Axiom Of Choice

We have the following indirect implication of form equivalence classes:

393 \(\Rightarrow\) 38 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
393 \(\Rightarrow\) 165	clear
165 \(\Rightarrow\) 32	clear
32 \(\Rightarrow\) 5	clear
5 \(\Rightarrow\) 38	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.

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

Howard-Rubin Number	Statement
393:	\(C(LO,WO)\): Every linearly ordered set of non-empty well orderable sets has a choice function.
165:	\(C(WO,WO)\): Every well ordered family of non-empty, well orderable sets has a choice function.
32:	\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets has a choice function.
5:	\(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function.
38:	\({\Bbb R}\) is not the union of a countable family of countable sets.

Comment:

Back