Implications

We have the following indirect implication of form equivalence classes:

86-alpha \(\Rightarrow\) 98
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
86-alpha \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 9 Was sind und was sollen die Zollen?, Dedekind, [1888]
9 \(\Rightarrow\) 98 clear

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

Howard-Rubin Number Statement
86-alpha:

\(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(|X| = \aleph_{\alpha }\), then \(X\) has a choice function.

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.

98:

The set of all finite subsets of a Dedekind finite set is Dedekind finite. Jech [1973b] p 161 prob 11.5.

Comment:

Back