# Chapter 2. Mathematical Truth

## Chapter 2. Mathematical Truth

Mathematics depends on precise notions of truth, but truth appears in more than one form. A statement can be true in a structure, and it can be provable within a formal system. These two notions are related, but they are not identical. This chapter gives an overview of how they interact.

At the syntactic level, mathematics is built from formal systems. A language specifies symbols and rules for forming expressions. Axioms fix initial assumptions. Inference rules determine how new statements can be derived. Proofs are sequences of steps that follow these rules. This perspective treats mathematics as symbol manipulation governed by strict constraints.

At the semantic level, statements are interpreted in structures. A statement is true when it holds in a given model. The same statement may be true in one structure and false in another, depending on how its symbols are interpreted. This introduces a separation between meaning and derivation.

Soundness and completeness connect these two levels. Soundness ensures that anything provable is true in the intended interpretation. Completeness, when it holds, ensures that every true statement can be proved. In many basic logical systems, these properties align. In stronger systems, especially those capable of expressing arithmetic, completeness fails and true but unprovable statements appear.

This leads to the notion of independence. A statement is independent of a theory when it can neither be proved nor refuted from the given axioms. Independence shows that axioms do not determine all truths. Different consistent extensions of a theory may assign different truth values to the same statement.

These ideas are not confined to logic. They appear across mathematics. Geometric statements depend on chosen axioms. Algebraic results may rely on additional principles. Set-theoretic assumptions can change what can be proved about infinite structures.

The purpose of this chapter is to make these distinctions explicit. It clarifies what it means for a statement to be true, what it means to prove it, and what it means when a system cannot decide it. The sections that follow develop formal systems, semantic interpretation, consistency, completeness, and independence, with examples drawn from different areas of mathematics.