Axiom Of Choice

We have the following indirect implication of form equivalence classes:

50 \(\Rightarrow\) 344 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
50 \(\Rightarrow\) 14	A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
14 \(\Rightarrow\) 123	Variants of Rado's selection lemma and their applications, Rav, Y. 1977, Math. Nachr.
123 \(\Rightarrow\) 344	Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal

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

Howard-Rubin Number	Statement
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.
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 \(x\).
344:	If \((E_i)_{i\in I}\) is a family of non-empty sets, then there is a family \((U_i)_{i\in I}\) such that \(\forall i\in I\), \(U_i\) is an ultrafilter on \(E_i\).

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