I'm attempting to unpack the following paragraph from a proof of the Model Existence Theorem (p.44 on this set of notes):$\def\f{\phi}$
Suppose we have a consistent $S$ in the language $L = L(Ω, Π)$. Extend $S$ to a consistent $S_1$ such that $p ∈ S_1$ or $(¬p) ∈ S$ for each sentence $p ∈ L$ (by applying Zorn’s lemma to get a maximal consistent $S_1$). In particular, $S_1$ is complete, meaning $S_1 ⊢ p$ or $S_1 ⊢ ¬p$ for all $p$. Then for each sentence of the form $(∃x)p$ in $S_1$, add a new constant $c$ to $L$ and add $p[c/x]$ to $S_1$ — obtaining $T_1$ in language $L_1 = L(Ω ∪ C1, Π)$. It is easy to check that $T_1$ is consistent. Extend $T_1$ to a complete theory $S_2 ⊆ L_1$, and add witnesses to form $T_2 ⊆ L_2 = L(Ω ∪ C1 ∪ C2, Π)$. Continue inductively. Let $\overline{S} = S_1 ∪ S_2 ∪ · · ·$ in language $\overline{L} = L_1 ∪ L_2 ∪ · · ·$ (i.e. $\overline{L} = L(Ω ∪ C_1 ∪ C_2 ∪ · · · , Π))$.
Here is what I have so far:
Definition: for any set of sentences $S$ in the language $L$, define $$w(S) := S\cup\bigg\{\phi[c/x] : \exists x\phi\in S\bigg\}$$ in the language $w(L)$ consisting of $L$ with enough constants $c$ added (one such constant $c$ per $\exists x\phi$ statement in $S$ is sufficient). Abbreviate $w(w(S))$ as $w^2(S)$ and so on.
Proof: $S$ can be extended to a complete, witnessing, consistent set.
Step 1: $S$ can be extended to a complete, consistent set of sentences $S_1$. We apply Zorn's Lemma on
$$X := \{T\in S : T \text{ is consistent and } T\supseteq S\}.$$
There is some maximal element $S_1\in X$ (with regards to the relation $\subseteq$). To prove this, we use \textbf{Zorn's Lemma}, which in turn requires us to show that any chain has an upper bound: let $\{T_i\}_{i\in I}$ be one such chain, and consider $T=\bigcup_{i\in I} T_i$. Certainy $T\supseteq S$, and if $T$ was inconsistent there would be $\{\phi_1,\ldots,\phi_n\}\in T$ such that $\phi_1,\ldots,\phi_n\vdash\bot$. Then some $T_m$ contains all such $\phi_k$, implying $T_m$ is inconsistent, a contradiction.
$S_1$ is completene: if $S_1$ was incomplete, we could find $\phi\in L$ independent from $S_1$, so that $S_1\cup\{\phi\}$ is consistent and a member of $X$, contradicting the maximality of $S_1$.
$S_1$ is consistent: trivial as $S_1\in X$.
Step 2: add witnesses to $S_1$ extending it to a consistent of sentences $T_1$. Let $$L_1 := \bigcup_{n\in\mathbb{N}}w^n(L) \ \ \ \ \text{ and } \ \ \ \ T_1 := \bigcup_{n\in\mathbb{N}}w^n(S_1).$$
- $T_1$ is consistent: suppose we could show $w$ preserves consistency i.e. if $S$ is consistent, so is $w(S)$ (I do not know how to prove this). Then $w^n(S)$ is consistent for all $n$ by induction. Finally we show, by contradiction, that $T_1$ is consistent. If not, there are $\f_1,\ldots,\f_r$ all in $T_1$ such that $\{\f_1,\ldots,\f_r\}\vdash \bot$. As there is some $w^n(S)$ containing all these $\f_i$, this $w^n(S)$ is also inconsistent, a contradiction.
Step 3: apply Step 1 and Step 2 alternatingly ad infinitum (I currenly do not need help with this).
Am I unpacking the proof correctly?
How can the consistency of $T_1$ be proven?
