Independence (mathematical logic)

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In mathematical logic, a sentence σ is called independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that σ is false.

Sometimes, σ is said (synonymously) to be undecidable from T; however, this usage risks confusion with the distinct notion of the undecidability of a decision problem.

Many interesting statements in set theory are independent of Zermelo-Fraenkel set theory(ZF). It is possible for the statement "σ is independent from T" to be itself independent from T. This reflects the fact that statements about proofs of mathematical statements when represented in mathematics become themselves mathematical statements.

Usage note

Some authors say that σ is independent of T if T simply cannot prove σ, and do not necessarily assert by this that T cannot refute σ. These authors will sometimes say "σ is independent of and consistent with T" to indicate that T can neither prove nor refute σ.

Theorems relevant to independence

Kurt Gödel proved the completeness theorem and the incompleteness theorem

The completeness theorem states (Assuming ZFC) A theory T is consistent iff T has a model.

The incompleteness theorem states (Assuming ZF) In any consistent formalization of mathematics that is sufficiently strong to define the concept of natural numbers, one can construct a statement that can be neither proved nor disproved within that system.

Independence results in set theory

The following statements in set theory are known to be independent of ZF, granting that ZF is consistent (see also the list of statements undecidable in ZFC):

The axiom of choice
The continuum hypothesis and The generalised continuum hypothesis
The Souslin conjecture
The Kurepa hypothesis

The following statements (none of which have been proved false) cannot be proved in ZFC to be independent of ZFC, even if the added hypothesis is granted that ZFC is consistent. However, they cannot be proved in ZFC (granting that ZFC is consistent), and few working set theorists expect to find a refutation of them in ZFC.

The existence of strongly inaccessible cardinals
The existence of large cardinals

The following statements are inconsistent with the axiom of choice, and therefore with ZFC. However they are probably independent of ZF, in a corresponding sense to the above: They cannot be proved in ZF, and few working set theorists expect to find a refutation in ZF. However ZF cannot prove that they are independent of ZF, even with the added hypothesis that ZF is consistent.

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