Implications

We have the following indirect implication of form equivalence classes:

264 \(\Rightarrow\) 306
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
264 \(\Rightarrow\) 202 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
202 \(\Rightarrow\) 91 note-75
91 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 203 clear
203 \(\Rightarrow\) 306 clear

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

Howard-Rubin Number Statement
264:

\(H(C,P)\): Every connected relation \((X,R)\) contains a \(\subseteq\)-maximal partially ordered set.

202:

\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function.

91:

\(PW\):  The power set of a well ordered set can be well ordered.

79:

\({\Bbb R}\) can be well ordered.  Hilbert [1900], p 263.

203:

\(C\)(disjoint,\(\subseteq\Bbb R)\): Every partition of \({\cal P}(\omega)\) into non-empty subsets has a choice function.

306:

The set of Vitali equivalence classes is linearly orderable. (Vitali equivalence classes are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow (\exists q\in{\Bbb Q})(x-y = q)\).).

Comment:

Back