Like logic, the subject of sets is rich and interesting for its own sake. We will need only a few facts about sets and techniques for dealing with them, which we set out in this section and the next.

While logic gives a language and rules for doing mathematics, set theory provides the material for building mathematical structures. Set theory is not the only possible framework.

The purpose of this book is to present mathematical logic and set theory to prepare the reader for more advanced courses that deal with these subjects either directly or indirectly.

Since we want to use theorems of set theory in doing model theory (and for other reasons concerning 220C), we adopt the following purely set theoretic definition as our official one.

Recall that a group is a set A, equipped with functions m : A2 → A (of ‘arity’ 2), i : A1 → A (arity 1), and a constant e ∈ A (arity 0, i.e. e : A0 → A), satisfying:

Since the publication of the first volume of Principia Mathematica, it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, …

Definition 2.6. Let E and F be sets. The (Cartesian) product of E and F, denoted E F, is the set of all ordered pairs (x; y), where x is an element of E and y is an element of F.

May 4, 2025 · Sentential logic, also called propositional logic, studies how prime sentences (atomic) combine using logical connectives. A prime sentence is a basic statement that is either true or false.

The language of propositional logic consists of a set P of primitive propositions and the set L = L(P) of propositions (or compound propositions), which is de ned inductively as follows. Examples. We often …

The role of set theory is two-fold: On the one hand, (a fragment of) it serves as the meta-theory. On the other hand, it can be viewed as a formal theory to which we can apply the logical methods we are …