Axiom Of Choice

We have the following indirect implication of form equivalence classes:

112 \(\Rightarrow\) 396 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
112 \(\Rightarrow\) 395	clear
395 \(\Rightarrow\) 396	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 \(y\).
395:	\(MC(LO,LO)\): For each linearly ordered family of non-empty linearly orderable sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is a non-empty, finite subset of \(x\).
396:	\(MC(LO,WO)\): For each linearly ordered family of non-empty well orderable sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is a non-empty, finite subset of \(x\).

Comment:

Back