Implications

We have the following indirect implication of form equivalence classes:

123 \(\Rightarrow\) 221
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
123 \(\Rightarrow\) 63 clear
63 \(\Rightarrow\) 221 clear

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

Howard-Rubin Number Statement
123:

\(SPI^*\): Uniform weak ultrafilter principle: For each family \(F\) of infinite sets \(\exists f\) such that \(\forall x\in F\), \(f(x)\) is a non-principal ultrafilter on \(x\).

63:

\(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.
Jech [1973b], p 172 prob 8.5.

221:

For all infinite \(X\), there is a non-principal measure on \(\cal P(X)\).

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