Axiom Of Choice

We have the following indirect implication of form equivalence classes:

430-p \(\Rightarrow\) 315 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
430-p \(\Rightarrow\) 67	clear
67 \(\Rightarrow\) 89	On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
89 \(\Rightarrow\) 90	The Axiom of Choice, Jech, 1973b, page 133
90 \(\Rightarrow\) 51	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
51 \(\Rightarrow\) 25	Choice and cofinal well-ordered subsets, Morris, D.B. 1969, Notices Amer. Math. Soc.
25 \(\Rightarrow\) 315	clear

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

Howard-Rubin Number	Statement
430-p:	(Where \(p\) is a prime) \(AL21\)\((p)\): Every vector space over \(\mathbb Z_p\) has the property that for every subspace \(S\) of \(V\), there is a subspace \(S'\) of \(V\) such that \(S \cap S' = \{ 0 \}\) and \(S \cup S'\) generates \(V\) in other words such that \(V = S \oplus S'\). Rubin, H./Rubin, J [1985], p.119, AL21.
67:	\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).
89:	Antichain Principle: Every partially ordered set has a maximal antichain. Jech [1973b], p 133.
90:	\(LW\): Every linearly ordered set can be well ordered. Jech [1973b], p 133.
51:	Cofinality Principle: Every linear ordering has a cofinal sub well ordering. Sierpi\'nski [1918], p 117.
25:	\(\aleph _{\beta +1}\) is regular for all ordinals \(\beta\).
315:	\(\Omega = \omega_1\), where \(\Omega = \{\alpha\in\hbox{ On}: (\forall\beta\le\alpha)(\beta=0 \vee (\exists\gamma)(\beta=\gamma+1) \vee\) there is a sequence \(\langle\gamma_n: n\in\omega\rangle\) such that for each \(n\), \(\gamma_n<\beta\hbox{ and } \beta=\bigcup_{n<\omega}\gamma_n.)\} \)

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