[275]Consider again Proposition [26J] that states that \(ℕ_\text {ZF}\) is well ordered by the relation \(⊆\).
We know by [255] and [257] that \(ℕ_\text {ZF}\) is an ordinal; we may be tempted to see Proposition [26J] as a corollary of the previous result [26V].
This is unfortunately not a well posed way of proving this result, because of this cascade of dependencies:
the result [263] in turn needs a definition by recurrence of a function: this is Theorem [08Z]
the proof of Theorem [08Z] uses the fact that the induction principle holds on \(ℕ\).
So we need to first prove the properties of \(ℕ_\text {ZF}\) independently of the theory of ordinals, and then prove the results in Sec. [1X9], and then eventually we can prove the result [26V], that states that any ordinal is well ordered by the relation \(⊆\).