Implications

We have the following indirect implication of form equivalence classes:

147 \(\Rightarrow\) 93
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
147 \(\Rightarrow\) 91 The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic
note-26
91 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 70 clear
70 \(\Rightarrow\) 93 The Axiom of Choice, Jech, 1973b, page 7 problem 10

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

Howard-Rubin Number Statement
147:

\(A(D2)\):  Every \(T_2\) topological space \((X,T)\) can be covered by a well ordered family of discrete sets.

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.

93:

There is a non-measurable subset of \({\Bbb R}\).

Comment:

Back