1. The theory of classes presented in this paper is a simplification of that presented by J. von Neumann in his paper Die Axiomatisierung der Mengenlehre. However, this paper is written so that it can be read independently of von Neumann's. The principal modifications of his system are the following.(1) The idea of ordered pair is defined in terms of the other primitive concepts of the system. (See Axiom 4.3 below.)(2) A much simpler proof of the well-ordering theorem, based on (...) von Neumann's equivalence axiom (Axiom 2.2 below) is given. (More exactly, the theory of ordinal numbers, on which the proof of the well-ordering theorem is based, is simplified—see §8.)(3) Functions are not assumed to be defined for all arguments. In place of “ = A” we have “is undefined.” This makes possible a constructive interpretation of the system. (See §2.)2. We wish first to give a rough picture of the system which we are trying to construct. Suppose that we have the idea “class,” but no material from which to construct classes. Nevertheless, we can construct the class 0 having no elements. And then the class 1 having 0 as its only element. And then the classes {1}, and {0, 1}, etc., using always the previously constructed classes as elements. To extend this method to infinite classes, we must give rules telling what elements are to be included. Since the nature of the rules which we can use is not altogether clear, we try to formalize the whole system; that is, to set up a system of axioms which seems to characterize this system of classes. Now in this set of axioms we find it necessary to use the idea of function (in the “Axiom of Replacement”). (shrink)

Consider an axiomatic set theory in which there is a distinction between “sets” and “classes,” only sets being allowable as elements. How can one define a finite sequence of classes? This problem was proposed to me by A. Tarski, and a solution is given in this note. We shall assume the axiom system Σ used by Godei in his study of the continuum hypothesis, and shall use the same notation.1.

The aim of this study was to analyze the association between physical activity and eating habits during the COVID-19 pandemic among Brazilian adults. A sample of 1,929 participants answered an online survey, however 1,874 were included in the analysis. The impact of the COVID-19 pandemic on eating habits was assessed inquiring about participants' intake of fruits, vegetables, fried foods, and sweets during the pandemic. Physical activity was assessed by asking participants about their weekly frequency, intensity and number of minutes/hours engaging (...) in structured physical activities per week. Participants were then stratified into categories based on moderate-to-vigorous intensity and into active or inactive. Increased sweets consumption was the most commonly reported change to eating habits, followed by an increase in the consumption of vegetables, fruits, and fried foods. Physical activity practice was related to lower consumption of fried foods and sweets. A cluster analysis revealed subjects with higher the level of physical activity was more likely to follow a healthy diet. Thus, physical activity was positively associated with healthier eating habits. Health authorities must recommend regular physical as a strategy to improve overall health during the COVID-19 pandemic. Future studies should address the physical activity interventions to improve health status during a pandemic. (shrink)

BackgroundThe outbreak of novel coronavirus disease 2019 has caused a global panic and public concern due to its mortality ratio and lack of treatments/vaccines. Reduced levels of physical activity have been reported during the outbreak, affecting the normal daily pattern.ObjectiveTo investigate the relationship of physical activity level with sleep quality and the effects of reduction physical activity levels on sleep quality.MethodsA Google form was used to address personal information, COVID-19 personal care, physical activity, and mental health of 1,907 adult volunteers. (...) Binary logistic regression was used to verify the association of physical activity parameters and sleep quality.ResultsInsufficient physical activity levels were a risk factor to have disturbed sleep pattern [OR: 1.28, 95% CI ]; however, when the BMI was added to the analysis, there was no more statistical difference [OR: 1.23, 95% CI ]. On the other hand, we found that the reduction of physical activity levels was associated with negative changes in sleep quality [OR: 1.73, 95% CI ], regardless all the confounders [OR: 1.30, 95% CI ], unless when feeling of depression was added in Model 6 [OR: 1.28, 95% CI ].ConclusionDisruption in daily physical activity routine, rather than physical activity level, negatively influences sleep quality during the COVID-19 quarantine. (shrink)

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A lawyer explains the difference between law and ethics--what you can do versus what you should do--and how leaders can create more ethical, satisfying, and profitable workplaces.