Implications

We have the following indirect implication of form equivalence classes:

57 \(\Rightarrow\) 127
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
57 \(\Rightarrow\) 64 Classes of Dedekind finite cardinals, Truss, J. K. 1974a, Fund. Math.
64 \(\Rightarrow\) 127 Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung

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

Howard-Rubin Number Statement
57:

If \(x\) and \(y\) are Dedekind finite sets then either \(|x|\le |y|\) or \(|y|\le |x|\).
Mathias [1979], p 125.

64:

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

127:

An amorphous power of a compact \(T_2\) space, which as a set is well orderable, is well orderable.

Comment:

Back