Bernstein’s Division Theorem

Abstract

AbstractBernstein’s Division Theorem (BDT) states that if an infinite cardinal number is divisible by a finite number then the quotient is unique, namely, if \( {\text{km}} = {\text{kn}} \) then \( {\text{m = n}} \), where k is a natural number, m, n cardinal numbers. The theorem is included (p 122) in Bernstein’s doctorate dissertation of 1901 (published in 1905). It was reproduced in Hobson 1907 pp 159–162. BDT is sometimes called Bernstein’s theorem but since there are other results that bear Bernstein’s name we use ‘Bernstein’s Division Theorem’.KeywordsCardinal NumberEquality SignEquivalence TheoremInverse ElementSimple ChainThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.