Hypothesis: HR 77:
A linear ordering of a set \(P\) is a well ordering if and only if \(P\) has no infinite descending sequences. Jech [1973b], p 23.
Conclusion: HR 316:
If a linearly ordered set \((A,\le)\) has the fixed point property then \((A,\le)\) is complete. (\((A,\le)\) has the fixed point property if every function \(f:A\to A\) satisfying \((x\le y \Rightarrow f(x)\le f(y))\) has a fixed point, and (\((A,\le)\) is complete if every subset of \(A\) has a least upper bound.)
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N47\) Höft/Howard Model II | This model is similar to \(\cal N33\).The atoms \(A\) are ordered by \(\le\) so that they have order type that ofthe real numbers \(\Bbb R\) (\(|A| = 2^{\aleph_0}\)) |
Code: 5
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