Implications

We have the following indirect implication of form equivalence classes:

8 \(\Rightarrow\) 378
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
8 \(\Rightarrow\) 9 Was sind und was sollen die Zollen?, Dedekind, [1888]
9 \(\Rightarrow\) 376 clear
376 \(\Rightarrow\) 377 Weak choice principles, De la Cruz, O. 1998a, Proc. Amer. Math. Soc.
377 \(\Rightarrow\) 378 clear

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

Howard-Rubin Number Statement
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.

376:

Restricted Kinna Wagner Principle:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) and a function \(f\) such that for every \(z\subseteq Y\), if \(|z| \ge 2\) then \(f(z)\) is a non-empty proper subset of \(z\).

377:

Restricted Ordering Principle:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) such that \(Y\) can be linearly ordered.

378:

Restricted Choice for Families of Well Ordered Sets:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) such that the family of non-empty well orderable subsets of \(Y\) has a choice function.

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