Notes

Description:

Form 31 (\(UT(\aleph^0,\aleph^0,\aleph^0)\)) does not imply Form 9 (Dedekind finite = finite).

Content:

Either \(\cal N1\) or \(\cal N3\) may be used to prove that Form 31 (\(UT(\aleph^0,\aleph^0,\aleph^0)\)) does not imply Form 9 (Dedekind finite = finite). Form 9 is known to be false in these models and a proof that Form 31 holds is given below. The result is transferable to \(ZF\) by Pincus [1972a]. Form 31 is true in both \(\cal N1\) and \(\cal N3\). Let \(\cal N\) be either \(\cal N1\) or \(\cal N3\) with set of atoms \(A\) and \(\cal G\) the group of permutations used in defining the model.  Assume that in \(\cal N\), \(B\) is a denumerable set of denumerable sets.  Let \(e\subseteq A\) be a finite support for every \(b \in B\).  Such an \(e\) exists since \(B\) is well orderable in \(\cal N\).  Assume \(b \in B\) and \(x\in b\).  The proof will be completed by showing that \(e\) is a support of \(x\).  We argue by contradiction.  Assume that \(e\) is not a support of \(x\).  We will show that under this assumption \(b\) has a subset of the same cardinality as an infinite subset of \(A\) which contradicts the countability of \(b\).
Let \(e'\supseteq e\) be a support of \(x\).  Then there is a \(t \in e'\setminus e\) and a \(\phi \in \cal G\) which fixes \(e' \setminus \{ t\}\) pointwise and such that \(\phi(x) \neq x\).  (It follows that \(\phi(t)\neq t\).)  It can be verified that \[\left\{\langle \psi(t),\psi(x) \rangle: \psi \in \cal G \mathrm{\; and \;} \psi \mathrm{\; fixes\;} e' \setminus \{ t \} \mathrm{\; pointwise\;} \right\}\] is a one to one function from an infinite subset of \(A\) into \(b\) which is in \(\cal N\).

Howard-Rubin number: 88

Type: Theorem

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