Some Propositions Equivalent to the Continuum Hypothesis

Let 8 denote the real line. If TQ& and r £ 8 , we set {t+r: / £ J H } = r [ r ] . In [ l ] we have proved these two theorems: (BK) Let S C S , T C 8 , S be at most enumerable and T be of first category. Then 8 contains a residual subset R such that SC\T\r\ is empty for every rÇzR. (BM) Let S C S , 2"C8, S be at most enumerable and T be of measure zero. Then 8 contains a subset R such that 8— Ris of measure zero and Sr^T[r] is empty f or every rÇ^RWe introduce the following propositions: (53K) Let S C S , T C 8 , S be of power less than 2*° and T be of first category. Then 8 contains a residual subset R such that SC\T[r] is empty f or every rÇ-R(33M) Let S C S , TC.&, S be of power less than 2**° and T be of measure zero. Then 8 contains a subset R such that & — Ris of measure zero and S(^T[r] is empty f or every r £ i £ . (93K) Let 5 C 8 , T C 8 , S be of power less than 2*« and T be of first category. Then there exists an r £ 8 such that Sf~\T[r] is empty, (93M) Let S C S , 3TCS, S be of power less than 2**° and T be of measure zero. Then there exists an r £ 8 such that SC\T[r] is empty. Clearly (93K) implies (93|) and (93M) implies (93M). The following five propositions are discussed at some length in [2 ] : (H) 2*0 = 6^. ($) The union of less than 2**° subsets of 8 of first category is of first category, (9JÏ) The union of less than 2**° subsets of 8 of measure zero is of measure zero, ($*) 8 is not the union of less than 2**° subsets of 8 of first category, (9Ji*) 8 is not the union of less than 2**° subsets of 8 of measure zero. Evidently (H) implies ($) and (2tt), (®) implies ($*) , and (9tt) implies (2ft*). By examining the proofs of (BK) and (BM), it is easy to see that the following lemma is true.