Inmathematical logic,independence is the unprovability of some specificsentence from some specific set of other sentences. The sentences in this set are referred to as "axioms".

Asentence σ isindependent of a givenfirst-order theoryT ifT neither proves nor refutes σ; that is, it is impossible to prove σ fromT, and it is also impossible to prove fromT that σ is false. Sometimes, σ is said (synonymously) to beundecidable fromT. (This concept is unrelated to the idea of "decidability" as in adecision problem.)

A theoryT isindependent if no axiom inT is provable from the remaining axioms inT. A theory for which there is an independent set of axioms isindependently axiomatizable.

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

Many interesting statements in set theory are independent ofZermelo–Fraenkel set theory (ZF). The following statements in set theory are known to be independent of ZF, under the assumption that ZF is consistent:

• Theaxiom of choice
• Thecontinuum hypothesis and thegeneralized continuum hypothesis
• TheSuslin conjecture

The following statements (none of which have been proved false) cannot be proved in ZFC (the Zermelo–Fraenkel set theory plus the axiom of choice) to be independent of ZFC, under the added hypothesis that ZFC is consistent.

• The existence ofstrongly inaccessible cardinals
• The existence oflarge cardinals
• The non-existence ofKurepa trees

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.

• Theaxiom of determinacy
• Theaxiom of real determinacy
• AD+

Since 2000, logical independence has become understood as having crucial significance in the foundations of physics.^[1][2]

• List of statements independent of ZFC
• Parallel postulate for an example ingeometry

• Mendelson, Elliott (1997),An Introduction to Mathematical Logic (4th ed.), London:Chapman & Hall,ISBN 978-0-412-80830-2
• Monk, J. Donald (1976),Mathematical Logic, Graduate Texts in Mathematics, Berlin, New York:Springer-Verlag,ISBN 978-0-387-90170-1
• Stabler, Edward Russell (1948),An introduction to mathematical thought, Reading, Massachusetts:Addison-Wesley

• Mathematical logic
• Proof theory

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