Implications

We have the following indirect implication of form equivalence classes:

419 \(\Rightarrow\) 108
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
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 \(\Rightarrow\) 38 The Axiom of Choice, Jech, [1973b]
38 \(\Rightarrow\) 108 clear

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

Howard-Rubin Number Statement
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.

38:

\({\Bbb R}\) is not the union of a countable family of countable sets.

108:

There is an ordinal \(\alpha\) such that \(2^{\aleph _{\alpha}}\) is not the union of a denumerable set of denumerable sets.

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