Axiom Of Choice

We have the following indirect implication of form equivalence classes:

264 \(\Rightarrow\) 374-n given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
264 \(\Rightarrow\) 202	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
202 \(\Rightarrow\) 40	clear
40 \(\Rightarrow\) 39	clear
39 \(\Rightarrow\) 8	clear
8 \(\Rightarrow\) 9	Was sind und was sollen die Zollen?, Dedekind, [1888]
9 \(\Rightarrow\) 10	Zermelo's Axiom of Choice, Moore, 1982, 322
10 \(\Rightarrow\) 423	clear
423 \(\Rightarrow\) 374-n	clear

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

Howard-Rubin Number	Statement
264:	\(H(C,P)\): Every connected relation \((X,R)\) contains a \(\subseteq\)-maximal partially ordered set.
202:	\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has a choice function.
40:	\(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.
39:	\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.
8:	\(C(\aleph_{0},\infty)\):
9:	Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
10:	\(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function.
423:	\(\forall n\in \omega-\{o,1\}\), \(C(\aleph_0, n)\) : For every \(n\in \omega - \{0,1\}\), every denumerable set of \(n\) element sets has a choice function.
374-n:	\(UT(\aleph_0,n,\aleph_0)\) for \(n\in\omega -\{0,1\}\): The union of a denumerable set of \(n\)-element sets is denumerable.

Comment:

Back