I have a question about Godel's Constructible Universe. I think the best way to ask the question is to refer directly to the book that I'm using: Set Theory, Introduction To Independence Proofs by Kenneth Kunen.

My question is about the proof of 1.9 Lemma on p167.

In the second paragraph of the proof it is stated that "x is an ordinal" is expressible as a _delta_0 formula and is therefore absolute for transitive classes.

My problem is that, in Chapter IV 5.1, all that is proved is that "x is an ordinal" is ZF-P equivalent to a _delta_0 formula. So surely the statement is only absolute for transitive models of ZF-P? So then how are we able to apply absoluteness in this case? L_(_alpha) is not a model of ZF-P, so the statement is not necessarily absolute for L_(_alpha)?

Any help would be greatly appreciated!