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.)

Introduction · Zermelo-Fraenkel (ZF) Axioms