Need help rigorously stating the claim in my proof (lemma about models concerning the pure theory of equality)

Question

Trying to solve an exercise, I've come with a possible lemma that would furnish the result:

If a finite set of formulas $\Gamma$ in the language of the pure theory of equality has a countable normal model $M$, then it has a finite normal model.

(A normal model is a model in which $=$ is an identity relation over domain.)

And that's how I proved it:

Let $x_{i_1},\dots,x_{i_n}$ be the terms (all of which are variables) occuring in all the formulas of $\Gamma$. Take any $t_1,\dots,t_n$ distinct objects from the domain of $M$. Any sequence that places $t_1,\dots,t_n$ under $x_{i_1},\dots,x_{i_n}$ in any manner satisfies each $\mathscr C\in\Gamma$. We are not required to have any more objects, since the language only expresses equality between $x_{i_1},\dots,x_{i_n}$. So, let $\{t_1,\dots,t_n\}$ be a domain of $M'$ and propose an identity relation over them for $=$. All sequences of $M'$ must satisfy each $\mathscr C\in\Gamma$, thus it is a finite normal model for $\Gamma$.

Marked steps are clearly hand-wavy and I can't come up with a rigorous justification, yet they follow intuitively. The worst-case scenario is clearly when some subformula would express the existence of distinct objects, yet it could not express that there are more than $n$ of them. Any ideas?

Answer 1

I don't think the proof idea you sketched can be made to work. Instead, try this:

1. Show that any two infinite sets are elementarily equivalent in the language of equality.
2. Now assume for contradiction that some sentence $\varphi$ has infinite normal models but no finite model. If $\theta_n$ asserts "there are at least $n$ distinct elements", show that $\{\lnot\varphi\}\cup\{\theta_n\mid n\in \mathbb{N}\}$ is inconsistent (using 1).
3. Apply the compactness theorem to find a natural number $N$ such that $\lnot\varphi\land \theta_N$ is inconsistent.
4. Conclude not just that $\varphi$ has a finite normal model, but that in fact every set of size at least $N$ is a model of $\varphi$.