Description:
Form 10 is equivalent to [10 F]
Content:
Form 10 (\(C(\aleph_0,<\aleph_0,\aleph_0)\)) is equivalent to [10 F] (Every \(\omega \)-tree has an infinite chain.) The proof that Form 10 implies [10 F] uses the fact that Form 10 is equivalent to \(UT(\aleph_0,<\aleph_0,\aleph_0)\) which implies an \(\omega\)-tree is denumerable. For the argument that [10 F] implies Form 10, let \(A\) be a denumerable set of finite sets and \(T\) be the set of all choice functions on initial segments of \(A\). Define \(<\) on \(T\) by \(f < g\) iff \(f\) is a proper subset of \(g\). \(T\) is an \(\omega\)-tree and an infinite chain in \(T\) induces a choice function on \(A\). This proof is due to Arthur and Jean Rubin. A proof of the equivalence of Form 10 and [10 F] also occurs in Pelc [1978].
Howard-Rubin number: 35
Type: Equivalents
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