Implications

We have the following indirect implication of form equivalence classes:

202 \(\Rightarrow\) 222
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
202 \(\Rightarrow\) 91 note-75
91 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 70 clear
70 \(\Rightarrow\) 222 The strength of the Hahn-Banach theorem, Pincus, D. 1972c, Lecture Notes in Mathematics

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
202:

\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function.

91:

\(PW\):  The power set of a well ordered set can be well ordered.

79:

\({\Bbb R}\) can be well ordered.  Hilbert [1900], p 263.

70:

There is a non-trivial ultrafilter on \(\omega\). Jech [1973b], prob 5.24.

222:

There is a non-principal measure on \(\cal P(\omega)\).

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