Axiom Of Choice

We have the following indirect implication of form equivalence classes:

99 \(\Rightarrow\) 93 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
99 \(\Rightarrow\) 70	Handbook of Analysis and its Applications, Schechter, [1996a]
70 \(\Rightarrow\) 93	The Axiom of Choice, Jech, 1973b, page 7 problem 10

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

Howard-Rubin Number	Statement
99:	Rado's Selection Lemma: Let \(\{K(\lambda): \lambda \in\Lambda\}\) be a family of finite subsets (of \(X\)) and suppose for each finite \(S\subseteq\Lambda\) there is a function \(\gamma(S): S \rightarrow X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))\). Then there is an \(f: \Lambda\rightarrow X\) such that for every finite \(S\subseteq\Lambda\) there is a finite \(T\) such that \(S\subseteq T\subseteq\Lambda\) and such that \(f\) and \(\gamma (T)\) agree on S.
70:	There is a non-trivial ultrafilter on \(\omega\). Jech [1973b], prob 5.24.
93:	There is a non-measurable subset of \({\Bbb R}\).

Comment:

Back