Definition

A precise specification of the meaning of a mathematical term or symbol. Definitions establish how mathematical language is used within a theory or context.

Mathematical language: The symbols, expressions, and conventions used to state mathematical ideas. Definitions assign precise meanings to expressions within this shared vocabulary.
Explicit definition: A definition that directly states an expression's meaning using already understood terms. This is the most direct way to introduce a new mathematical term.
Theorem: A mathematical statement established by proof from definitions, axioms, and prior results. A theorem proves a claim; a definition fixes the meaning of its terms.
Set-builder notation: A notation for specifying a set by the properties its members satisfy. It turns a defining condition into a precise description of a set.
Definability: The study of which objects or properties can be specified in a given formal language or theory. It asks what a language can express through its available definitions.
Mathematical logic: The study of formal reasoning, proof, and the logical structure of mathematics. Logic clarifies how definitions support valid statements and inferences.
Implicit definition: A specification of a concept through the relations it must satisfy rather than an explicit formula. Mathematics can characterize objects by their defining relations.
Conjecture: A mathematical statement believed to be true but not yet proved. A conjecture makes a truth claim rather than introducing terminology.
Function definition: A specification assigning each input in a domain exactly one output in a codomain. Functions show how definitions encode existence and uniqueness requirements.
Implicit definability: The characterization of a symbol by the uniqueness of its interpretation in a theory. It tests when axioms determine a concept even without an explicit definition.

Linked from 19 pages