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

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

We will not worry too much about the formal semantics in this book; however, our intended interpretation of the symbol $\in$ 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 $\varphi$ is a formula, then so is $(\neg \varphi )$
• if $\varphi$ and $\psi$ are formulas, then so are $(\varphi \Leftrightarrow \psi )$, $(\varphi \Rightarrow \psi )$, $(\varphi \wedge \psi )$, and $(\varphi \vee \psi )$
• if $\varphi$ is a formula and $x$ is a variable symbol, then $\forall x:\varphi$ and $\exists x:\varphi$ are formulas
• finally, we have $\bot$ 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.)