We have the following indirect implication of form equivalence classes:
| 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 \(\Rightarrow\) 311 | The Banach-Tarski Paradox, Wagon, [1985] |
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. |
| 311: | Abelian groups are amenable. (\(G\) is amenable if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G)=1\) and \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
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