Axiom Of Choice

We have the following indirect implication of form equivalence classes:

130 \(\Rightarrow\) 142 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
130 \(\Rightarrow\) 79	clear
79 \(\Rightarrow\) 70	clear
70 \(\Rightarrow\) 142	The Axiom of Choice, Jech, 1973b, page 7 problem 11

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

Howard-Rubin Number	Statement
130:	\({\cal P}(\Bbb R)\) is well orderable.
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.
142:	\(\neg PB\): There is a set of reals without the property of Baire. Jech [1973b], p. 7.

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