Non‐discrete metrics in and some notions of finiteness. Issue 4 (12th July 2016)

• Record Type: Journal Article
• Authors: Keremedis, Kyriakos
• Is Part Of: Mathematical logic quarterly. Volume 62:Issue 4/5(2016)
• Page Start: 383
• Page End: 390
• Publication Date: 2016-07-12
• Subjects: Mathematics -- Periodicals; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1002/malq.201300054
• Languages: English
• ISSNs: 0942-5616

Abstract

Abstract : 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 ℘ ( X ) ; and (iv) an infinite set X is weakly Dedekind‐infinite if and only if ( ℘ ( X ), ⊆ ) has infinite towers if and only if X has countable partitions.