Description:
A proof that dependent choice (Form 43) holds
in \(FM\) models of \(ZF^{0}\) in which the set of supports \(S\) is closed under
countable unions.
Content:
Dependent choice (Form 43) holds
in \(FM\) models of \(ZF^{0}\) in which the set of supports \(S\) is closed under
countable unions.
Note: This includes models where \(S\) consists of
all countable subsets of the set \(A\) of atoms.
Assume that \(\cal N\) is
such a model and that \(R\) is a relation in \(\cal N\) such that for all
\(x \in \hbox{dom}(R)\) there is a \(y\) such that \(x \mathrel R y\). Then, in
the ground model (where \(AC\) holds) there is a sequence
\((x_n)_{n \in \omega}\) such that
\(( \forall n \in \omega)(x_{n} \mathrel R x_{n+1})\). If for each \(n \in \omega\),
\(x_n\) has support \(E_n\), then \(\bigcup_{n \in \omega} E_n\) is a support for
\((x_n)_{n \in \omega}\). Hence, this sequence is in \(\cal N\).
Howard-Rubin number:
144
Type:
proof of result
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