Notes

Description: Form 191 implies Form 182

Content:

Form 191 is the statement:  There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\) and Form 182 is: There is an aleph whose cofinality is greater than \(\aleph_0\). In Blass [1979] it is proved, assuming Form 191, that \(\forall A\) there is a limit ordinal \(\alpha\) such that \(A\) cannot be mapped cofinally onto \(\alpha\). If \(A\) cannot be mapped cofinally onto \(\alpha\), then \(A\) cannot  be mapped cofinally onto \(\aleph_{\alpha}\).  Therefore, Form 191 implies Form 182.

Blass also shows that if Form 191 holds with a set \(S\) then Form 14 (the Boolean Prime Ideal Theorem) holds if and only if there is an ultrafilter \(U\)  on \(S^{<\omega}\) that is regular in the sense that \(\{p\in S^{<\omega} : s\in\) range \(p\}\) is in \(U\)  for every \(s\in S\).

With regard to (191,\(n\)), Blass shows that Form 191 holds in  most known models of \(ZF^0\) or \(ZF\).  In particular,

  1. Every permutation model whose class of atoms forms a set satisfies Form 191.
  2. Every symmetric model satisfies Form 191. (A symmetric model is a model constructed from a  ground  model \(M\) of \(ZFC\) by taking a complete Boolean algebra \(B\) in \(M\), an \(M\) generic filter \(G\) in \(B\), a group \({\cal G}\) of automorphisms of \(B\) and a normal filter of subgroups  of \({\cal G}\).  The model consists of members of \(M[G]\) that have  hereditarily symmetric names.)
  3. If \(V\) is any model of \(ZFC\) and \(A\in V\), then the submodel \(HOD(A)\) of sets hereditarily ordinal definable over A satisfies Form 191.

Howard-Rubin number: 59

Type: Implication

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