User:IssaRice/Computability and logic/Expresses versus captures: Difference between revisions
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* Expressing is done by a language. There is only one form of expressing; I think this follows from the [[wikipedia:Law of excluded middle]]. | * 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). | * 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). | ||
==Capturing functions== | |||
==Comparison of usage patterns== | |||
{| class="sortable wikitable" | {| class="sortable wikitable" | ||
Revision as of 06:08, 7 February 2019
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).
Capturing functions
Comparison of usage patterns
| Text | "Expresses" | "Captures" |
|---|---|---|
| Peter Smith. Godel book (see especially footnote 9 on p. 45) | expresses | captures |
| Leary & Kristiansen | defines | represents |
| Goldrei | defines (but the book also uses "represents")[1] | |
| Boolos, Burgess, Jeffrey | arithmetically defines[2] | defines (for sets), represents (for functions)[2] |
| Wikipedia | arithmetically defines | this page uses "represents", but I don't think there's a standalone article for the concept |