Let Γ=|σ1, σ2, σ3| be a set of 𝒮 𝒞 sentences. For i, j, k ∈N a t, define a sequence of sets of 𝒮

First, let's understand the definition of the sequence $Gamma_k$. It starts with $Gamma_0 = Gamm

Let Γ=|σ1, σ2, σ3| be a set of 𝒮 𝒞 sentences. For i, j, k ∈N...

Let $\Gamma=\left|\sigma_1, \sigma_2, \sigma_3\right|$ be a set of $\mathcal{S C}$ sentences. For $i, j, k \in N a t$, define a sequence of sets of $\mathcal{S C}$ sentences as follows: $$ \Gamma_0=\Gamma \text {, } $$ and, for any $k>0$, $\Gamma_k=\left\{(\phi \wedge \psi) \mid\right.$ there exist $i, j<k$ such that $\phi \in \Gamma_i \& \psi \in \Gamma_j \mid \cup$ $\left\{(\phi \vee \psi) \mid\right.$ there exist $i, j<k$ such that $\left.\phi \in \Gamma_i \& \psi \in \Gamma_j\right\}$. Now let $\Delta=\cup\left\{\Gamma_k \mid k \in N a t\right\}$. Then any interpretation I that satisfies $\Gamma$ also satisfies $\Delta$. [Hint. If $\chi \in \Delta$, then it must be in $\Gamma_m$ for some $m$. Why? Now use CVI to prove that I satisfies $\Gamma_k$, for any $k \in N a t$.]

Let $\Gamma=\left|\sigma_1, \sigma_2, \sigma_3\right|$ be a set of $\mathcal{S C}$ sentences. For $i, j, k \in N a t$, define a sequence of sets of $\mathcal{S C}$ sentences as follows:
$$
\Gamma_0=\Gamma \text {, }
$$
and, for any $k>0$,
$\Gamma_k=\left\{(\phi \wedge \psi) \mid\right.$ there exist $i, j<k$ such that $\phi \in \Gamma_i \& \psi \in \Gamma_j \mid \cup$
$\left\{(\phi \vee \psi) \mid\right.$ there exist $i, j<k$ such that $\left.\phi \in \Gamma_i \& \psi \in \Gamma_j\right\}$.
Now let $\Delta=\cup\left\{\Gamma_k \mid k \in N a t\right\}$. Then any interpretation I that satisfies $\Gamma$ also satisfies $\Delta$. [Hint. If $\chi \in \Delta$, then it must be in $\Gamma_m$ for some $m$. Why? Now use CVI to prove that I satisfies $\Gamma_k$, for any $k \in N a t$.]

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