Axiom Of Choice

We have the following indirect implication of form equivalence classes:

147 \(\Rightarrow\) 211 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
147 \(\Rightarrow\) 91	The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic note-26
91 \(\Rightarrow\) 79	clear
79 \(\Rightarrow\) 203	clear
203 \(\Rightarrow\) 211	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
147:	\(A(D2)\): Every \(T_2\) topological space \((X,T)\) can be covered by a well ordered family of discrete sets.
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.
211:	\(DCR\): Dependent choice for relations on \(\Bbb R\): If \(R\subseteq\Bbb R\times\Bbb R\) satisfies \((\forall x\in \Bbb R)(\exists y\in\Bbb R)(x\mathrel R y)\) then there is a sequence \(\langle x(n): n\in\omega\rangle\) of real numbers such that \((\forall n\in\omega)(x(n)\mathrel R x(n+1))\).

Comment:

Back