Axiom Of Choice

We have the following indirect implication of form equivalence classes:

346 \(\Rightarrow\) 108 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 \(\Rightarrow\) 5	clear
5 \(\Rightarrow\) 38	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
38 \(\Rightarrow\) 108	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.
5:	\(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function.
38:	\({\Bbb R}\) is not the union of a countable family of countable sets.
108:	There is an ordinal \(\alpha\) such that \(2^{\aleph _{\alpha}}\) is not the union of a denumerable set of denumerable sets.

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