Axiom Of Choice

We have the following indirect implication of form equivalence classes:

264 \(\Rightarrow\) 280 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
264 \(\Rightarrow\) 202	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
202 \(\Rightarrow\) 91	note-75
91 \(\Rightarrow\) 79	clear
79 \(\Rightarrow\) 70	clear
70 \(\Rightarrow\) 142	The Axiom of Choice, Jech, 1973b, page 7 problem 11
142 \(\Rightarrow\) 280	clear

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

Howard-Rubin Number	Statement
264:	\(H(C,P)\): Every connected relation \((X,R)\) contains a \(\subseteq\)-maximal partially ordered set.
202:	\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has a choice function.
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.
70:	There is a non-trivial ultrafilter on \(\omega\). Jech [1973b], prob 5.24.
142:	\(\neg PB\): There is a set of reals without the property of Baire. Jech [1973b], p. 7.
280:	There is a complete separable metric space with a subset which does not have the Baire property.

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