded both twin mothers and twin offspring from the analyses. Thus, the objective of this study was to investigate the intergenerational effect of preterm birth and SGA births among twins and singletons. A prospective population-based register study of mother–firstborn offspring pairs recorded in the Swedish Medical Birth Register was performed. The study included 4,073 twins and 264,794 singletons born from 1973–1983 and their firstborns born from 1986–2009. Preterm birth was defined as birth at <37 weeks of gestation, and SGA as birth weight <2 standard deviations of the Swedish standard. Logistic regressions were performed to estimate the intergenerational effect of each birth characteristic. Adjustments were made for maternal grandmothers’ and mothers’ socio-demographic factors, in addition to maternal birth characteristics. Among mothers born as singletons, being born preterm was associated with an increased risk of delivering a preterm child (adjusted odds ratio (OR) 1.39, 95% Confidence Interval (CI) = 1.29–1.50), while being born SGA increased the likelihood of having an SGA child (adjusted OR 3.04, 95% CI = 2.80–3.30) as well as a preterm child (adjusted OR 1.30, 95% CI = 1.20–1.40). In twin mothers, the corresponding ORs tended to be lower, and the only statistically significant association was between an SGA mother and an SGA child (adjusted OR 2.15, 95% CI = 1.40–3.31). A statistically significant interaction between twinning and mother's size for gestational age was identified in a multivariate linear regression analysis, indicating that singleton mothers born SGA were associated with a lower birth weight compared with mothers not born SGA. Preterm birth and SGA appear to be transferred from one generation to the next, although not always reaching statistical significance. These effects seem to be less evident in mothers born as twins compared with those born as singletons.

Recent studies comparing cognitive abilities between contemporary twins and singletons in developed countries have suggested that twin deficits in cognitive abilities no longer exist. We examined cognitive abilities in a sample of twins and singletons born recently in Nigeria to determine whether recent findings can be replicated in developing countries. Our sample consisted of 413 pairs of twins and 280 singletons collected from over 45 public schools in Abuja and its neighboring states in Nigeria. The ages of twins and singletons ranged from 9 to 20 years with a mean (SD) of 14.6 years (2.2 years) for twins and 16.1 years (1.8 years) for singletons. Zygosity of the same-sex twins was determined by analysis of 16 deoxyribonucleic acid markers. We asked participants to complete a questionnaire booklet that included Standard Progressive Matrices-Plus Version (SPM+), Mill-Hill Vocabulary Scale (MHV), Family Assets Questionnaire, and demographic questions. The data were corrected for sex and age and then analyzed using maximum likelihood model-fitting analysis. Although twins and singletons were comparable in family social class indicators, singletons did better than twins across all the tests (d = 0.10 to 0.35). The average of d for SPM+ total [0.32; equivalent to 4.8 Intelligence Quotient (IQ) points] and d for MHV (0.24; equivalent to 3.6 IQ points) was 0.28 (equivalent to 4.2 IQ points), similar to the twin–singleton gap found in old cohorts in developed countries. We speculate that malnutrition, poor health, and educational systems in Nigeria may explain the persistence of twin deficits in cognitive abilities found in our sample.

Children born with non-optimal birth characteristics — that is, are small for gestational age and/or preterm — have an increased risk for several long-term effects such as neurological sequelae and chronic disease. The purpose of this study was to examine whether twins exhibited a different outcome, compared with singletons, in terms of hospitalization during adolescence and early adulthood, and to what extent differences remain when considering the divergence in birth characteristics between singletons and twins. Persons born between 1973 and 1983 in Sweden and surviving until age 13 were included and followed until the end of 2006. Data on birth characteristics, parental socio-demographic factors, and hospitalizations were collected from national registers. Adjusting for parental socio-demographic factors, twins had a higher risk of being hospitalized than singletons (odds ratio, OR = 1.17, 95% confidence interval, CI = 1.10–1.25) and more often due to ‘Congenital anomalies’ (OR = 1.18, 95% CI = 1.06–1.28), ‘Infections’ (OR = 1.14; 95% CI = 1.08–1.20), ‘External causes of illness’ (OR = 1.10, 95% CI = 1.06–1.15), and ‘Diseases of the nervous system’ (OR = 1.18, 95% CI = 1.10–1.26). Stratifying for birth characteristics, this difference diminishes, and for some diagnoses non-optimal twins seem to do slightly better than non-optimal singletons. Thus, twins with non-optimal birth characteristics had a lower risk of hospitalization than non-optimal singletons on, for example, ‘Congenital anomalies’ and ‘Diseases of the nervous system’ (OR = 0.86, 95% CI = 0.77–0.96; OR = 0.88, 95% CI = 0.81–0.97, respectively) and Total (any) hospitalization (OR = 0.87, 95% CI = 0.83–0.92). Among those with optimal birth characteristics, twins had an increased hospitalization due to ‘External causes of illness’ (OR = 1.07, 95% CI = 1.02–1.13) compared with optimal singletons. Twins have higher hospitalization rates than singletons. In stratifying for birth characteristics, this difference diminishes, and for some diagnoses, non-optimal twins seem to do less poorly than non-optimal singletons.

Parental adjustment following the death of a premature singleton or multiple birth infant has hitherto been studied by mailed questionnaires or telephone survey. In the present study, using an in-depth personal interview, grief reactions and adjustment patterns of nine families who lost a singleton premature infant (‘Single Group’) were compared with those of nine families who lost one of a premature multiple birth cohort (‘Multiple Group’). The interview was conducted 1–4 years after the death of the infant and evaluated specific areas or ‘scales’ of life adjustment, including individual feelings, relationship between husband and wife, and functioning at home and at work. There was no significant difference between the paternal and maternal level of adjustment of the two groups in any of the studied scales. A positive correlation was found between maternal and paternal grief reaction of the same family in the scales of individual feelings (r = 0.65), relationships between husband and wife (r = 0.70), and functioning at home (r = 0.57). Comparing the father's scale with the mother's scale revealed a significant difference only in the area of ‘individual feelings’. The gestational age, maternal bonding during hospitalisation of the infant and the parental attendance at the event of death were significantly associated with the process of parental adjustment. The results of this study support previous reports of similar parental reactions following the demise of a premature singleton or multiple birth infant. Since societal environment may not recognise the need for consolation of these families, care, compassion, and sensitivity should be encouraged in dealing with these parents at the time of their infant' death, and for a long time thereafter.

We analyzed whether motor development in early life is different between singletons and triplets in Japan. The motor development was reported by mothers by postal questionnaire for 1,121 triplet children and in regular health check-ups for 13,906 singleton children. Children who were suspected of having neurological abnormality or disability were excluded from the analysis. The ages of milestone achievements were significantly higher in triplets for each outcome compared to singletons. Further, after adjustment for gestational age, birthweight, and birth length, the differences were significant for maintaining head, sitting alone and standing holding on. In children with birthweight of 2 kg or more, the ages of milestone achievements were significantly higher in triplets for each outcome compared to singletons, except walking holding on. Moreover, after adjustment for the confounding factors, the differences were significant for sitting alone and walking independently. On the contrary, singletons attained motor development facilitating crawling, walking holding on, and walking independently slower than triplets among those children with birthweight of 2 kg or less after adjustment for gestational age. In conclusion, triplets are overall at higher risk for the delay of gross motor milestones as compared to singletons independently of birth-related factors. In contrast, among children with a birthweight of less than 2 kg, singletons showed slower motor development than triplets after adjusting for gestational age. There is an obvious need to apply developmental standards that consider at least both multiple birth status (singleton, twin or triplet) and birthweight.

We attempted to define parental anxiety in a population of parents of high-risk premature twins (mean birth weight 1.493 ± 227 kg; mean gestational age 33 ± 3.5 weeks), admitted to III level NICU. We specifically examined the following factors: gestational age of the twins, whether or not the twins had ventilatory support, pulmonary sequelae, major malformations or intra-ventricular hemorrhage, parental gender and highest level of education obtained by the parent. In the immediate pre-discharge period and a month later, a questionnaire (State-Trait Anxiety Inventory) was given to all parents of premature twins presenting for the discharge. The parents of 30 twins entered the study twice, at the discharge of their first twin (mean postnatal age 40 ± 32 days), and one month later. They included 15 mothers and 11 fathers, aged 33 ± 5.5 and 33 ± 4.2 years, and at the second evaluation 11 mothers and 10 fathers, respectively. As case-controls we examined parental anxiety of fifteen consecutive singleton high-risk prematures, with equal gestational age, discharged immediately after. Our results indicate that the parents of high-risk twin and singleton prematures present an elevated, lasting state-trait anxiety level. Pre- and post-discharge parental anxiety is more elevated (not significant) in twinning with respect to the prematurity alone. When assessed separately by parental gender, in both these groups an increased (not significant) anxiety was persistently found in the mothers. We recommend that, although neonatologists generally define the discharge of the high-risk premature based upon the acquired stabilization of vital parameters, they pay special attention to the twin group we have identified which is at increased risk for predischarge parental anxiety.

Twin gestation is associated with higher rate of neonatal hypoglycemia than do singletons. We examined the role of specific risk factors associated with neonatal hypoglycemia of 216 premature twins and 1284 premature singletons, consecutively born in the years 1994-1996 in the Department of Pediatrics of Padua University, Italy. Significantly higher risk of hypoglycemia (Dextrostix <40 and <20 mg%) was found in twins vs singletons (54% vs 32%, OR 2.49, CI 1.77-3.56; 19% vs 8%, OR 2.65, CI 1.59-4.19, respectively). Gestational age of 34-37 weeks increased hypoglycemia risk for the premature twins (77% vs 51%, OR 3.20 CI 1.49-6.88). Twin deliveries statistically differed from those of singletons in several perinatal characteristics. More twins were born by cesarean section (85% vs 55%, OR 4.15, CI 2.48-6.95), and the birth weight of twins was much lower related to prematures with BW < 1.0 kg (12% vs 6%, OR 2.06, CI 1.11-3.82) and SGA (20% vs 10%, OR 2.41, CI 1.46-3.98). The risk of twin deliveries was increased at 30-33 weeks gestational age (25% vs 15%, OR 1.84; CI 1.17-2.90). Twins were found to have higher rates of hospitalization (50% vs 40%, OR 1.52, CI 1.04-2.23) and showed an increased risk of cardiorespiratory resuscitation (51% vs 31%, OR 2.36, CI 1.61-3.47), hypothermia (11% vs 4%, OR 3.02, CI 2.333.91), BPD (25% vs 19%, OR 2.55, CI 1.10-5.91), and PVL (4% vs 1%, OR 4.08, CI 1.23-13.5). Mortality was found more often (not significant) in premature twins. The risk for intrapartum and early neonatal morbidity was however, mostly reduced in hypoglycemic twins, while it was comparable between smaller or smaller weight discordant twins and larger twins. Similarly, SGA twins, and smaller or smaller weight-discordant twins did not show increased hypoglycemia risk. In conclusion, our findings suggest that the multiple gestation per se is the single most important relative risk factor of hypoglycemia in premature twins.