First order theory - define a finite domain

Question

I'm trying to write a first-order theory that defines a domain containing a maximum of 2 elements. Up to this point, I've only been working with theories that use functions, constants, and relations, so I'm having a tough time wrapping my head around one that (supposedly) shouldn't use any of these.

What I've come up with so far is the following: $$∃x\ (x=x) ∨ ∃x\ ∃y \ ((x=y) ∨ (¬x=y))$$

So, there exists some x such that it simply equals itself OR there exists some $x$ and some $y$ such that they are the same or not the same. This should account for a domain that has one element and a domain that has two elements. However, I have no idea how to explicitly account for the case in which the domain is empty.

Is this at all on the right track? This is pretty new to me and most online resources are far more complex than anything I've seen so far.

Thanks in advance!

Answer 1

Your sentence basically boils down to asserting that the domain is non-empty: both conjuncts simply ensure the existence of at least one element. You want to say something very different: that the domain contains at most two elements, not that it contains at least one element.

HINT: You can do it with a single sentence of the form $\exists x\,\exists y\,\forall z\,(\text{something})$. Can you see what the $\text{something}$ should be?

Answer 2

Outline: For your "at most $2$" it is enough to say that for all $x$, $y$, and $z$ we have $x=y$ or $x=z$ or $y=z$.

By the way, it is a standard convention in Model Theory that empty domains are not allowed. However, even if we do allow such a thing, the above suggestion will work.