Implications

We have the following indirect implication of form equivalence classes:

51 \(\Rightarrow\) 316
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
51 \(\Rightarrow\) 316 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
51:

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

316:

If a linearly ordered set \((A,\le)\) has the fixed point property then \((A,\le)\) is complete. (\((A,\le)\)  has the fixed point property if every function \(f:A\to A\) satisfying \((x\le y \Rightarrow f(x)\le f(y))\) has a fixed point, and (\((A,\le)\) is complete if every subset of \(A\) has a least upper bound.)

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