Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
49 \(\Rightarrow\) 30 clear
30 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 64 Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung
64 \(\Rightarrow\) 390 clear

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

Howard-Rubin Number Statement
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.

62:

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

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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