User:IssaRice/Computability and logic/Expresses versus captures
From Machinelearning
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).
Text | "Expresses" | "Captures" |
---|---|---|
Peter Smith. Godel book | expresses | captures |
Leary & Kristiansen | defines | represents |
Goldrei | ||
Boolos, Burgess, Jeffrey | arithmetically defines^{[1]} | |
Wikipedia |
References
- ↑ George S. Boolos; John P. Burgess; Richard C. Jeffrey. Computability and Logic. p. 199 for "arithmetically defines".