Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
346 \(\Rightarrow\) 126	The vector space Kinna-Wagner Principle is equivalent to the axiom of choice, Keremedis, K. 2001a, Math. Logic Quart.
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
346:	If \(V\) is a vector space without a finite basis then \(V\) contains an infinite, well ordered, linearly independent subset.
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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