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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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