Implications

We have the following indirect implication of form equivalence classes:

27 \(\Rightarrow\) 182
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
27 \(\Rightarrow\) 31 clear
31 \(\Rightarrow\) 34 clear
34 \(\Rightarrow\) 104 clear
104 \(\Rightarrow\) 182 clear

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

Howard-Rubin Number Statement
27:

\((\forall \alpha)( UT(\aleph_{0},\aleph_{\alpha}, \aleph_{\alpha}))\): The  union of denumerably many sets each of power \(\aleph_{\alpha }\) has power \(\aleph_{\alpha}\). Moore, G. [1982], p 36.

31:

\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem:  The union of a denumerable set of denumerable sets is denumerable.

34:

\(\aleph_{1}\) is regular.

104:

There is a regular uncountable aleph. Jech [1966b], p 165 prob 11.26.

182:

There is an aleph whose cofinality is greater than \(\aleph_{0}\).

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