Axiom Of Choice

We have the following indirect implication of form equivalence classes:

95-F \(\Rightarrow\) 169 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
95-F \(\Rightarrow\) 67	Some theorems on vector spaces and the axiom of choice, Bleicher, M. 1964, Fund. Math. The Axiom of Choice, Jech, 1973b, page 148 problem 10.4
67 \(\Rightarrow\) 89	On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
89 \(\Rightarrow\) 90	The Axiom of Choice, Jech, 1973b, page 133
90 \(\Rightarrow\) 91	The Axiom of Choice, Jech, 1973b, page 133
91 \(\Rightarrow\) 79	clear
79 \(\Rightarrow\) 272	Models of set theory containing many perfect sets, Truss, J. K. 1974b, Ann. Math. Logic
272 \(\Rightarrow\) 169	clear

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

Howard-Rubin Number	Statement
95-F:	Existence of Complementary Subspaces over a Field \(F\): If \(F\) is a field, then every vector space \(V\) over \(F\) has the property that if \(S\subseteq V\) is a subspace of \(V\), then there is a subspace \(S'\subseteq V\) such that \(S\cap S'= \{0\}\) and \(S\cup S'\) generates \(V\). H. Rubin/J. Rubin [1985], pp 119ff, and Jech [1973b], p 148 prob 10.4.
67:	\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).
89:	Antichain Principle: Every partially ordered set has a maximal antichain. Jech [1973b], p 133.
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.
272:	There is an \(X\subseteq{\Bbb R}\) such that neither \(X\) nor \(\Bbb R - X\) has a perfect subset.
169:	There is an uncountable subset of \({\Bbb R}\) without a perfect subset.

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