Implications

We have the following indirect implication of form equivalence classes:

133 \(\Rightarrow\) 145
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
133 \(\Rightarrow\) 90 Dedekind-Endlichkeit und Wohlordenbarkeit, Brunner, N. 1982a, Monatsh. Math.
90 \(\Rightarrow\) 91 The Axiom of Choice, Jech, 1973b, page 133
91 \(\Rightarrow\) 145 P-Raüme and Auswahlaxiom, Brunner, N. 1984c, Rend. Circ. Mat. Palermo.

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

Howard-Rubin Number Statement
133:  

Every set is either well orderable or has an infinite amorphous subset.

90:

\(LW\):  Every linearly ordered set can be well ordered. Jech [1973b], p 133.

91:

\(PW\):  The power set of a well ordered set can be well ordered.

145:

Compact \(P_0\)-spaces are Dedekind finite. (A \(P_0\)-space is a topological space in which the intersection of a countable collection of open sets is open.)

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