Independence (mathematical logic)
A sentence σ is 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; this is not the same meaning of "decidability" as in a decision problem.
A theory T is independent if each axiom in T is not provable from the remaining axioms in T. A theory for which there is an independent set of axioms is independently axiomatizable.
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
Many interesting statements in set theory are independent of Zermelo-Fraenkel set theory (ZF). The following statements in set theory are known to be independent of ZF, granting that ZF is consistent:
- The axiom of choice
- The continuum hypothesis and the generalised continuum hypothesis
- The Suslin 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, 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 non-existence of Kurepa 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.
Applications in Physical Theory
Since 2000, Logical Independence has become understood as having crucial significance in the Foundations of Physics. Logical Independence has been shown, empirically, to be significant at the foundational level, to Quantum Indeterminacy.
- Tomasz Paterek, Johannes Kofler, Robert Prevedel, Peter Klimek, Markus Aspelmeyer, Anton Zeilinger, and Caslav Brukner, Logical independence and quantum randomness, New Journal of Physics 12 (2010), no. 013019, 1367–2630
- Gergely Szekely The Existence of Superluminal Particles is consisitent with the Kinementics of Einstein's Special Relativity http://arxiv.org/pdf/1202.5790v2.pdf
- 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
- Edward Russell Stabler, An introduction to mathematical thought, Addison-Wesley Publishing Company Inc., Reading Massachusetts USA, 1948.