Axiom Of Choice

We have the following indirect implication of form equivalence classes:

391 \(\Rightarrow\) 137-k given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
391 \(\Rightarrow\) 112	clear
112 \(\Rightarrow\) 90	Equivalents of the Axiom of Choice II, Rubin/Rubin, 1985, page 79
90 \(\Rightarrow\) 91	The Axiom of Choice, Jech, 1973b, page 133
91 \(\Rightarrow\) 79	clear
79 \(\Rightarrow\) 139
139 \(\Rightarrow\) 137-k	Cancellation laws for surjective cardinals, Truss, J. K. 1984, Ann. Pure Appl. Logic

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

Howard-Rubin Number	Statement
391:	\(C(\infty,LO)\): Every set of non-empty linearly orderable sets has a choice function.
112:	\(MC(\infty,LO)\): For every family \(X\) of non-empty sets each of which can be linearly ordered there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
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.
139:	Using the discrete topology on 2, \(2^{\cal P(\omega)}\) is compact.
137-k:	Suppose \(k\in\omega-\{0\}\). If \(f\) is a 1-1 map from \(k\times X\) into \(k\times Y\) then there are partitions \(X = \bigcup_{i \le k} X_{i} \) and \(Y = \bigcup_{i \le k} Y_{i} \) of \(X\) and \(Y\) such that \(f\) maps \(\bigcup_{i \le k} (\{i\} \times X_{i})\) onto \(\bigcup_{i \le k} (\{i\} \times Y_{i})\).

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