Implications

We have the following indirect implication of form equivalence classes:

100 \(\Rightarrow\) 58
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
100 \(\Rightarrow\) 347 Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
347 \(\Rightarrow\) 40 Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
40 \(\Rightarrow\) 208 note-69
208 \(\Rightarrow\) 58 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 \(y\prec x\).

347:

Idemmultiple Partition Principle: If \(y\) is idemmultiple (\(2\times y\approx y\)) and \(x\precsim ^* y\), then \(x\precsim y\).

40:

\(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.

208:

For all ordinals \(\alpha\), \(\aleph_{\alpha+1}\le 2^{\aleph_\alpha}\).

58:

There is an ordinal \(\alpha\) such that \(\aleph(2^{\aleph_{\alpha }})\neq\aleph_{\alpha +1}\). (\(\aleph(2^{\aleph_{\alpha}})\) is Hartogs' aleph, the least \(\aleph\) not \(\le 2^{\aleph _{\alpha}}\).)
Mathias [1979], p 126.

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