Implications

We have the following indirect implication of form equivalence classes:

66 \(\Rightarrow\) 127
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
66 \(\Rightarrow\) 67 Existence of a basis implies the axiom of choice, Blass, A. 1984a, Contemporary Mathematics
67 \(\Rightarrow\) 126 clear
126 \(\Rightarrow\) 82 note-76
82 \(\Rightarrow\) 83 Definitions of finite, Howard, P. 1989, Fund. Math.
83 \(\Rightarrow\) 64 The Axiom of Choice, Jech, 1973b, page 52 problem 4.10
64 \(\Rightarrow\) 127 Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung

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

Howard-Rubin Number Statement
66:

Every vector space over a field has a basis.

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).

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 \(y\).

82:

\(E(I,III)\) (Howard/Yorke [1989]): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.)

83:

\(E(I,II)\) Howard/Yorke [1989]: \(T\)-finite is equivalent to finite.

64:

\(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)

127:

An amorphous power of a compact \(T_2\) space, which as a set is well orderable, is well orderable.

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