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Also, it was shown recently by Omar de la Cruz that $\Delta_3$-finite
and III-finite are independent, neither implies the other in ZF. In
$\Cal N3$, Mostowski's linearly ordered model, $\Cal P(A)$, where $A$
is the set of atoms, is Dedekind finite, but can be linearly ordered.
Therefore, $\Cal P(A)$ is III-finite, but not $\Delta_3$-finite.\par
On the other hand, in Fraenkel's Basic Model, $\Cal N1$, the
set of all finite subsets of A, is III-infinite,
but it contains no infinite linearly ordered subset. For suppose
that $L$ is such an infinite linearly ordered subset and $E$ is a
support for $L$ and its linear order $R$. Then, there exists
$n\in\omega$ such that $L'=\{x\in L: |x|=n\}$ is infinite; otherwise
we can well order $L$ by
$$ x