Implications

We have the following indirect implication of form equivalence classes:

67 \(\Rightarrow\) 125
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
67 \(\Rightarrow\) 144 Axioms of multiple choice, Levy, A. 1962, Fund. Math.
144 \(\Rightarrow\) 125 P-Raüme and Auswahlaxiom, Brunner, N. 1984c, Rend. Circ. Mat. Palermo.

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

Howard-Rubin Number Statement
67:

\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).

144:

Every set is almost well orderable.

125:

There does not exist an infinite, compact connected \(p\) space. (A \(p\) space is a \(T_2\) space in which the intersection of any well orderable family of open sets is open.)

Comment:

Back