# 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 ${\displaystyle {\mathcal {L}}}$, is the language of first-order logic with the symbol ${\displaystyle \in }$.

Our alphabet includes variable symbols ${\displaystyle v,w,\ldots }$, and the symbols ${\displaystyle \vee \;\wedge \;\Rightarrow \;\Leftrightarrow \;\neg \;\forall \;\exists \;\bot \;=\;\in }$.

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

Our syntax is (informally) described by the following

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