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 $ℒ$, is the language of first-order logic with the symbol $\in$.

Our alphabet includes variable symbols $v,w,\dots$, and the symbols $\vee \wedge \Rightarrow \iff \mathrm{\neg }\mathrm{\forall }\mathrm{\exists }\mathrm{\perp }=\in$.

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

Our syntax is (informally) described by the following

• if $x$ and $y$ are variable symbols, then $(x=y)$ and $(x\in y)$ are formulas
• if $\phi$ is a formula, then so is $(\mathrm{\neg }\phi )$
• if $\phi$ and $\psi$ are formulas, then so are $(\phi \iff \psi )$, $(\phi \Rightarrow \psi )$, $(\phi \wedge \psi )$, and $(\phi \vee \psi )$
• if $\phi$ is a formula and $x$ is a variable symbol, then $\mathrm{\forall }x:\phi$ and $\mathrm{\exists }x:\phi$ are formulas
• finally, we have $\mathrm{\perp }$ 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.)