Description:
\(C(\infty,2)\) (form 88) is false in \(\cal N10\).
Content:
\(C(\infty,2)\) (form 88) is false in \(\cal N10\).
Let \(C\) be the set of all order preserving
functions \(f:\omega \to A\) such that \(\hbox{rng}(f)\) has an upper bound
in \(A\) but no least upper bound in \(A\). This means that rng\((f)\) is
a support and therefore \(f\) is in \(\cal N10\).
\(C\) is in \(\cal N10\) since
it has empty support. Now define an equivalence relation \(\sim\) on \(C\)
by \(f\sim g \leftrightarrow (\exists n\in \omega)(f(k) = g(k+n)\) for all
\(k\in\omega\) or \(g(k) = f(k+n)\) for all \(k\in\omega)\). Let \([f]_\sim\)
denote the \(\sim\) equivalence class of \(f\) and let \(X\) be the set of
all pairs of elements of \(\left\{[f]_\sim: f\in C\right\}\). The set
\(X\) has no choice function in \(\cal N10\) for suppose that \(E\) is a
support of such a choice function. Choose a sequence \(\langle a_n \rangle_
{n\in\omega}\) which has order type \(\omega\) in the ordering on \(A\) and
which lies in one of the intervals determined by \(E\) in the ordering on
\(A\). There is a permutation \(\psi\in\cal G\) which fixes \(E\)
pointwise and which satisfies \((\forall n\in\omega)(\psi(a_n)=a_{n+1})\).
Therefore, if we define \(f:\omega\to A\) by \(f(n) = a_{2n}\) and \(g:
\omega\to A\) by \(g(n)=a_{2n + 1}\), then \(\psi([f]_\sim) = [g]_\sim\)
and \(\psi([g]_\sim)= [f]_\sim\). Therefore, no choice function with
support \(E\) for a set containing \(\{ [f]_\sim, [g]_\sim \}\) exists.
Howard-Rubin number:
113
Type:
proof of result
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