Hypothesis: HR 191:
\(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\).
Conclusion: HR 59-le:
If \((A,\le)\) is a partial ordering that is not a well ordering, then there is no set \(B\) such that \((B,\le)\) (the usual injective cardinal ordering on \(B\)) is isomorphic to \((A,\le)\).
Mathias [1979], p 120.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N11\) Jech's Model II | Let \((I,\precsim)\) be a partially ordered set inthe kernel (in the base model without atoms) |
| \(\cal N19(\precsim)\) Tsukada's Model | Let \((P,\precsim)\) be a partiallyordered set that is not well ordered; Let \(Q\) be a countably infinite set,disjoint from \(P\); and let \(I=P\cup Q\) |
Code: 5
Comments: