Platonisme, konseptualisme en nomimalisme in die versamelingsteorie
Platonism, Conceptualism and Nominalism in the set theory
In the first part of the article I explain how the bounded variable assumes the ontology of the language user. If the variables extend over concrete objects, we are dealing with a nominalistic point of view. When values include abstract objects, a platonistic stance is taken. The article points out that nominalism cannot really give up on some platonistic areas.
In the investigation into the transition from predicate letters to set theory, the predicates receive the status of variables that adopt/accept classes as values. A reality (universals), therefore, is postulated behind forms of language. Russell’s antinomy may be deduced from this, but is evaded by classes being constructed according to a specific principle of rank. In this regard conceptualistic constructivism is involved. The constructivist system ?, a ranking hierarchy that reaches into the transfinite, is an example of this.
Studying Platonism, nominalism and conceptualism involves the idea of the finite and the infinite, abstract as well as concrete. In the present article these fundamental issues are discussed. Introductory remarks will be made regarding iterative models in the set theory. Inter alia the article reaches the conclusion that the Christian philosophy smooths the way to regard the set theory as a deepened numerical theory, more or less in line with Kanamori’s viewpoint that the set theory is an “open-ended framework for mathematics” (Kanamori, 2007:52).
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