Axiom (general principle)

An axiom is a statement accepted as a starting point for reasoning, either because it seems self-evident or because a system stipulates it.

Proposition: A statement that can be evaluated as true or false. An axiom is a proposition granted a foundational role in reasoning.
Postulate: A foundational assumption, especially one proposed within a particular theory or discipline. The word often names axioms in geometry and natural science.
Euclidean geometry: The geometry of flat space developed from Euclid's definitions, postulates, and propositions. Its parallel postulate became a famous example of a geometric axiom.
Independence (mathematical logic): A property of a statement that cannot be proved or disproved from a specified axiom system. Independent statements show that a chosen axiom set may leave claims undecided.
Inference rule: A rule specifying how conclusions may be derived from premises. Inference rules determine what follows from accepted axioms.
Hypothesis: A proposed explanation or assumption that can be examined through reasoning or evidence. A hypothesis is typically investigated, not simply installed as a foundation.
Zermelo–Fraenkel set theory: A foundational axiomatic theory describing sets and their relationships. Its axioms provide a widely used foundation for modern mathematics.
Gödel's incompleteness theorems: Results showing limits on proof and completeness in sufficiently expressive, consistent formal systems. They constrain what any suitable arithmetic axiom system can establish internally.
Deductive reasoning: Reasoning that derives conclusions from premises through rules of inference. Axioms serve as premises in deductive arguments.
Empirical evidence: Information obtained through observation or experiment. Empirical claims rely on observation, unlike stipulated starting principles.

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