It is well known that the Continuum Hypothesis is independent from the axioms of set theory by the results of Kurt Godel and Paul Cohen. However, there are some set theorists who do not consider that this independency of the hypothesis is the ultimate answer of the continuum problem and who think the validity of the hypothesis should be determined. Recently, several ideas have been proposed to decide the hypothesis. In this paper, we introduce two ideas to determine the truth of the Continuum Hypothesis, which are related to the concept of forcing absoluteness. One is the concepet of forcing axioms, and the other is Ω-logic.

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We use varieties of cognitive structures in cognitive activities. Classes and collections are two kinds of cognitive structures for class-inclusion reasoning introduced by Markman to analyze the Inhelder and Piaget's class-inclusion problem. In this paper, we shall review investigations on classes and collections in developmental psychology, and propose mathematical definitions and models of classes and collections. We shall show also interpretations of these two models.

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A numerical model (individual-based model) to describe ecosystem dynamics is proposed. The ecosystem is partitioned into the same time and space scale levels and the partitions at different levels are connected through transfer of mass. The partitioned system is expressed as aggregates of three-dimensional cells. Within the cells, nutrient intake, excretion, growth, reproduction, death, and movement of individual organisms are executed in accordance with the model rules. The mass transfer due to metabolic processes of organisms, including energy dissipation (efficiency) is described by difference equations. Movement to neighboring cells occurs through diffusion for substances, and through expansion of habitat range and migration in organisms. This individual-based model was applied to an experimental microecosystem (microcosm) derived from an aquatic ecosystem. This model was demonstrated to be applicable to descriptions from the micro behavior of individuals to macro system formation.

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A hierarchical view of natural systems provides an effective framework to describe a given process of interest effectively by focusing a particular hierarchical level. However, this raises a serious problem, how is the focal level restricted or determined by the upper and lower levels? This problem is seen in the theory of natural selection. The theory explains the evolution of a population of biological entities as an automatic consequence of differential heritable fitness among variants, in which the complex biotic/abiotic interactions are put into a black box ("environment"), and the existence of the variants is presupposed; no interactions are explicitly incorporated into this explanation between the focal (population) and the upper/lower levels. In this paper I analyze the inter-level interactions involved in the process of adaptive evolution, and show that it proceeds under upward determination (i.e. generation of novel genetic variants) from the lower level, and under downward one from the upper level such as ecosystem (i.e. supply of new genetic information, creation of selective environments, and dynamic mechanism of replacement).

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