Implications

We have the following indirect implication of form equivalence classes:

347 \(\Rightarrow\) 196-alpha
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
347 \(\Rightarrow\) 40 Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
40 \(\Rightarrow\) 86-alpha clear
86-alpha \(\Rightarrow\) 196-alpha Successive large cardinals, Bull Jr., E. L. 1978, Ann. Math. Logic

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

Howard-Rubin Number Statement
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.

86-alpha:

\(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(|X| = \aleph_{\alpha }\), then \(X\) has a choice function.

196-alpha:

\(\aleph_{\alpha}\) and \(\aleph_{\alpha+1}\) are not both measurable.

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