Implications

We have the following indirect implication of form equivalence classes:

28-p \(\Rightarrow\) 84
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
28-p \(\Rightarrow\) 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 \(\Rightarrow\) 51 Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
51 \(\Rightarrow\) 77 Well ordered subsets of linearly ordered sets, Howard, P. 1994, Notre Dame J. Formal Logic
77 \(\Rightarrow\) 185 Well ordered subsets of linearly ordered sets, Howard, P. 1994, Notre Dame J. Formal Logic
185 \(\Rightarrow\) 84 Well ordered subsets of linearly ordered sets, Howard, P. 1994, Notre Dame J. Formal Logic

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

Howard-Rubin Number Statement
28-p:

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

51:

Cofinality Principle: Every linear ordering has a cofinal sub well ordering.  Sierpi\'nski [1918], p 117.

77:

A linear ordering of a set \(P\) is a well ordering if and only if \(P\) has no infinite descending sequences. Jech [1973b], p 23.

185:

Every linearly ordered Dedekind finite set is finite.

84:

\(E(II,III)\) (Howard/Yorke [1989]): \((\forall x)(x\) is \(T\)-finite  if and only if \(\cal P(x)\) is Dedekind finite).

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