Description:
Form 9 (Dedekind finite = finite) and [9 D] (The union of a Dedekind finite family of finite sets is Dedekind finite)
are equivalent.
Content:
Form 9 (Dedekind finite = finite) and [9 D] (The union of a Dedekind finite family of finite sets is Dedekind finite)
are equivalent.
It is clear that Form 9 implies [9 D]. Since Form 9 and
[9 A] are equivalent (see note 8)) it suffices to show that [9 D]
finite such that \(\cal P(x)\) is Dedekind infinite. Then there is a one to one function \(f :\omega\rightarrow \cal P (x)\) and, as in note 11, we may assume that \((\forall i,j\in\omega)(i\neq j\) implies \(f(i) \cap f(j)=\emptyset)\). It is also no loss of generality
to assume \((\forall i\in\omega)(f(i)\neq\emptyset)\). Let
\[W =\left\{\{i,t\}: i\in\omega\wedge t\in f(i)\right\}.\]
Then \(W\) is Dedekind finite since \(x\) is. But \(\omega
\subseteq\bigcup W\), a contradiction.
(See Jech [1973b]
p 161 prob 11.4.)
Howard-Rubin number:
12
Type:
proof
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