Implications

We have the following indirect implication of form equivalence classes:

303 \(\Rightarrow\) 222
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
303 \(\Rightarrow\) 50 Some propositions equivalent to the Sikorski extension theorem for Boolean algebras, Bell, J.L. 1988, Fund. Math.
50 \(\Rightarrow\) 14 A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
14 \(\Rightarrow\) 63 clear
63 \(\Rightarrow\) 70 clear
70 \(\Rightarrow\) 222 The strength of the Hahn-Banach theorem, Pincus, D. 1972c, Lecture Notes in Mathematics

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

Howard-Rubin Number Statement
303:

If \(B\) is a Boolean algebra, \(S\subseteq B\) and \(S\) is closed under \(\land\), then there is a \(\subseteq\)-maximal proper ideal \(I\) of \(B\) such that \(I\cap S= \{0\}\).

50:

Sikorski's  Extension Theorem: Every homomorphism of a subalgebra \(B\) of a Boolean algebra \(A\) into a complete Boolean algebra \(B'\) can be extended to a homomorphism of \(A\) into \(B'\). Sikorski [1964], p. 141.

14:

BPI: Every Boolean algebra has a prime ideal.

63:

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

70:

There is a non-trivial ultrafilter on \(\omega\). Jech [1973b], prob 5.24.

222:

There is a non-principal measure on \(\cal P(\omega)\).

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