Implications

We have the following indirect implication of form equivalence classes:

387 \(\Rightarrow\) 390
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
387 \(\Rightarrow\) 64 "Dense orderings, partitions, and weak forms of choice", Gonzalez, C. 1995a, Fund. Math.
64 \(\Rightarrow\) 390 clear

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

Howard-Rubin Number Statement
387:

DPO:  Every infinite set has a non-trivial, dense partial order.  (A partial ordering \(<\) on a set \(X\) is dense if \((\forall x, y\in X)(x \lt y \to (\exists z \in X)(x \lt z \lt y))\) and is non-trivial if \((\exists x,y\in X)(x \lt y)\)).

64:

\(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)

390:

Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements. Ash [1983].

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