Axiom Of Choice

We have the following indirect implication of form equivalence classes:

90 \(\Rightarrow\) 74 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
90 \(\Rightarrow\) 91	The Axiom of Choice, Jech, 1973b, page 133
91 \(\Rightarrow\) 79	clear
79 \(\Rightarrow\) 94	clear
94 \(\Rightarrow\) 74	note-10

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

Howard-Rubin Number	Statement
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.
94:	\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals has a choice function. Jech [1973b], p 148 prob 10.1.
74:	For every \(A\subseteq\Bbb R\) the following are equivalent: \(A\) is closed and bounded. Every sequence \(\{x_{n}\}\subseteq A\) has a convergent subsequence with limit in A. Jech [1973b], p 21.

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