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Distinction between semantic truth and syntactic provability, with examples and limits.

Statements that cannot be proved or refuted from a chosen axiom system, and what independence means in mathematical practice.

Distinction between finite and infinite objects, methods of reasoning, and consequences across mathematics.

Overview of truth, provability, formal systems, and independence in mathematics.

Comparison of constructive and classical mathematics, including existence, proof, logic, and computation.

Different notions of sameness in mathematics: strict equality, structural identity, and equivalence relations.

Three ways to organize a domain of discourse for mathematics: sets, types, and universes — and how they relate.

How mathematics treats objects through the rules they satisfy, the relations they support, and the transformations that preserve them.

How this volume defines the ground layer of mathematics: language, structure, and method before specialization.

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