Implications

We have the following indirect implication of form equivalence classes:

161 \(\Rightarrow\) 198
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
161 \(\Rightarrow\) 9 Defining cardinal addition by \(le\)-formulas, Haussler, A. 1983, Fund. Math.
9 \(\Rightarrow\) 198 clear

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

Howard-Rubin Number Statement
161:

Definability of cardinal addition in terms of \(\le\): There is a first order formula whose only non-logical symbol is \( \le \) (for cardinals) that defines cardinal addition.

9:

Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.

198:

For every set \(S\), if the only linearly orderable subsets of \(S\) are the finite subsets of \(S\), then either \(S\) is finite or \(S\) has an amorphous subset.

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