Axiom Of Choice

We have the following indirect implication of form equivalence classes:

100 \(\Rightarrow\) 377 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
100 \(\Rightarrow\) 9	On the existence of large sets of Dedekind cardinals, Tarski, A. 1965, Notices Amer. Math. Soc. The Axiom of Choice, Jech, 1973b, page 162 problem 11.8
9 \(\Rightarrow\) 376	clear
376 \(\Rightarrow\) 377	Weak choice principles, De la Cruz, O. 1998a, Proc. Amer. Math. Soc.

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

Howard-Rubin Number	Statement
100:	Weak Partition Principle: For all sets \(x\) and \(y\), if \(x\precsim^* y\), then it is not the case that \(y\prec x\).
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.

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