Implications

We have the following indirect implication of form equivalence classes:

385 \(\Rightarrow\) 223
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
385 \(\Rightarrow\) 70 note-150
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
385:

Countable Ultrafilter Theorem:  Every proper filter with a countable base over a set \(S\) (in \({\cal P}(S)\)) can be extended to an ultrafilter.

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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