Implications

We have the following indirect implication of form equivalence classes:

239 \(\Rightarrow\) 306
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
239 \(\Rightarrow\) 427 clear
427 \(\Rightarrow\) 67 clear
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 \(\Rightarrow\) 305 Equivalents of the Axiom of Choice II, Rubin, 1985, theorem 5.7
305 \(\Rightarrow\) 306 Cardinality of the set of Vitali equivalence classes, Kanovei, V.G. 1991, Mat. Zametki

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

Howard-Rubin Number Statement
239:

AL20(\(\mathbb Q\)):  Every vector \(V\) space over \(\mathbb Q\) has the property that every linearly independent subset of \(V\) can be extended to a basis. Rubin, H./Rubin, J. [1985], p.119, AL20.

427: \(\exists F\) AL20(\(F\)): There is a field \(F\) such that every vector space \(V\) over \(F\) has the property that every independent subset of \(V\) can be extended to a basis.  \ac{Bleicher} \cite{1964}, \ac{Rubin, H.\/Rubin, J \cite{1985, p.119, AL20}.
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.

305:

There are \(2^{\aleph_0}\) Vitali equivalence classes. (Vitali equivalence classes are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in{\Bbb Q})(x-y=q)\).). \ac{Kanovei} \cite{1991}.

306:

The set of Vitali equivalence classes is linearly orderable. (Vitali equivalence classes are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow (\exists q\in{\Bbb Q})(x-y = q)\).).

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