Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
418 \(\Rightarrow\) 419 Metric spaces and the axiom of choice, De-la-Cruz-Hall-Howard-Keremedis-Rubin-2002A[2002A], Math. Logic Quart.
419 \(\Rightarrow\) 420 Metric spaces and the axiom of choice, De-la-Cruz-Hall-Howard-Keremedis-Rubin-2002A[2002A], Math. Logic Quart.
420 \(\Rightarrow\) 34 Metric spaces and the axiom of choice, De-la-Cruz-Hall-Howard-Keremedis-Rubin-2002A[2002A], Math. Logic Quart.
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
418:

DUM(\(\aleph_0\)): The countable disjoint union of metrizable spaces is metrizable.

419:

UT(\(\aleph_0\),cuf,cuf): The union of a denumerable set of cuf sets is cuf. (A set is cuf if it is a countable union of finite sets.)

420:

\(UT(\aleph_0\),\(\aleph_0\),cuf): The union of a denumerable set of denumerable sets is cuf.

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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