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.)