Description: Proof of Form [0 AL]
Content:
In this note we give a proof of Form [0 AL] (For every set \(A\), there is an \(\aleph\) which cannot be covered by fewer than \(|A|\) sets each of cardinality \(< |A|\).) in ZF\(^0\).Form [0 AL] was suggested by K. Keremedis.
Let \(A\) be a set and suppose that \(K\) is an aleph which is covered by fewer than \(|A|\) sets of cardinality \(< |A|\). Let \(\le\) be the usual ordering on \(K\) and let \(\bigcup_{b\in B} C_b = K\) where \(|B| < |A|\) and for each \(b\in B\), \(|C_b| < |A|\). Since each \(C_b\) is well orderable, \(|C_b| < H(A)\). (The function \(H\) is Hartogs' aleph function, \(H(A)\) is the least aleph that is not \(\le |A|\).) Therefore the ordering \((C_b,\le)\) has order type that of an ordinal \(\alpha_b < H(A)\). Let \(f_b\) be the unique order automorphism from \(\alpha_b\) onto \(C_b\). Let \(\gamma\) be a fixed element of \(K\) and define \(F : A\times H(A) \to K\) by \(F(b,\beta) = f_b(\beta)\) if \(b\in B\) and \(\beta\in\alpha_b\) and \(F(b,\beta) = \gamma\) otherwise. \(F\) is onto and \(|K| = |\cal E|\) where \(\cal E\) is the set of equivalence classes \(\subseteq A\times H(A)\) under the relation \(q \sim r \Leftrightarrow F(q) = F(r)\). Therefore \(|K| = \aleph_\sigma\) where \(A\times H(A)\) has a partition of cardinality \(\aleph_\sigma\). Therefore, if we let \(\aleph_\rho\) be the smallest well ordered cardinal greater than every cardinal in the set \(\{\aleph_\sigma : A\times H(A)\) has a partition of cardinality \( \aleph_\sigma\}\), then \(\aleph_\rho\) cannot be covered by fewer than \(|A|\) sets each of cardinality \(|A|\).
Howard-Rubin number: 131
Type: Proof
Back