Axiom Of Choice

We have the following indirect implication of form equivalence classes:

292 \(\Rightarrow\) 170 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
292 \(\Rightarrow\) 90	The axiom of choice and linearly ordered sets, Howard, P. 1977, Fund. Math.
90 \(\Rightarrow\) 91	The Axiom of Choice, Jech, 1973b, page 133
91 \(\Rightarrow\) 79	clear
79 \(\Rightarrow\) 92	clear
92 \(\Rightarrow\) 170	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.

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

Howard-Rubin Number	Statement
292:	\(MC(LO,\infty)\): For each linearly ordered family of non-empty sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is non-empty, finite subset of \(x\).
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.
92:	\(C(WO,{\Bbb R})\): Every well ordered family of non-empty subsets of \({\Bbb R}\) has a choice function.
170:	\(\aleph_{1}\le 2^{\aleph_{0}}\).

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