In this note we list implications that hold in every Fraenkel-Mostowski model. None of the implications listed are known to be theorems of \(ZF\). In each case we list the implication and a reference if one is available.
Number | Hypothesis | Relationship | Conclusion | Reference(s) |
---|---|---|---|---|
1 | \(\) 231 + 0 | \(\to\) | 23 | This is reasonably clear |
2 | \(\) 40 + 0 | \(\to\) | 1 | Howard, P. [1995b] |
3 | \(\) 62 + 0 | \(\to\) | 60 | Howard, P. [1973] |
4 | \(\) 14 + 0 | \(\to\) | 317 | Howard, P. [1973] |
5 | \(\) 175 + 0 | \(\to\) | 231 |
|
6 | \(\) 323 + 0 | \(\to\) | 60 | Because \(323\) \(\to\) \(62\) is a theorem of \(ZF^{0}\), and, by implication number 3. above, \(62\) \(\to\) \(60\)in every Fraenkel-Mostowski model. ( Howard [1973]) |
7 | \(\) 8 + 67 | \(\leftrightarrow\) | 1 | Rubin, J. [1986] |
8 | \(\) 175 + 0 | \(\to\) | 24 | K. Keremedis. (In every \(FM\) model \(\cal P(\Bbb R)\) can be well ordered, Form 130 is true, and \(|\cal P(\Bbb R)| = 2^{(2^{\aleph_0})}\).) |
9 | \(\) 175 + 0 | \(\to\) | 16 | (In every \(FM\) model, \(\Bbb R\) can be well ordered, Form 79 is true, and \(|\Bbb R| = 2^{\aleph_0}\).) |
10 | \(\) 175 + 0 | \(\to\) | 212 | (See item 9. above) |