Why is the symbol ∈ classified as a binary relation in first order set theory?

Question

Aren't binary relations themselves defined using the symbol ∈ ? Also, what are the rules for writing down statements in first order set theory? If someone could point me towards a simple text, that'd be great.

Thanks.

Answer 1

You are dealing with two essentially different uses of "binary relation."

1. The meaning that is applied to primitive binary predicates in any first order theory.

2. The definition, inside of the first order theory of sets, of a binary relation, as a subset of a product of two sets.

These are two different things, but they are closely related, semantically.

The first can be applied to any first order theory. For example, the first order theory of ordered fields has a binary relation, $\leq$, but that same theory has no "internal" notion of binary relation.

Also, the $\in$ binary relation has no internal representation as a binary relation (usually.) It is not necessarily true that $\exists R:\forall x,y\,(x\in y\iff (x,y)\in R$). That is $\in$ is not a binary relation of the type (2).

There is a third definition of "binary relation" which is an expansion of (1), namely "binary predicates:"

3. A binary relation is a predicate in a first order theory of the form $P(x,y)$ which has no other unbound variables.

For example, $\subseteq$ in the theory of sets is a binary relation of this sort, because it can be written as:

$$\forall z\,(z\in x\implies z\in y)$$

As with $\in$, this binary relation cannot be expressed internally as a binary relation of type (2).