Independence (mathematical logic)

A sentence is independent of an axiom system when neither it nor its negation is provable from that system, assuming the system is consistent.

Relative consistency: A proof that one theory is consistent if another theory is consistent. Independence proofs often show that adding either the sentence or its negation preserves consistency.
Continuum hypothesis: The assertion that no cardinality lies strictly between the natural numbers and the real numbers. It is independent of the usual Zermelo–Fraenkel set theory with choice, if that theory is consistent.
Formal proof: A finite sequence of statements, each justified by axioms or inference rules, ending in a conclusion. Independence concerns the absence of proofs of both a sentence and its negation.
Completeness: A property of a theory that proves every sentence in its language or its negation. A complete theory has no independent sentences in its language.
Model theory: The study of mathematical structures and the sentences true in them. Models of the axioms with opposite truth values can establish independence.
Axiom of choice: The set-theoretic principle that every family of nonempty sets has a choice function. It is independent of Zermelo–Fraenkel set theory, assuming that theory is consistent.
Formal theory: A collection of sentences specified by axioms and closed under formal deduction. The theory supplies the axioms against which a sentence’s independence is assessed.
Decidability: The property of a formal problem for which an algorithm always determines the correct yes-or-no answer. Independence of one sentence is distinct from whether a theory’s consequence problem is algorithmically decidable.
Completeness theorem: Gödel’s theorem that a first-order sentence is provable exactly when it is true in every model of its premises. It connects syntactic nonprovability with models that demonstrate independence.
Goodstein's theorem: A statement about sequences formed by hereditary base notation that is true but unprovable in Peano arithmetic. It illustrates independence from a particular arithmetic system without being independent of ordinary mathematics.

Linked from 10 pages