Axiom Of Choice

We have the following indirect implication of form equivalence classes:

388 \(\Rightarrow\) 182 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
388 \(\Rightarrow\) 106
106 \(\Rightarrow\) 126	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
126 \(\Rightarrow\) 94	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
94 \(\Rightarrow\) 34	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
34 \(\Rightarrow\) 104	clear
104 \(\Rightarrow\) 182	clear

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

Howard-Rubin Number	Statement
388:	Every infinite branching poset (a partially ordered set in which each element has at least two lower bounds) has either an infinite chain or an infinite antichain.
106:	Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.
126:	\(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
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.
34:	\(\aleph_{1}\) is regular.
104:	There is a regular uncountable aleph. Jech [1966b], p 165 prob 11.26.
182:	There is an aleph whose cofinality is greater than \(\aleph_{0}\).

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