zbMATH — the first resource for mathematics

Arithmetic of cuts and cuts of classes. (English) Zbl 0658.03032
In Alternative Set Theory infinite objects (proper classes) are considered as undetermined parts of formally finite large objects (infinitely large sets). This approach seems to be closer to reality than that of Cantor’s set theory. If we want to determine the size of a class, we can use the class of cardinalities of subsets of the investigated class and the class of its supersets. For classes not being sets the classes determined above describe complete parts of N (the class of natural numbers) not being natural numbers. By this, natural numbers (cardinalities) are determined for sets. Similar as in classical set theory it is reasonable to extend arithmetical operations also on cardinal numbers; it is worth extending these operations to cuts (complete parts of N). The properties of extended operations can be used for determining the size of classes. For four basic operations \(+\), -, \(\times\), /, there are two ways of extension; these two versions are equal if e.g. one of the considered cuts is a natural number. (In classical ordinal numbers \(\alpha +\beta \neq \beta +\alpha\) and \(\alpha \times \beta \neq \beta \times \alpha.)\) The properties of extended operations are investigated, examples where the two versions differ are described and a classification of cuts relative to the investigated operations is given. Moreover, the infinite sum of cuts and a nontrivial notion of nearness of cuts are examined.
Reviewer: K.Čuda

03E70 Nonclassical and second-order set theories
03H99 Nonstandard models
03H15 Nonstandard models of arithmetic
03E10 Ordinal and cardinal numbers
Full Text: EuDML