User:IssaRice/Computability and logic/Expresses versus captures
The expresses versus captures distinction is an important one in mathematical logic, but unfortunately the terminology differs wildly between different texts. The following table gives a comparison.
- Expressing is done by a language. There is only one form of expressing; I think this follows from the wikipedia:Law of excluded middle.
- Capturing is done by a theory or by axioms. There are two forms of capturing: strong capture (corresponding to deciding), and weak capture (corresponding to recognizing, or semi-deciding).
|Peter Smith. Godel book||expresses||captures|
|Leary & Kristiansen||defines||represents|
|Boolos, Burgess, Jeffrey||arithmetically defines|
- George S. Boolos; John P. Burgess; Richard C. Jeffrey. Computability and Logic. p. 199 for "arithmetically defines".