Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
152 \(\Rightarrow\) 4	Russell's alternative to the axiom of choice, Howard, P. 1992, Z. Math. Logik Grundlagen Math. note-27 note-27 note-27
4 \(\Rightarrow\) 9	clear
9 \(\Rightarrow\) 64	The independence of various definitions of finiteness, Levy, A. 1958, Fund. Math. clear
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
152:	\(D_{\aleph_{0}}\): Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. (See note 27 for \(D_{\kappa}\), \(\kappa\) a well ordered cardinal.)
4:	Every infinite set is the union of some disjoint family of denumerable subsets. (Denumerable means \(\cong \aleph_0\).)
9:	Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
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