Implications

We have the following indirect implication of form equivalence classes:

345 \(\Rightarrow\) 83
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
345 \(\Rightarrow\) 14 Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal
14 \(\Rightarrow\) 49 A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
49 \(\Rightarrow\) 30 clear
30 \(\Rightarrow\) 83 note-81

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

Howard-Rubin Number Statement
345:

Rasiowa-Sikorski Axiom:  If \((B,\land,\lor)\) is a Boolean algebra, \(a\) is a non-zero element of \(B\), and \(\{X_n: n\in\omega\}\) is a denumerable set of subsets of \(B\) then there is a maximal filter \(F\) of \(B\) such that \(a\in F\) and for each \(n\in\omega\), if \(X_n\subseteq F\) and \(\bigwedge X_n\) exists then \(\bigwedge X_n \in F\).

14:

BPI: Every Boolean algebra has a prime ideal.

49:

Order Extension Principle: Every partial ordering can be extended to a linear ordering.  Tarski [1924], p 78.

30:

Ordering Principle: Every set can be linearly ordered.

83:

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

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