Description:
Form 269 is false in the first Fraenkel model \(\cal N1\) of \(ZF^{0}\)
Content:
In this note we argue that
Form 269 is false in the first Fraenkel model \(\cal N1\) of ZF\(^0\).
Form 269 is: "For every cardinal \(m\), there is a set \(A\) such that \(2^{|A|^2} \ge m\) and there is a choice function on the collection of 2 element subsets of \(A\)." It suffices to showthat for any set \(A\) in \(\cal N1\) if the set \([A]^{2}\) of two element subset of \(A\) has a choice function, then \(A\) is well orderable in \(\cal N1\). Assume that \(A\) is such a set and assume that \(E\) is a (finite) support of \(A\) and a choice function \(f\) on \([A]^{2}\). Assume \(x \in A\). We show by contradiction that \(E\) must be a support of \(x\). Suppose not and let \(F\) be a support of \(x\) containing \(E\). We claim there is a permutation \(\psi \) in fix\(_{G}(E)\) and an element \(y \in A\) such that \(x \neq y, \psi (x) = y\) and \(\psi (y) = x\). This would contradict our choice of \(E\) as a support for a choice function on\([A]^{2}\) since \(\psi \) fixes \(\{x,y\}\) but moves both of its elements. To get such a \(\psi \), let \(\phi \in \) fix\(_{G}(E)\) satisfy \(\phi (x)\neq x\) and \(\phi \) moves only finitely many atoms. Assume \(F - E = \{ t_{1},\ldots ,t_{n}\}\) and choose a set of atoms \(\{ s_{1}, \ldots , s_{n} \}\) disjoint from \(F \cup\{t \in U:\phi(t)\neq t\}\). Let \(\psi \) be the product of transpositions \(\prod_{i=1}^n (t_{i},s_{i})\). The element \(\psi(x)\) is different from \(x\) since \(\phi(x)\neq x\) and \(\phi(\psi(x))= \psi (x)\). It is also clear that if \(y = \psi (x)\), then \(\psi (y)= x\).
Howard-Rubin number:
91
Type:
Theorem
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