Hypothesis: HR 30:
Ordering Principle: Every set can be linearly ordered.
Conclusion: HR 49:
Order Extension Principle: Every partial ordering can be extended to a linear ordering. Tarski [1924], p 78.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M3\) Mathias' model | Mathias proves that the \(FM\) model <a href="/models/Mathias-Pincus-1">\(\cal N4\)</a> can be transformed into a model of \(ZF\), \(\cal M3\) |
| \(\cal M45\) Pincus' Model VII | This is the model of <a href="/articles/Pincus-1977a">Pincus [1977a]</a>, Theorem 2.1 \((C)\) |
| \(\cal N5\) The Mathias/Pincus Model II (an extension of \(\cal N4\)) | \(A\) iscountably infinite; \(\precsim\) and \(\le\) are universal homogeneous partialand linear orderings, respectively, on \(A\), (See <a href="/articles/Jech-1973b">Jech [1973b]</a>p101 for definitions.); \(\cal G\) is the group of all order automorphismson \((A,\precsim,\le)\); and \(S\) is the set of all finite subsets of \(A\) |
| \(\cal M14\) Morris' Model I | This is an extension of Mathias' model, <a href="/models/Mathias-1">\(\cal M3\)</a> |
Code: 3
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