Question about passage in Halbeisen's book I am looking at the following passage in Halbeisen's book "Combinatorial Set Theory" (p 260 at the bottom): 

What is the role of $\Phi$? It seems to me that a finite fragment is the same as a finite subset of axioms. Then the sentence saying that for a finite subset $\Phi \subset ZFC$ there is a finite subset $ZFC^\ast \subset ZFC$ seems to contain redundancy. What am I missing? Or is this a "writo"? Many thanks for your help.
 A: (This is basically exactly what Brian mentioned in his comment, but in way more than 600 characters.)
Note that if $\mathsf{ZFC}$ refutes the sentence $\varphi$, then there must be a finite fragment $\Phi$ of $\mathsf{ZFC}$ which refutes $\varphi$.  So what we are aiming for is to show that (relative to the consistency of $\mathsf{ZFC}$) this cannot happen.
So we begin with a finite fragment $\Phi$ of $\mathsf{ZFC}$, and we have in mind a forcing notion that should produce generic extensions satisfying $\Phi + \varphi$.  Unfortunately, even demonstrating that the desired forcing notion $\mathbb{P}$ is an element of an arbitrary set model $\mathsf{M}$ of $\Phi$ might require axioms not in $\Phi$.  Furthermore, the demonstration that the generic extension satisfies $\Phi + \varphi$ might also require axioms of $\mathsf{ZFC}$ not in $\Phi$ (because we will have to construct the required names, which will in all likelihood require, for example, instances of Replacement not in $\Phi$).
We must then analyse exactly what we need so that the above process can be carried out, and get a suitable finite fragment $\mathsf{ZFC}^*$ of $\mathsf{ZFC}$ such that if you begin with a set model $M$ of $\mathsf{ZFC}^*$ the forcing notion $\mathbb{P}$ is an element of $\mathsf{M}$, and, moreover, constructing a generic extension $\mathsf{M}[X]$ results in a model of $\Phi + \varphi$.  This then shows (relative to the consistency of $\mathsf{ZFC}$) that the finite fragment $\Phi$ cannot refute $\varphi$.
This analysis can be carried out for any finite fragment $\Phi$ of $\mathsf{ZFC}$, leading to an appropriate finite fragment $\mathsf{ZFC}^*$ so that the above works.  In this manner we can demonstrate the relative consistency of $\varphi$ with $\mathsf{ZFC}$.
Note that there are many relative consistency results that begin not with finite fragments of $\mathsf{ZFC}$, but rather of stronger theories, such as $\mathsf{ZFC} + \exists \text{ inaccessibles}$.  The above description, mutatis mutandis, will handle those cases as well.
(But in practice we don't tend to worry about the particulars, and think of forcing over models of $\mathsf{ZF(C)}$ -- or stronger theories.  Even more, (and perhaps far more often than the formalist in me would like to admit) we generally think of forcing over the entire von Neumann universe $\mathsf{V}$.)
A: Reading pages 260 and 261, I believe that Halbeisen's idea is to workaround the limitation of having definable models (sets) for only finite part of ZFC.
The role of $\Phi$ is to allow us (by using the Compactness theorem) to move from a finite sub-theory $T$ of ZFC, to the whole theory.
(When using forcing you analyze "metamathematics" within mathematics. You assume some countable model $M$ for a specific finite list of axiom (the exact list of axioms is built during your forcing construction) and you show how $M$ can be "extended" to provide the desired new "model" (a new set). )
The only problem I have with Halbeisen's approach is that the Compactness theorem 3.7 was proved for meta-mathematical objects (real formulas etc.) not with their mathematical counterparts (Gödel encoding of formulas, proofs etc.), and there needs to be at least a note saying that the same proof can be used with the encoded objects.
