Implications

We have the following indirect implication of form equivalence classes:

100 \(\Rightarrow\) 342-n
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 \(\Rightarrow\) 342-n clear

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.

342-n:

(For \(n\in\omega\), \(n\ge 2\).) \(PC(\infty,n,\infty)\):  Every infinite family of \(n\)-element sets has an infinite subfamily with a choice function. (See Form 166.)

Comment:

Back