Axiom Of Choice

We have the following indirect implication of form equivalence classes:

149 \(\Rightarrow\) 367 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
149 \(\Rightarrow\) 67	The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic note-26
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 \(\Rightarrow\) 367	Eine Basis aller Zahlen und die unstetigen Losungen der Functionalgleichung: \(f(x+y) = f(x) + f(y)\), Hamel, G. 1905, Math. Ann.

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

Howard-Rubin Number	Statement
149:	\(A(F)\): Every \(T_2\) topological space is a continuous, finite to one image of an \(A1\) space.
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.
367:	There is a Hamel basis for \(\Bbb R\) as a vector space over \(\Bbb Q\).

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