The Nature of Numbers: Russell explores the concept of numbers, distinguishing between natural numbers, real numbers, and complex numbers, and explains their philosophical significance.

Logic and Mathematics: The book emphasizes the role of logic as the basis of mathematical principles, arguing that mathematics can be reduced to logical axioms and theorems.

The Concept of Infinity: Russell discusses the different types of infinity in mathematics, such as countable and uncountable infinities, and the philosophical implications of these concepts.

Sets and Classes: He introduces the idea of sets and classes, which are fundamental to modern mathematics, and addresses the paradoxes that arise from naive set theory.

Propositional Functions: Russell explains propositional functions and their role in the formation of mathematical statements, highlighting the importance of functions in mathematical logic.

Types and Hierarchies: The book addresses the hierarchy of types to avoid logical paradoxes, such as Russell's own paradox, which arises from considering the set of all sets that do not contain themselves.

The Axiom of Choice: Russell discusses the controversial Axiom of Choice, its applications in mathematics, and the debates surrounding its acceptance.

The Theory of Descriptions: He elaborates on his theory of descriptions, which provides a way to handle denoting phrases in logic and philosophy, resolving ambiguities and paradoxes in language.

Formal Systems and Consistency: The book explores the need for formal systems in mathematics and the importance of proving the consistency of these systems to ensure they do not lead to contradictions.

Philosophical Implications: Russell emphasizes the broader philosophical implications of mathematical concepts, arguing that understanding the foundations of mathematics can lead to deeper insights into the nature of reality and knowledge.