Hypothesis: HR 57:
If \(x\) and \(y\) are Dedekind finite sets then either \(|x|\le |y|\) or \(|y|\le |x|\).
Mathias [1979], p 125.
Conclusion: HR 9:
Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M32\) Sageev's Model II | Starting with a model \(\cal M\) of \(ZF + V =L\), Sageev constructs a sequence of models \(\cal M\subseteq N_0 \subseteq N_1\subseteq\cdots\subseteq N_{\kappa}\) where \(\kappa\) is an inaccessible cardinal, \(N_0\) is Cohen's model <a href="/models/Cohen-1">\(\cal M1\)</a>, and \(N_{\kappa}\) is \(\cal M32\) |
| \(\cal N49\) De la Cruz/Di Prisco Model | Let \(A = \{ a(i,p) : i\in\omega\land p\in {\Bbb Q}/{\Bbb Z} \}\) |
Code: 3
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