Implications

We have the following indirect implication of form equivalence classes:

147 \(\Rightarrow\) 223
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 \(\Rightarrow\) 206 clear
206 \(\Rightarrow\) 223 clear

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.

206:

The existence of a non-principal ultrafilter: There exists an infinite set \(X\) and a non-principal ultrafilter on \(X\).

223:

There is an infinite set \(X\) and a non-principal measure on \(\cal P(X)\).

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