Completeness Theorem (Version 1)

Theorem

Suppose $\Sigma$ is a set of sentences in language $\mathcal{L}$, and $\Sigma /⊢\perp$. Then $\Sigma$ has a model.

Step 1 Let $\mathcal{C}=\{c_1,c_2,c_3\dots \}$ a countably infinite set of constant symbols not in $\mathcal{L}$. Introduce a language ${\mathcal{L}}^{\ast }=\mathcal{L}\cup C$ which extends $\mathcal{L}$. Step 2 Extend $\Sigma$ to ${\Sigma }^{\ast }$ with C a Set of Henkin Witnesses

Step 4 An equivalence relation on $\mathcal{C}$ by $c\sim c^{\prime }⟺{\Sigma }^{\ast }⊢c=c^{\prime }$

Step 5

${\Sigma }^{\ast }⊢d=c$. We have $⊢d=d$ because $⊢u=u⟹⊢\forall u(u=u)⟹d=d⟹\exists v(v=d)$ But Henkin Witeness implies $⊢\exists v(v=d)\to c=d$ for some $c$. ${\Sigma }^{\ast }⊢c=d$ by Modus Ponens. Define $d^{{\mathfrak{A}}^{\ast }}=[c]$ where $c$ (iff ${\Sigma }^{\ast }⊢d=c$) Suppose $f$ is a function symbol in $\mathcal{L}$. How to interpret $f^{{\mathfrak{A}}^{\ast }}([c_1],\dots ,[c_n])=[c]$ Suppose $c_1\dots c_n\in \mathcal{C}$ We want to show ${\Sigma }^{\ast }⊢f(c_1,\dots c_n)=c$ for some $c\in \mathcal{C}$. We claim ${\Sigma }^{\ast }⊢\exists v(f(c_1,\dots ,c_n)=v)$ We have $⊢f(c_1,\dots ,c_n)=f(c_1,\dots ,c_n)$ This gives $⊢\exists v(f(c_1,\dots c_n)=v)$ ${\Sigma }^{\ast }⊢\exists v(f(c_1\dots ,c_n)=v)\to f(c_1,\dots ,c_n)=c$ for some $c$ by Henkin.