I am trying to understand the use of the Axiom of Choice in the proof of Lemma 2.1.8 (page 37 below) in David Marker's Model Theory: An Introduction
Pages 35 and 36 can be found here. Lemma 2.1.8 is part of a series of lemmas that begin on page 35 culminating in the proof of Theorem 2.1.11 (page 38). I identify two fundamental uses of Choice in the proof of this theorem:
One clear in Corollary 2.1.10 (page 38), although Zorn's Lemma can be avoided by using the Ultrafilter theorem. Then two apparent in the proof of Lemma 2.1.8. The first in choosing the model $\mathcal{M}$ satisfying $\Delta_0$, and the second in the assignment of constant symbols to elements in the domain $M$ of $\mathcal{M}$. It is these last apparent uses of Choice that I am trying to understand:
Is it full Choice being used here? Can it be avoided?
I though that the strength of Choice needed to prove Theorem 2.1.11 (Compactness Theorem) is only that of the ultrafilter theorem, which is used in Corollary 2.1.10.
Thank you in advance.
Note that in line 3, p.38 below there appears to be a typo. Where $\phi$ appears it should be $\phi_i$.
