Complete theory
From Wikipedia, the free encyclopedia
In mathematical logic, a first-order theory is complete, if it contains either <math>S</math> or <math>\neg S</math> as a theorem for every sentence <math>S</math> in its language.
Theories that are rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's incompleteness theorem.
This sense of complete is distinct from the notion of a complete logic, which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). Gödel's completeness theorem is about this latter kind of completeness.
[edit] References
- Mendelson, Elliott (1997). Introduction to Mathematical Logic, Fourth edition, Chapman & Hall, p. 86. ISBN 978-0-412-80830-2.
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