I'm trying to write a first-order theory that defines a domain of even size (containing an even number of elements). I'm told that I can use a single binary predicate symbol, R, that can be defined as I see fit. I can also assume that infinite sets are "even size".
What I've come up with is the following:
$$∀x\ ∀y\ ∃z\ (R(x,y) ↔ R(z,y))$$
where R(x,y) means x is less than y.
However, after looking at this more closely, it seems that all I'm saying is that each element has an immediate successor. How can I modify this to explicitly say that for every element, there is exactly one element following it? I'm struggling to see how I can state that the pattern continues on the element after that, rather than linking all elements in one chain.
Any help would be appreciated. Thanks!
HINT: Write a sentence that says that $R$ satisfies the following condition: $R$ is symmetric, and for each $x$ there is a unique $y\ne x$ such that $R(x,y)$. (And of course explain why this does what you want.)