# Set Theory/The Language of Set Theory

Recall that a *language* consists of an *alphabet* (i.e., a collection of symbols), a *syntax* (i.e., rules to form formulas), and *semantics* (i.e., the interpretation of the formulas).

The *Language of Set Theory*, denoted as , is the language of first-order logic with the symbol .

Our *alphabet* includes variable symbols , and the symbols
.

We will not worry too much about the formal semantics in this book; however, our intended interpretation of the symbol is as a set membership relation, i.e., means set is a member of set .

Our *syntax* is (informally) described by the following

- if and are variable symbols, then and are formulas
- if is a formula, then so is
- if and are formulas, then so are , , , and
- if is a formula and is a variable symbol, then and are formulas
- finally, we have is also a formula

Note that to formally define the syntax, we need to use the notion of `recursion'. However, recursion is soon to be defined within the theory (**ZF** theory), so we will refrain from using theorems in **ZF** as meta-theorems for **ZF**.

Also note that we quantify over the *universal set*, i.e., the set of all sets. (**Fun fact**: the universal set is not a set, *ipso facto* by two axioms we will soon see.)