Notes

Listing items from note 2

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 If \(X\) is a set of well orderable sets in a Fraenkel-Mostowski model \(\cal N\), then for each \(y\in X\), the set \(y'\) of well orderings of \(y\) is well orderable in \(\cal N\) since any permutation fixing \(y\) pointwise also fixes \(y'\) pointwise. Therefore, by Form 165 (\(C(WO,WO)\)), the set \(X' = \{y' : y\in X\}\) has a choice function. Using this choice function and a well ordering of \(X\) a well ordering of \(\bigcup X\) can be constructed.
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)