We have the following indirect implication of form equivalence classes:

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

Implication Reference
112 \(\Rightarrow\) 90 Equivalents of the Axiom of Choice II, Rubin/Rubin, 1985, page 79
90 \(\Rightarrow\) 51 Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
51 \(\Rightarrow\) 25 Choice and cofinal well-ordered subsets, Morris, D.B. 1969, Notices Amer. Math. Soc.
25 \(\Rightarrow\) 34 clear
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
112:

\(MC(\infty,LO)\): For every family \(X\) of non-empty sets each of which can be linearly ordered there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).

90:

\(LW\):  Every linearly ordered set can be well ordered. Jech [1973b], p 133.

51:

Cofinality Principle: Every linear ordering has a cofinal sub well ordering.  Sierpi\'nski [1918], p 117.

25:

\(\aleph _{\beta +1}\) is regular for all ordinals \(\beta\).

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.

Comment:

Back