Proof. (i) $⟺$ (ii): Obvious.

(ii) $⟹$ (iii): Assume $\phi (\overline{x})$ isolates $p$. Fix an $L_A$-formula $\psi (\overline{x})$. We want to show $M\models \forall \overline{x}(\phi (\overline{x})\to \psi (\overline{x}))$. So suppose $M\models \phi (\overline{a})$. Then ${tp}^M(\overline{a}∕A)\in [\phi (\overline{x})]=\{p\}$.

So ${tp}^M(\overline{a}∕A)=p$, hence $M\models \psi (\overline{a})$.

(iii) $⟹$ (ii): By assumption, for every $L_A$-formula we have $\psi (\overline{x})\in p$, $[\phi (\overline{x})]\subseteq [\psi (\overline{x})]$. Thus if $q\sin [\phi (\overline{x})]$, $a\in [\psi (\overline{x})]$. So $\psi (\overline{x})\in q$, so $p\subseteq q$, so $p=q$. □

Proof. Fix $p\in S_n(T)$, isolated by $\phi (\overline{x})$. Fix $M\models T$.

By Proposition 13.4, there is some $N\succcurlyeq M$ realising $p$.

So $N\models \exists \overline{x}\phi (\overline{x})$, so $M\models \exists \overline{x}\phi (\overline{x})$.

Fix $\overline{a}\in M^n$ such that $M\models \phi (\overline{a})$. Then $\overline{a}\models p$ as for any $\phi (\overline{x})\in p$ we have

So $M\models \phi (\overline{a})$. □

Proof. Let $L^{\ast }=L\cup C$, with $C$ a countably infinite set of new constants.

An $L^{\ast }$-theory has the witness property if for any $L^{\ast }$-formula $\phi (\overline{x})$ there is a constant $c\in C$ such that $T^{\ast }\models \exists x\phi (x)\to \psi (c)$.

Fact: Suppose $T^{\ast }$ is a complete, satisfiable $L^{\ast }$-theory with the witness property.

Define $\sim$ on $C$ such that $c\sim d$ if and only if $T^{\ast }\models c=d$. Let $M=C∕\sim$, and define an $L^{\ast }$-structure on $M$ such that:

• $c^M=[c]$

• $f^M([c_1],\dots ,[c_n])=[d]$ if and only if

  $$T^{\ast }\models f(c_1,\dots ,c_n)=d.$$

• $R^M=\{([c_1],\dots ,[c_n])\in M^n:T^{\ast }\models R(c_1,\dots ,c_n)\}$.

Then $M$ is a well-defined $L^{\ast }$-structure and $M\models T^{\ast }$.

Note we have $M\models \phi ([c_1],\dots ,[c_n])$ if and only if $T^{\ast }\models \phi (c_1,\dots ,c_n)$. We call $M$ the Henkin model of $T^{\ast }$.

Fix $p\in S_n(T)$ non-isolated.

Aim: build a complete, satisfiable $L^{\ast }$-theory $T^{\ast }\supseteq T$, with the witness property, .

Such that for all $c_1,\dots ,c_n\in C$ there is some $\phi (\overline{x})\in p$ such that $T^{\ast }\models \neg \phi (c_1,\dots ,c_n)$. Then the Henkin model of $T^{\ast }$ omits $p$.

Enumerate all the $L^{\ast }$-sentences ${\phi }_0,{\phi }_1,\dots$ and all $c^{(}n=\{{\overline{c}}_1,{\overline{c}}_2,\dots \}$. We build a satisfiable $L^{\ast }$-theory $T\cup \{{𝜃}_1,{𝜃}_2,\dots \}$ such that

• (0) $\models {𝜃}_i\to {𝜃}_j$ for all $i>j$.
• (0) (Completeness): Either $\models {𝜃}_{3i+1}\to {\phi }_i$ or $\models {𝜃}_{3i+1}\to \neg {\phi }_i$.
• (0) (Witnessing property): If ${\phi }_i$ is $\exists v\psi (v)$ for some $\psi$ and $\models {𝜃}_{3i+1}\to {\phi }_i$ then $\models {𝜃}_{3i+2}\to \psi (c)$ for some $c$. (check this does ensure the witness property).
• (0) (Omit $p$): $\models {𝜃}_{3i+3}\to \neg \psi (c_i)$ for some $\psi (\overline{x})\in p$.

Let ${𝜃}_0$ be $\forall v(v=v)$, and suppose we have ${𝜃}_0,\dots ,{𝜃}_m$.

Case 1: $m+1=3i+1$ for some $i$.

If $T\cup \{{𝜃}_m,{\phi }_i\}$ is satisfiable then ${𝜃}_{m+1}={𝜃}_m\wedge {\phi }_i$. Otherwise ${𝜃}_{m+1}={𝜃}_m\wedge \neg \phi$.

So $T\cup \{{𝜃}_{m+1}\}$ is satisfiable by construction.

Case 2: $m+1=3i+2$ for some $i$.

Suppose ${\phi }_i$ is $\exists v,\psi (v)$ for some $\psi$ an $L^{\ast }$-formula, and $\models {𝜃}_i\to {\phi }_i$ (otherwise, let ${𝜃}_{m+1}={𝜃}_m$).

Choose a $c\in C$ not used in ${𝜃}_m$. Let ${𝜃}_{m+1}$ be ${𝜃}_m\wedge \psi (i)$.

Exercise: check $T\cup \{{𝜃}_{m+1}\}$ is satisfiable.

Case 3: $m+1=3i+3$ for some $i$.

Let ${\overline{c}}_i=(c_1,\dots ,c_n)$. Without loss of generality assume $x_1,\dots ,x_n$ not used in ${𝜃}_m$. We build an $L$-formula as follows:

• Replace $c_t$ by $x_t$ ($t\in \{1,\dots ,n\}$).

• Replace any $c\in C\setminus \{c_1,\dots ,c_n\}$ by new variables $v_c$ and add $\exists v_c$ to the front.

Then $\phi (\overline{x})$ doesn’t isolated $p$.

By Proposition 16.2, there is some $\psi (\overline{x})\in p$ with

Let ${𝜃}_{m+1}$ be ${𝜃}_m\wedge \neg \psi (c_1,\dots ,c_n)$. Check ${𝜃}_{m+1}$ is satisfiable.

TODO □