Numerosities of labelled sets: A new way of counting
In mathematics, it is very important to develop the mathematical theory of quantities (magnitudes). Classical theory was mainly developed by German and Italian mathematicians in the 19th century. Following the progress of mathematics, many new facts have been appended to this field. In this paper, the authors consider the case of counting. For counting we need some basic restrictions to characterize the concept of quantities, namely we must suppose that the range of the counting function \(\nu\) is a linearly ordered set. (1) If sets \(A\) and \(B\) are in one-to-one correspondence, then the counting results \(\nu(A)\), \(\nu(B)\) are the same. (2) If \(A\) is a proper subset of \(B\), then properly \(\nu(A) <\nu(B)\). The conditions needed are (3) the finite additivity of \(\nu\) and (4) \(\nu(A)= \nu(A')\), \(\nu(B)= \nu(B')\to\nu (A\times B)= \nu(A' \times B')\). The most interesting result in this paper is Theorem 4.3: such a counting function exists \(\Leftrightarrow\) there exists a selective ultrafilter. By a selective ultrafilter, we mean: let \({\mathcal U}\) be a nonprincipal ultrafilter on \(\mathbb{N}\). Then for any countably infinite partion \(\{X_n:n \in\mathbb{N}\}\) with \(X_n\notin {\mathcal U}\), there exists a selective set \(X \in {\mathcal U}\) that each \(X\cap X_n\) contains at most one element. The authors give three statements which are equivalent to the existence of a selective ultrafilter. From this result, we have an exciting result: The existence of counting functions is independent of ZFC.
scientific article; zbMATH DE number 1888335