- Independence (mathematical logic)
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
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.
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.
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):
axiom of choice
continuum hypothesisand the generalised continuum hypothesis
*The Souslin conjecture
*The existence of a
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
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.
Axiom of determinacy
axiom of real determinacy
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