Implications

We have the following indirect implication of form equivalence classes:

14 \(\Rightarrow\) 198
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
14 \(\Rightarrow\) 49 A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
49 \(\Rightarrow\) 30 clear
30 \(\Rightarrow\) 198 clear

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

Howard-Rubin Number Statement
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.

198:

For every set \(S\), if the only linearly orderable subsets of \(S\) are the finite subsets of \(S\), then either \(S\) is finite or \(S\) has an amorphous subset.

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