Implications

We have the following indirect implication of form equivalence classes:

430-p \(\Rightarrow\) 415
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
430-p \(\Rightarrow\) 67 clear
67 \(\Rightarrow\) 144 Axioms of multiple choice, Levy, A. 1962, Fund. Math.
144 \(\Rightarrow\) 415 Constructive order theory, Ern'e, M. 2001, Math. Logic Quart.

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

Howard-Rubin Number Statement
430-p:

(Where \(p\) is a prime) \(AL21\)\((p)\): Every vector space over \(\mathbb Z_p\) has the property that for every subspace \(S\) of \(V\), there is a subspace \(S'\) of \(V\) such that \(S \cap S' = \{ 0 \}\) and \(S \cup S'\) generates \(V\) in other words such that \(V = S \oplus S'\).   Rubin, H./Rubin, J [1985], p.119, AL21.

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).

144:

Every set is almost well orderable.

415:

Every \(\cal W\)-compactly generated complete lattice is algebraic.  \ac{Ern\'e} \cite{2000}.

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