Axiom Of Choice

We have the following indirect implication of form equivalence classes:

67 \(\Rightarrow\) 197 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
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 \(\Rightarrow\) 91	The Axiom of Choice, Jech, 1973b, page 133
91 \(\Rightarrow\) 79	clear
79 \(\Rightarrow\) 197	The plane is the union of three rectilinearly accessible sets, Davies, R. O. 1978, Real Anal. Exchange.

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

Howard-Rubin Number	Statement
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.
91:	\(PW\): The power set of a well ordered set can be well ordered.
79:	\({\Bbb R}\) can be well ordered. Hilbert [1900], p 263.
197:	\({\Bbb R}^{2}\) is the union of three sets \(C\) with the property that for all \(x\in C\) there is a straight line \(L\) such that \(L\cap C = \{x\}\).

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