Relationship between FOL and ZFC

Now that we have an idea of what first-order logic is, how can we use it to build set theory? We must extend our logic by adding the axioms of ZFC, and the special symbols use in those axioms. We call this extension a theory, and since we’re working in first-order logic, we’ll call it a first-order theory.

Definition

A first-order theory consist of

• A signature specifying the non-logical symbols we’ll use; and

• A collection of sentences known as our axioms

Signature and Axioms

ZFC has only one non-logical symbol in its signature, a binary predicate denoted \in. This symbol stands for set membership, and we read a \in b as a is a member in the set b.

The real substance in our theory comes from our axioms. While I have mentioned many times that ZFC consist of 9 axioms, this is not strictly true. Our theory contains a few axiom schema, which are an infinite list of axioms. These will be discussed more when they appear. Each of our axioms specify some way that our nonlogical symbol \in functions.

And with that, we can finally begin to study the consequences of our axioms and set theory as a whole!