Non-discrete metrics in ZF and some notions of finiteness

摘要

We show that (i) it is consistent with ZF that there are infinite sets X on which every metric is discrete; (ii) the notion of real infinite is strictly stronger than that of metrically infinite; (iii) a set X is metrically infinite if and only if it is weakly Dedekind-infinite if and only if the cardinality of the set of all metrically finite subsets of X is strictly less than the size of P (X); and (iv) an infinite set X is weakly Dedekind-infinite if and only if (P (X), subset of) has infinite towers if and only if X has countable partitions. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim