Axiom Of Choice

We have the following indirect implication of form equivalence classes:

50 \(\Rightarrow\) 102 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 \(\Rightarrow\) 49	A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
49 \(\Rightarrow\) 30	clear
30 \(\Rightarrow\) 62	clear
62 \(\Rightarrow\) 102	The Axiom of Choice, Jech, 1973b, page 162 problem 11.12

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.
49:	Order Extension Principle: Every partial ordering can be extended to a linear ordering. Tarski [1924], p 78.
30:	Ordering Principle: Every set can be linearly ordered.
62:	\(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function.
102:	For all Dedekind finite cardinals \(p\) and \(q\), if \(p^{2} = q^{2}\) then \(p = q\). Jech [1973b], p 162 prob 11.12.

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